10
Aug 15

@Mathemagicianme It's time for a revolution. It's time for liberals to realize the second amendment applies to them as well.

— The Mathemagician (@Mathemagicianme) August 10, 2015

@Mathemagicianme Funny how that works,isn't it? You'd think legally the 2nd amendment only applied to conservatives.

— The Mathemagician (@Mathemagicianme) August 10, 2015

9
Aug 15


Monoids,R-Modules And Nonassociative Rings-These Are Some of My Favorite Things: An Updated and Expanded Suggested Reading List For Honors Undergraduate and Graduate Algebra  Part II of II
Here we are finally at the second part of this now-three part bibliography of algebra texts. Yeah, I know, I said it would be 2 parts, but I got carried away. Sue me. Besides, for those of you who have been clamoring for a review of Paul Cohn’s classic texts from me, I’ve written a lengthy one and that’s why the list ran over. I’d rather split it then try and cram everything in at the end. That’s what Dummit and Foote try and do and you can see how well it works there (dig).
Anywho- on with part II!

The Big Three: These are the 3 textbooks that up until about 15 years ago, were the standard texts at the top graduate programs in the U.S. to use for first year graduate algebra courses and for qualifying exams at PHD programs in algebra. Of course, at such programs, the line between graduate and undergraduate coursework is somewhat ambiguous. But I think most mathematicians would agree with me on this assessment.

Algebra 3rd edition by Serge Lang Ok, let’s get the elephant in the room out of the way first.

Lang is a good example of the kind of strange “canonization’ of textbooks in academia which I’ve mentioned before at this blog and other places. It’s funny how some mathematicians-particularly algebracists at the more prestigious programs- that get very self-righteous and uppity when you question whether or not Lang should be used as a first-year graduate text anymore with all the new choices. I can’t help but use some of the past remarks of once-frequent poster at Math Overflow in this regards. Let’s call him Mr. G.
Mr. G was a talented undergraduate at one of the more prominent universities to study mathematics in the Midwestern United States. (  I have no doubt that since that exchange, Mr. G has gone on to bigger and better things-assuming nothing tragic has occurred to block his way-and is working on his PhD in mathematics at a prestigious university. I also have no doubt he treats most of his students like Neanderthals with Down Syndrome and uses Baby Rudin to teach calculus. )   Like the author of this blog, he also was occasionally slammed for shooting off his big mouth on MO by the moderators.He and I had several heated exchanged about his Bourbaki-worship: G believed that the Bourbaki texts are sacred tomes that are the only “real” texts for mathematicians and applications are for nonmathematicans. But I’ll let his own words state his position far better than I can. I have in mind a specific exchange that took place in 2010 between Mr. G and 2 mathematicians who were also at the time  frequent posters at MO: let’s call them Dr. H and Dr. L. This was a question regarding the presentation of graduate algebra. (I obviously can’t be more specific then that-to do so would identify the participants.)
@Dr. H: The first graduate algebra course is often going to be the student's first introduction to algebra. It's supposed to be abstract and intense! If you muddy the waters with applications, your students will never get to that level of Zen you achieve after stumbling around in an algebra course. It's like point-set topology, except the rabbit-hole called algebra goes much deeper and is much more important. –Mr.G
@Mr.G: After my first algebra course I still didn't understand why I should actually much about Galois theory from a practical point of view until I saw $GF(2^n)$ in all sorts of applications. My experience has been that most students--even graduate students studying algebra--are not going to be interested in abstraction for its own sake. Mechanics can help to motivate calculus. The same can be true of information theory and algebra. –Dr.H
@Dr.H: Graduate math students shouldn't be taught things "from a practical point of view". This isn't a gen. ed. class, and the abstract perspective one gains by really engaging algebra "as it is practiced) is completely worth the "journey in the desert", as it were. This is the "Zen" I was talking about. Also, I think that characterizing algebra as "abstraction for abstraction's sake" is really missing the point tremendously. – Mr.G.
@Mr.G. The journey through algebra does not necessarily have to go through the desert, nor is that necessarily the best or most ideal path. It might be so for you, but it is certainly not the best path for everybody. There are numerous other paths to take, most of which can lead and have lead people to mathematical understanding and success. Please take a moment to consider, for instance, Richard Borcherds' recent algebraic geometry examples post. – Dr. L.

Speaking for myself, I firmly believe in heeding Lebesgue’s warning about the state of the art in mathematics: “ Reduced to general theories, mathematics would become a beautiful form without content: It would quickly die.” Generality in mathematics is certainly important, but it can and often is, overly done. But I digress. My point is that Mr.G’s attitude is typical of the Lang-worshipper: That if you can’t deal with Lang, you’re not good enough to be a graduate student in mathematics. Or to use Mr.G’s own words on another thread on the teaching of graduate algebra: ”Lang or bust.” Many feel the “journey in the desert” of Lang is a rite of passage for graduate students, much as Walter Rudin’s Principles of Mathematical Analysis is for undergraduates.
Well, there’s no denying Lang’s book is one of a kind and it’s very good in many respects. People ask me a lot how I feel about Lang’s remarkable career as a textbook author. It’s important to note I never met the man, sadly-and everything I know about him is second hand.
Reading Lang’s books brings to mind a quote from Paul Halmos in his classic autobiography I Want To Be A Mathematician .which was said in reference to another famous Hungarian mathematician, Paul Erdos: “I don’t like the kind of arithmetic-geometric-combinatorial problems Erdos likes, but he’s so good at them, you can’t help but be impressed. “
I didn’t like the kind of ultra-abstract, application-devoid, Bourbakian, minimalist presentations Lang was famous for-but he was SO good at writing them, you couldn’t help but be impressed.
And Algebra is his tour de force.
The sheer scope of the book is stunning. The book more or less covers everything that’s covered in the later editions of van der Waerden-all from a completely categorical, commutative-diagram-with-functors point of view. There’s also a generous helping of algebraic number theory and algebraic geometry from this point of view as well, much more so then any other texts at this level. His proofs are incredibly concise and with zero fat, but quite clear if you take the effort to follow them and fill in the blanks. The chapters on groups and fields are particularly good. Lang also is an amazingly thorough and responsible scholar; each chapter is brimming with references to original proofs and their source papers. This is a book by one of the giants in the field and it’s clearly a field he had enough respect for to know his way around the literature remarkably well-and he believed in giving credit where credit is due. Quite a few results and proofs-such as localization of rings, applications of representation theory to functional analysis and the homology of derivations, simply don’t appear in other texts. The last point is one I think Lang doesn’t get a lot of credit for, without which the book would be all but unreadable: He gives many, many examples for each concept-many nonstandard and very difficult to ferret out of the literature. Frankly, the book would be worth having just for this reason alone.

So fine, why not go with the party line then of “Lang or bust”?

Because the book is absurdly difficult, that’s why.

First of all, it’s ridiculously terse. It takes 2-3 pages of scrap paper sometimes to fill in the details in Lang’s proofs. Imagine doing that for OVER 911 PAGES. And worse, the terseness increases as one progresses in the book. For the easier topics, like basic group theory and Galois theory, it’s not so bad. But the final sections on homological algebra and free resolutions are almost unreadable. You actually get exhausted working through them.

AND WE HAVEN’T EVEN TALKED ABOUT THE EXERCISES YET.

“Yeah, I’ve heard the horror stories about Lang’s exercises in the grad algebra book. C’mon, they’re not THAT bad are they?”

You’re right, they’re not.

They’re WORSE.

I mean, it’s just ludicrous how hard some of these exercises are.

I’ll just describe 2 of the more ridiculous exercises and it pretty much will give you an idea what I’m talking about. Exercise 30 on page 256 asks for the solution of an unsolved conjecture by Emil Artin. That’s right, you read correctly. Lang puts in parentheses before it: “The solution to the following exercise is not known.” No shit? And you expect first year graduate students-even at Yale-to have a chance solving it? I’m sorry, that’s not a reasonable thing to do! (One does wonder if a brilliant student in one of Lang’s courses had actually succeeded in solving the problem whether or not they’d get credit for it. Or if suddenly a few months later there’d be a new interview with Lang in the New York Times science section about the brilliant Lang’s new discovery solving Artin’s conjecture while the student scrambles to pay his or her student bills while eating  ramen noodles. We’ll never know-thankfully.)

Then there’s the famous-or more accurately, infamous-exercise in the chapter on homological algebra: “Take any textbook on homological algebra and try and prove all the results without looking up the proofs.”

I know in principle, that’s what we’re all supposed to do with any mathematical subject we’re learning.

But HOMOLOGICAL ALGEBRA?!? SERIOUSLY?!?

(An aside: I actually had a rather spirited discussion via email with Joseph Rotman, Professor Emeritus of the University of Illinois at Urbana-Champaign, over this matter-a guy that knows a thing or 3 about algebra. Rotman felt I was too hard on Lang for assigning this problem. He thought Lang was trying to make a point with the exercise, namely that homological algebra just looks harder than any other subject, it really isn’t. Well, firstly, that’s a debatable point Lang was trying to make if so. Secondly, I seriously doubt any graduate student who’s given this as part of his or her final grade is going to be as understanding as Rotman was. Actually, it’s kind of ironic Rotman thinks that since I know many a graduate student who would have failed the homology part of their Lang-based algebra course without Rotman’s book on the subject! )

My point is I don’t care how good your students are, it’s educational malpractice to assign problems like that for mandatory credit. And even if you don’t and just leave them as challenges for the best students-isn’t that rubbing salt in the wounds inflicted by this already Draconian textbook?

These exercises are why so many mathematicians have bitter memories of Lang from their student days.

A lot of you may be whining now that I just don’t like hard books. That’s just not true. Herstien is plenty difficult for any student and it’s one of my favorites. In fact, quite the contrary. You’re really supposed to labor over good mathematics texts anyway-math isn’t supposed to be EASY. An easy math textbook is like a workout where you’re not even winded at the end-it’s doubtful you’re going to get any benefits from it.

But Lang isn’t just hard; I don’t just mean students have to labor over the sections before getting them.

The average graduate student learning algebra from Lang’s book is like a fat guy trying to get in shape by undergoing a 3 month U.S. Marine Corp boot camp and having a steady diet of nothing but vitamins, rice cakes and water. Assuming he doesn’t drop dead of a heart attack halfway through, such a regimen will certainly have the desired effect-but it will be inhumanely arduous and unpleasant.

And there are far less Draconian methods of obtaining the same results.

So unless one is a masochist, why in God’s name would you use Lang for a first year graduate course in algebra?

Is it a TERRIBLE book? No-as I said above, it has many good qualities and sections. As a reference for all the algebra one will need in graduate school unless becoming an algebraicist, the book is second to none.

Would I use it as a text for a first year graduate course or qualifying exam in algebra?

HELL NO. 

Algebra by Thomas Hungerford : This has become a favorite of a lot of graduate students for their algebra courses and it’s pretty easy to see why-at least at first glance. It’s nearly as demanding as Lang-but it’s much shorter and more selective, has a lot more examples of elementary difficulty and the exercises are tough but manageable.
The main problem with this book occurs in the chapter on rings and it boils down to a simple choice. Hungerford-for some strange reason-decides to define rings without a multiplicative identity.

I know in some ring theoretic cases, this is quite useful. But for most of the important results in basic ring and module theory, this results in proofs that are much more complicated since this condition needs to be “compensated” for by considering left and right R-modules as separate cases. Hungerford could alleviate this considerably by giving complete, if concise, proofs as Lang does in most cases.

But he doesn’t. He only sketches proofs in more than half the cases.

The result is that every section on rings and modules is very confusing. In particular, the parts on modules over commutative rings and homological algebra-which students really need for qualifying exams-are all over the place.

And the field theory chapter is a train wreck, frankly.

Still, the book has a lot of really nicely presented material from a totally modern, categorical point of view. The first chapter on category theory is probably the best short introduction there is in the textbook literature and the section on group theory is very nice indeed. So I ultimately have mixed feelings about Hungerford. I’d definitely give it a look and if I could borrow a copy, great.  But I’m really not sure I’d want to shell out money for my own copy. I’d rather get a copy of Grillet or Rotman.

Basic Algebra, 2nd edition by Nathan Jacobson, volumes I and II: I say we should nominate Dover Books for a Nobel Peace Prize for their recent reissue of this classic. The late Nathan Jacobson, of course, was one of the giants of non-commutative ring theory in the 20th century.

He was also a remarkable teacher with an awesome record of producing PhDs at Yale, including Charles Curtis, Kevin Mc Crimmon, Louis H.Rowen, George Seligman, David Saltman and Jerome Katz. His lectures at Yale on abstract algebra were world famous and had 2 incarnations in book form: The first, the 3 volume Lectures In Abstract Algebra, was for a generation the main competition for van der Waerden as the text for graduate algebra courses. Basic Algebra is the second major incarnation- the first edition came out in the 1970’s and was intended as an upgraded course in algebra for the extremely strong mathematics students entering Yale from high school during the Space Age. The first volume-covering classical topics like groups, rings, modules, fields and geometric constructions-was intended as a challenging undergraduate course for such students. The second volume-covering an overview of categorical and homological algebra as well as the state-of-the-art (circa 1985) of non-commutative ring theory-was intended as a graduate course for first year students. The complete collapse of the American educational system in the 1990’s has rendered both volumes useless as anything but graduate algebra texts. Still, given that the second volume was going for nearly 400 dollars at one point online in good condition, its reissue by Dover in wonderfully cheap editions is a serious cause for celebration.

Both books are beautifully and authoritatively written with a lot of material that isn’t easily found in other sources, such as sections on non-associative rings , Jordan and Lie algebras, metric vector spaces and an integrated introduction to both universal algebra and category theory. They are rather sparse in examples compared with other books, but the examples they do have are very well chosen and described thoroughly. There are also many fascinating, detailed historical notes introducing each chapter, particularly in the first volume.

The main problem with both books is that Jacobson’s program here absolutely splits in half algebra into undergraduate and graduate level topics; i.e. without and with categorical and homological structures. This leads to several topics being presented in a somewhat disjointed and inefficient manner because Jacobson refuses to combine them in a modern presentation. It also results in a quantum leap in abstraction between the presentations of the same material in the 2 volumes. This is really where the lack of examples in the first volume hurts the total presentation-it would be doable but very tough for even strong students to use that first volume by itself as an adequate foundation for Volume II. Module theory in particular suffers from this organization. R-modules are first presented classically in volume I over a principal ideal domains (PIDs). In Volume II, a gargantuan step in generality is taken by discussing the categories of left and right modules over arbitrary noncommutative rings.  This is a bit like making the first half of an analysis course a pencil pushing calculus course with virtually no theory and then beginning with “Adult Rudin” in the second half. Ok, I’m exaggerating-it’s not quite that bad. But the leap in difficulty is pretty significant.  

Which is why Jacobson’s approach makes sense.  Doing modules over PIDs makes everything very clean and simple with important applications to groups and linear algebra. Also, PIDs are probably the most important example of commutative rings that aren’t fields-they’re very similar in structure to the integers. So it makes a lot of sense to introduce modules this way to beginners. Personally, I didn’t find it THAT big an issue with a little effort and providing you use the 2 books simultaneously as a single text, that’s the key. But a lot of other students have complained about it. Also, some of the exercises are quite difficult, rivaling Lang’s. Even so, the sheer richness of these books makes them true classics. If graduate students are willing to work a little to unify the various pieces of the vast puzzle that Jacobson presents here with astonishing clarity, he or she will be greatly rewarded by a master’s presentation and depth of understanding.

Algebra Volumes I- III Second Edition by Paul M. Cohn

Classic Algebra by Paul M Cohn
Basic Algebra: Groups, Rings And Fields by Paul M.Cohn
Further Algebra And Applications by Paul M. Cohn  : Ever since I posted the first version of this list at this blog, I’ve gotten comment after comment wondering how I could have left the most famous algebra text in Europe out. To be perfectly honest, I simply hadn’t carefully read Cohn’s books back then. I hadn’t seen them except in passing in the St. John’s University library and they were far too expensive for me to purchase. In preparing this revision, I realized it would be malpractice for me to not comment on them since they’re so widely used and respected.  Yes, I’m well aware the books have since been republished in new editions by Springer-Verlag as 3 separate books with the titles above. This is why I’m considering both the latest edition of the old “unified” text and the new versions simultaneously. Essentially, the new books are corrected and radically reorganized versions of the old 3 volume text.  Classic is a new version of volume I with some additions from volume II, such as group representations and tensor products. Basic and Further are similarly corrected and rearranged versions of volumes II and III. Since the reissued editions don’t differ significantly in subject matter, merely organization, I want to focus on the second edition of the original 3 volumes as they’ve traditionally been used and revered in the U.K.  The book is quite different from standard American algebra texts- which is why I’m going to go into some depth on the background of the book. I think this is important to fully understand the book and its’ intention.
   Most US trained mathematics students and teachers, if they’d never heard of it and picked it up, would find it kind of baffling. I know I did at first. I think for American students and mathematicians to understand it, it’s important to read Cohn’s original preface and understand his intention for writing it:   
{Algebra}’s changing role is reflected in the importance of algebra in the curricula, as well as in the many excellent textbooks that now exist. Most of these are designed for undergraduates at North American universities and are either (a) a very broad introduction to linear algebra, with a little groups and rings, for general students taking mathematics, or (b) a course for graduates, or junior—senior students majoring in mathematics, who have already taken a course of type (a). The pattern in Britain is a little different: the honours student specializing in mathematics takes algebra for two or three years (depending on his ultimate interests) and his need is for a textbook which combines (a) and (b) above and is somewhere between them in level. The object of the present work is to provide such a book: the present first volume includes most of the algebra taught in the first two years to undergraduates at British universities; this will be followed by a second volume covering the third year (and some graduate) topics. 
  It’s important to know some of the ways British undergraduate work differs from even strong American universities to fully understand the context.  First of all, in the U,K, university only lasts 3 years traditionally going back nearly to feudal times and the preparation of high school students entering university is generally much stronger than in the United States. Preparation for university in the U.K. is serious business- it’s required of many programs for high school students there planning on applying to university to be taking calculus and linear algebra before graduating. The other major difference that’s relevant to the background of Cohn is that practical applications are generally sneered upon. This is not because British academics don’t think it’s important, it’s that they don’t believe a university is the place for it. A university is a place for strictly formal training in pure theory-as much as you can cram into three years of coursework. The thinking is that only in a strictly academic setting can theory truly be learned in the depth and breadth needed for professional work. Once students have that background, they’re more then capable of picking practical stuff up after graduating. This is one of the reasons they’re able to effectively condense a Master’s degree level of training into 3 years while American students take twice that-it’s much more focused and rigidly structured then in the U.S.(There’s also the completely insane manner British universities grade students, which would give me a nervous breakdown. But that’s not really relevant here, so I’ll save it for another day.)  
  I’ve gone on at more length on this then I really intended. My major points here is that a) undergraduate mathematics curricula in the U.K. is much stronger than in the U.S. and consequently, honours (i.e. honors-England and America, 2 countries separated by a common language) are even better. Freshman honours students at top universities there would probably have a good working knowledge of US-“honors” level single variable calculus and linear algebra as well as basic proof-based mathematics. In other words, they would have the equivalent mathematical background of average third year undergraduate mathematics majors in the U.S. Also, their classes would generally be very abstract with little or no applications.
     Cohn wrote this massive text to serve the needs of such students, covering virtually all the algebra these students would need to know to begin doing research when they graduate to doctoral level work.
 Given all this backstory, what would mathematicians in the U.S expect such a text to look like?  Well, it would certainly be more sophisticated than a typical undergraduate US algebra text. Cohn’s texts are certainly that. The first volume is pitched at about the same level as Herstien or Dummit/Foote, but far more concise then even Herstien. It covers the standard material of a strong first algebra course: Sets and mappings, the integers and congruence, groups, rings, R-modules, vector spaces and linear maps, quadratic forms and advanced matrix theory Volume II moves on to topics beyond a first course, which would usually be covered in a first year graduate course in the US: General field theory, semisimple and free modules, general algebras over a ring, the Wedderburn structure theorems and the Jacobson radical, group representation theory, valuation theory and commutative algebra. Volume III covers topics beyond a first year US graduate course: Universal, multilinear and homological algebra, advanced group and field theory including commutators, derivations and field extensions, Galois and Hochschild cohomology and more.  Volumes II and III would no doubt be strong first year graduate texts in the States.  All three volumes are uncompromisingly modern in presentation. Commutative diagrams and category theory are both introduced very early in volume I and used throughout all 3 volumes. The topic selection in all three volumes is very similar to both of Jacobson’s books. But that doesn’t mean the books are interchangeable. Cohn is considerably terser then Jacobson and even Lang. Seriously-Lang is significantly harder than Cohn, but it’s quite a bit more detailed and readable. It’s also at a somewhat lower level then Jacobson and organized in a far different and completely unified manner. The one word description that comes to mind to describe the books collectively is efficient.It’s all there, but the author works hard to ensure it’s presented with the absolute minimum of verbiage. Interestingly, while there are not many explicit examples-3 or 4 at most in the first 2 texts and less in the last-there are many examples embedded in the discussions of theorem proofs. Also, the explicit examples are very well chosen. So the books are actually quite bit deeper than a skimming read would indicate. There are also many exercises that are quite diverse in both type and level. Cohn writes extremely well and he demonstrates enormous command of the material.  However, his style is very dry and dictatorial. If you’re looking to be inspired, look elsewhere. Personally, I’d rather use either Rotman or Rowen for a first graduate course. That being said, there’s a school of thought in mathematics that prefers texts that aren’t “too wordy”. This is the school of thought that thinks there’s no other real analysis text except “baby Rudin”. For them, I think they’ll find this book very much to their liking for both undergraduate and graduate algebra courses. It’ll also be of great use for students preparing for qualifying exams in algebra and need to review and/or learn most of the landscape of modern algebra relatively quickly and actively. It’ll also make quite a good graduate algebra text if used in conjunction with a source that supplies examples, such as Ash.  As I said, it wouldn’t be my first choice for an advanced algebra text. But I am happy I have a copy.  

The New Kids On The Block: As I said earlier, the adoption of Lang worldwide as the canonical graduate algebra text had a backlash effect that’s been felt with a slew of new graduate texts. I haven’t seen them all, but I’ve seen quite a few. Here’s my commentary on the ones I’m most familiar with.

Basic Algebra/Advanced Algebra by Anthony W. Knapp: This is probably the single most complete reference for abstract algebra that currently exists. It is also paradoxically, the single most beautiful, comprehensive textbook on it. This is the book Dummit and Foote should have been instead of trying to cram a whole graduate course on the back end without category theory. Knapp taught both undergraduate and graduate algebra at SUNY Stonybrook for nearly 3 decades-and these volumes are the finished product of the tons of lecture notes that resulted. The purpose of these books, according to Knapp, is to provide the basis for all the algebra a mathematician needs to know to be able to attend a conference on algebra and understand it. If so, he’s succeeded very well indeed. The main themes of both books are group theory and linear algebra (construed generally i.e. module theory and tensor algebra) . The first volume corresponds roughly to what could possibly be covered at the undergraduate level from basic number theory and linear algebra through all the standard undergraduate topics up to the beginning  of a first year graduate course in abstract algebra (groups, rings, fields, Galois theory, multilinear algebra, module theory over commutative rings). The second corresponds to a first and second year graduate syllabus focusing on topics in noncommutative rings, algebraic number theory and algebraic geometry( adeles and ideles, homological algebra, Wedderburn-Artin ring theory, schemes and varieties, Grobner bases, etc.)  This is the dream of what an advanced textbook should be-beautifully written, completely modern and loaded with both examples and challenging exercises that are both creative and not too difficult. In fact, the exercises are really extensions of the text where many topics and applications are in fact derived-such as Jordan algebras, Fourier analysis and Haar groups, Grothendieck groups and schemes, computer algebra and much, much more. The concept of group actions on sets is stressed throughout. Both books are completely modern in presentation regardless of level. Categorical arguments are given implicitly before categories are explicitly covered by giving many commutative diagram arguments as universal properties. (This avoids the trap Jacobson fell into. Knapp in many ways is a much more detailed version of Cohn’s opus.) The books are also supremely organized with each chapter independent to some degree from the others. This means the books are extremely versatile and many different kinds of courses can be based on them.  Best of all-there are hints and solutions to all the exercises in the back of each volume. My one complaint with the book is that several topics I feel are too important to be shortchanged are. The most glaring example is group representation and character theory. But this is really a minor quibble. In a book like this, the author has to make choices and such choices are never going to do everything perfectly in a way that makes everyone completely happy.  I would love to use this set to teach algebra courses one day at any level-either as the main texts, as supplements or just references. If you enjoy algebra, are learning it or plan to teach it, you have to have a copy. Hopefully, there will be many editions to come.

Abstract Algebra, 2nd edition by Pierre Grillet: The first edition of this book was simply called “Algebra” and it came out in 1999. To me, this book is what Lang should be. Grillet is an algebraist and award-winning teacher at Tulane University. Interestingly, he apparently carried out the revision in the aftermath of Katrina. The book covers all the standard and more modern topics for a graduate course in a concise, very modern manner-much like Lang. Unlike Lang, though, Grillet is extremely readable, selective in his content and highly structured with many digressions and historical notes. The sheer depth of the book is amazing. Unlike Lang, which focuses entirely on what-or Hungerford, which explains a great deal but also is very terse- Grillet focuses mainly on why things are defined this way in algebra and how the myriad results are interconnected. It also has the best one chapter introduction to category theory and universal algebra I’ve ever seen-and it occurs in Chapter 17 after the previous 16 chapters where commutative diagrams are constructed on virtually every other page. So by the time the student gets to category theory, he or she has already worked a great deal with the concepts implicitly in the previous chapters. This is very typical of the presentation. Also, Grillet doesn’t overload the book with certain topics and give the short shriff to others-many texts are half group theory and half everything else, for instance. Grillet gives relatively short chapters on very specific topics- which makes the book very easy to absorb. The exercises run the gamut from routine calculations to proofs of major theorems. Unfortunately, the book has 2 flaws that annoy me. First of all, like Cohn and Jacobson, it has relatively few examples. That being said, it is a graduate course and the examples the author does choose are very good ones and not so standard. For example, both semigroups are discussed at some length and this is the only text to my knowledge that mentions the Light Cayley table associativity test. More serious is the omission of a chapter on homological algebra, which is kind of strange given the length at which category theory is discussed.  The resulting text is a clinic in how to write “Bourbaki” style texts and it would be a great alternative to either Lang or Hungerford.

Algebra: A Graduate Course by I.Martin Issacs: This is a strange book and I have many mixed feelings about it. After being out of print for over a decade, it was recently reissued by the AMS. Isaacs claims this course was inspired by his teacher at Harvard, Lynn Loomis, whose first-year graduate course Issacs took in 1960 there. Like Loomis’ course, Issacs emphasizes noncommutative aspects first, focusing mainly on group theory and representation theory. He then goes on to commutative theory-discussing ring and ideal theory, Galois theory ,cyclotomy and many other topics. The book is one of the most beautifully written texts I’ve ever seen, with most theorems proved and most example constructions left as exercises. Unfortunately, the book is not without serious flaws. First of all, Issacs’ material choice seems to follow his memories of his graduate course in 1960 far too closely-this choice of topics would be a first year graduate course at a top university only before the 1960’s. Issacs omits completely multilinear algebra, category theory and homological algebra. How can you call such a book in 2015 a graduate course? More seriously, the book has virtually no concrete examples in it.  Almost none. Just a smattering in the discussion and exercises. I mean it practically makes Cohn look like it’s brimming with examples. For any book written at a lesser level, that would be the kiss of death for me. But Issacs is so wonderfully and deeply written-it reads almost like a novel, with so many wonderful insights.  To be able to use the book for a graduate course, one would have to supplement it extensively. The problem is that Issacs is so expensive- even now in the new AMS reissue of it-that I’m not sure if it’s worth it. This is why I was hoping the book would be reissued by Dover, but it didn’t happen. Because of the cost and the strange layout of the book, I’d be really hard pressed to assign the book to my students as the text for either an honors algebra or graduate algebra text. That being said-I certainly would put a copy on reserve for my students to browse.

Advanced Modern Algebra by Joseph J. Rotman:  Rotman may be the best writer of algebra textbooks alive. Hell, he may be the best writer of university-level mathematics textbooks period. (Well, ok, I wouldn’t go that far. Tom Korner, John Stillwell, the late George Simmons, Peter Cameron, Gilbert Strang and several others probably beat Rotman for that title. But solely for algebra texts, Rotman is probably king.)  So when his graduate textbook came out, I begged, borrowed and cajoled until I could buy it. And it was one of the best textbooks I ever bought. I’m talking about the first edition, but second edition is very similar and in many respects, even better. The contents of the book are, as the AMS’ blurb discusses:
“This book is designed as a text for the first year of graduate algebra, but it can also serve as a reference since it contains more advanced topics as well. This second edition has a different organization than the first. It begins with a discussion of the cubic and quartic equations, which leads into permutations, group theory, and Galois theory (for finite extensions; infinite Galois theory is discussed later in the book). The study of groups continues with finite abelian groups (finitely generated groups are discussed later, in the context of module theory), Sylow theorems, simplicity of projective unimodular groups, free groups and presentations, and the Nielsen-Schreier theorem (subgroups of free groups are free). The study of commutative rings continues with prime and maximal ideals, unique factorization, noetherian rings, Zorn's lemma and applications, varieties, and Grobner bases. Next, noncommutative rings and modules are discussed, treating tensor product, projective, injective, and flat modules, categories, functors, and natural transformations, categorical constructions (including direct and inverse limits), and adjoint functors. Then follow group representations: Wedderburn-Artin theorems, character theory, theorems of Burnside and Frobenius, division rings, Brauer groups, and abelian categories. Advanced linear algebra treats canonical forms for matrices and the structure of modules over PIDs, followed by multilinear algebra. Homology is introduced, first for simplicial complexes, then as derived functors, with applications to Ext, Tor, and cohomology of groups, crossed products, and an introduction to algebraic K-theory. Finally, the author treats localization, Dedekind rings and algebraic number theory, and homological dimensions. The book ends with the proof that regular local rings have unique factorization.”
That says what’s in the book. What it doesn’t tell you is what makes this incredible book so special and why it deserved a second edition so quickly with the AMS: Rotman’s gifted style as a teacher, lecturer and writer. The book is completely modern, amazingly thorough and contains discussions of deep algebraic matters completely unmatched in clarity. As one simple example, read the following excerpt of how Rotman explains the basic idea of category theory and its’ importance in algebra:
Imagine a set theory whose primitive terms, instead of set and element, are set and function.
How could we define bijection, cartesian product, union, and intersection? Category theory
will force us to think in this way. Now categories are the context for discussing general
properties of systems such as groups, rings, vector spaces, modules, sets, and topological
spaces, in tandem with their respective transformations: homomorphisms, functions, and
continuous maps. There are two basic reasons for studying categories: The first is that they
are needed to define functors and natural transformations (which we will do in the next
sections); the other is that categories will force us to regard a module, for example, not in
isolation, but in a context serving to relate it to all other modules (for example, we will
define certain modules as solutions to universal mapping problems).

     I dare you to find a description of category theory that would serve a novice better. The book is filled with passages like that-as well as hundreds of commutative diagrams, examples, calculations and proofs of astounding completeness and clarity.                Rotman’s book is often compared with Dummit/Foote, which is one of its’ main competitors as a graduate algebra text. It really isn’t a fair comparison in my opinion.  Dummit/ Foote begins from jump and presents things in greater generality. For example. D/F introduces noncommutative rings with the definition of rings. Rotman develops commutative rings in detail first. I think this approach, while more tedious, prepares a student better for a fuller discussion of both noncommutative algebra and

algebraic geometry. But Rotman’s book really is at a considerably higher level than D/F and his proofs of theorems tend to be much cleaner and clearer. While D/F has many clear chapters and a freight car of great detailed examples, as I’ve said, the chapters towards the end ramble and are incoherent in places. These considerations really make Rotman better for a full blown graduate course and subsequent advanced courses.
    Rotman presents algebra as a huge, beautiful puzzle of interlocking pieces-one he knows as well as anyone in the field. In the second edition, the index has been greatly improved and entire sections have been rewritten to emphasize noncommutative algebra-which is appropriate for a graduate course. The one common complaint in the first edition was book’ s exercises- they were considered too soft compared to the ones in Hungerford or Lang and of limited use for graduate students preparing for qualifiers. Rotman has made many additions and revisions to the exercises in the second edition –there are now many significantly difficult exercises-but fortunately none as difficult as the ridiculous ones in Lang. And the sheer size of the book-like D/F-is a bit daunting. ( Rotman joked with me via email that more than a few times, he mistakenly carried the first edition to his calculus class and had to go back to his office to switch books.) Would a graduate student with barely time to breathe between courses, teaching and research really be able spend enough time with this book to benefit from it? That of course is the big paradox about writing comprehensive graduate textbooks-you don’t really know how much of the book the students are going to have time to really use. As a student myself, I can tell you just knowing you have a source this complete and user-friendly handy takes a load of stress off regardless of how much you actually use it. These are really very minor quibbles in a book destined to become a classic. One of my absolute faves.

Basic Abstract Algebra for Undergraduate And Graduate Students by Robert B. Ash Ok, very quietly, this has become students’ secret weapon for their first year qualifying exams in algebra. It’s also the book that made the author nearly a household name in mathematics textbooks and it’s not hard to see why. The history of the book is remarkable-the book began life as an extensive set of lecture notes Ash began compiling nearly 20 years ago for the graduate algebra sequence at the University of Illinois at Urbana-Champaign when he was frustrated looking for a textbook his students could actually read and understand. He’d used Lang, Hungerford and Jacobson for the course with mixed results. His students really hated reading and working through any of them-especially Lang. Ash liked Lang’s choice of material and thought with a little more concrete discussion and explanation, it could be a terrific book. So he began writing a kind of “supplemental guide” to Lang with many more examples and a lot of rigorous, but concrete discussion. Over the next 2 decades, the notes evolved into the online version of the book and students all over the internet found it to be very  illuminating-especially when used in combination with Lang’s text. Gradually, there grew enormous demand for a printed version of the notes and Ash was able to get Dover to put out a nice, cheap paperback and the rest is history.  It’s not hard to see why this book in short order has become so beloved by struggling graduate students. The book has huge coverage of the subject and a plethora of examples in each section. (Indeed, there is a supplementary section to the first 4 chapters that allows an instructor to use this part of the book as a stand alone undergraduate text!) Groups, rings, fields, modules, algebraic geometry and algebraic number theory(!) and much more. The book is incredibly clear, comprehensive and illuminating with a style that is resolutely concrete but never lacking in either precision or rigor. The real apex of the book is a nearly 60 page concluding chapter on basic homological algebra that is one of the most pellucid and detailed presentations of the subject there is.  Frankly, if this was all there was in the book, it’d still be worth having. But there is so much more good stuff in here.  This includes brief sidebars into major topics like p-adic analysis and group representation theory, complete with references for further study- and Ash’s book contains complete, meticulous solutions to all the exercises. There’s simply no more any student could ask for in either a course text or for self-study. This book is a true classic in every sense of the word and Ash has given a huge boon to students of all levels and generations by making it available so widely, easily and cheaply. My one quibble with the book is I wish he’d included a more detailed section on group representation theory. That subject still doesn’t have a good introductory book in the same price range. Tragically, there will be no future editions. Ash died in April 2015 when he went for his usual afternoon walk and was plowed over by a car. A senseless death in an equally senseless world.   He left behind this wonderful text as part of his legacy and due to its low price, it’s guaranteed to be one of the go-to texts for serious students of algebra for generations to come.  

Algebra by Larry Grove: For over a generation in the U.S., this was the most popular alternative for a graduate class in algebra to either Lang or Hungerford unless you wanted to use one of the old classics like van der Waerden . ( In Europe, there were a few other options, but most of them weren’t in English except for the early editions of Cohn’s books.) The best description of the book’s contents is from Grove’s own preface to the book:
My own practice in teaching has been to treat the material in Chapters I-V (Groups, Rings, Fields And Galois Theory, Modules, Structure of Rings and Algebras ) as the basic course, and to include material from Chapter VI (Infinite Abelian Groups, P6lya- Redfield Enumeration, Integral Dependence and Dedekind Domains, Transcendental Field Extensions Valuations and padic Numbers, Real Fields and Sturm’s Theorem, Representations and Characters of Finite Groups Some Galois Groups)  as time permits. There are in Chapters I-V, however, several sections that can be omitted with little consequence for later chapters; examples include the sections on generators and relations, on norms and traces, and on tensor products. The selection of “further topics” in Chapter VI is naturally somewhat arbitrary. Everyone, myself included, will find unfortunate omissions, and further  topics will no doubt be inserted by many who use the book. The topics in Chapter VI are more or less independent of one another, but they tend to draw freely on the first five chapters.
Grove’s approach to the first year graduate course seems to be the old Jack Webb/”Dragnet” philosophy of mathematical exposition: “Just the facts.” Definitions, theorems and examples are all woven together with virtually no sidebar chit chat-that’s why the book’s so short compared to its’ brethren in the graduate algebra textbook family. Despite being somewhat dry, the book is quite well written with clear discussions, a decent number of good examples and quite a few nice exercises. The good thing about the choice of topics by Grove is that it can easily form the nucleus of a first year graduate course and be supplemented any way the instructor wishes-say, for example,  you want to add homological algebra to the course by adding the final 2 chapters of Ash. Grove supplies all the background. If you want something short and concise and covers all the basics, then Grove could be just what you’re looking for. At this price, it’s definitely worth a look.

Algebra Chapter 0 by Paulo Aluffi : Most of the books listed so far fit into one of 2 general types: They’re either  gigantic texts of exhaustive detail in one or multiple volumes that can’t possibly be entirely covered in a single year from cover to cover or they’re terse, slender volumes that leave the bulk of the proofs to the students.  What students really want and need is a single volume book of manageable length and depth that contains all the essential topics of modern algebra and can actually be read from cover to cover and mostly mastered in a year. It would be much easier for student and teacher alike to base a one year graduate course on such a text. We finally have such a text in Aluffi.  Several things make Aluffi’s text special and different among this whole gaggle of texts. First of all, it’s completely modern at a level somewhat lower than usual. It’s pitched at a slightly higher level than Dummit/Foote, but not quite as difficult as Lang. It’s about the same level as Rotman, but is nowhere near as exhaustive in coverage. It covers all the standard topics for a graduate algebra course: groups, rings, fields, R-modules, linear and multilinear algebra and a very lengthy chapter on homological algebra. It also covers some not so standard topics, such as homotopical algebra, derived functors and group cohomology.  Categories and commutative diagrams are introduced along with set theory at the very beginning and in a very gentle way. He doesn’t just dump the whole functor-object-composition machine on the student’s heads and then just start running diagram chases all over the place like Lang or Cohn.  He introduces the material slowly and with many examples-and most importantly, he only introduces the categorical machinery as needed. For example, although categories and commutative diagrams are both used from the very beginning, for the most of the text, he uses only standard functions in a naïve set-theory sense with formulas alongside the diagram chases. The key concept that’s used and stressed is universal properties. Functors are formally introduced much later when discussing tensor products and multilinear algebra.  In many ways, Aluffi’s book is a more advanced, streamlined and better organized version of D/F.  There are quite a few examples, but some sections have more than others. There are also many, many exercises, through which both the material in the book and additional main topics are developed. They range in diversity and level from additional examples to side lemmas to full blown developments of topics like the modular group, quaternions, nilpotent groups, Artinian rings, the Jacobson radical, localization, Lagrange's theorem on four squares and a lot more. The student really needs to work through as many of these as possible to get the full value of the book. The real strength of the book is the very tight unification through both category theory and the enormous number of exercises that are challenging, but not too difficult. The one omission that annoys me yet again is representation theory. In all fairness, though-it’s such a vast subject it’s very hard to construct a single chapter or set of notes that does it justice in a first year graduate course. Still, it’s an outstanding text, one of the very best I’ve seen in the batch-and it’s relative brevity and superior organization makes it much more ideal for a real world classroom setting then most of the others. If I had to choose one textbook for graduate algebra and its qualifying exams and couldn’t pick any others-this is the one I’d pick, hands down. As much as I love both Rotman and Knapp, Aluffi’s book would much more practical for a student.

I hope most of you either in graduate school or applying to one will find this list helpful. There’s actually is one more book I wanted to cover, but time and space ran out. It’s Louis Rowen’s epic 2 volume work, which I liked immensely. I plan to add my review of that text at some point in a future update.  I’d hate to let professor Rowen down again after he was nice enough to email me a nearly final draft of chapters 13-15 of volume 2-which sadly, I never got to. Consider a full review of these finished texts my penance.
I’m exhausted. Tomorrow, a change of pace as I’ll be turning my awesome genius and superior cognitive analysis to a far more important topic: A movie review of last year’s Godzilla film.  
I now return you to your regularly scheduled lives.

Thank you for your attention.

Peace.

5
Aug 15

@JohnFugelsang I don't.Howeer,I think you're right that he's not a Satanist.I think he's Lucifer himself. Which is why he'll be the nominee

— The Mathemagician (@Mathemagicianme) August 5, 2015

4
Aug 15

My friends think I'm nuts because I date really tall women. If u can't date a 6'2' woman bc she scares u,I'm not the one w/ the issue.

— The Mathemagician (@Mathemagicianme) August 4, 2015

2
Aug 15

Tables Chairs And Beermugs: The Mathemagician's Blog :                        Coming Attractions This i...

Tables Chairs And Beermugs: The Mathemagician's Blog :                        Coming Attractions

This i...:                        Coming Attractions This ia another short one-hopefully the last one for awhile.  BTW-let me know what size fon...

2
Aug 15

                       Coming Attractions


This ia another short one-hopefully the last one for awhile. 

BTW-let me know what size font you want me to use. I haven't decided yet and I'm still experimenting until I find the right one. Suggestions would be appreciated.

. Just wanted to let you all know things have been busy, but I haven't forgotten about this blog. Just the opposite-it'll be kicking into high gear soon as I expect to be putting virtually all my free time into it from now on.

Which isn't much, but should be enough to keep this blog alive and vibrant.

I'm currently finishing the second half of the"advanced student "abstract algebra textbook recommendation list.  It's taking longer then I thought because I ended up revising and extending it more then I thought I would. I hope to post it tomorrow. For those who kept nagging me about it, yes, I will finally comment on Paul Cohn's classic comprehensive textbook.  

Now get the fuck off my back about it...........



And coming up in the near future?


Well, I plan on ripping Hollywood a new one over last year’s Godzilla film.

 Why?

   Because Gareth Edwards had a potential 3 hour science fiction epic and he was probably afraid to even suggest making it because Hollywood doesn’t dare invest money in ideas anymore.

 So instead, we get a standard TBLB film. (TBLB = Thud, Blunder And Lizard Bashing).


Which no one should spend 10 bucks for a ticket to see no matter how good the CGI is.

What else?

I’ll be asking a magical, musical, mystery question that’s currently a stone in my shoe that’s driving me mad: Where have you gone, Keith Olbermann, the liberal world turns its lonely eyes to youuuuuuuuu, ooooo, oooo,oooooooooo……………….


   At some point, I’ll be reviewing and discussing Marvel’s Secret Wars series. If you haven’t been reading Jonathan Hickman’s Avengers titles and his 2 year Cosmic Epic To End All Cosmic Epics-well, shame on you. It’s been a thrill or three. Much more inventive, subtly written and unpredictable then any of DC’s 52 Crises on Infinite Flashpoints -or whatever their latest attempt to rip you off buying 537 crossover issues is called.

If you don't believe me, consider this:This story begins with the complete destruction of creation.
That's right. Everything ends-and then the story begins.

You have to admit-that's new. I think James Blish's The Day After Judgement is the only other story I'm aware of that does this. And Blish's story doesn't include all of time and space ending, just our little dirtball.

I not only have my commentary on this opus-I have my speculations about how it's gonna play out. I won't give anything away right now, but I will say this: I don't think the current scar faced Ruler of Creation has nearly as much control over the situation as he thinks he does.


And of course, there’ll be politics-eventually. I’ve decided to go light on this in the beginning since I’m trying to build an audience. But it’ll never be completely missing from the blog and eventually, it’ll be a main bone of content at it.

My primary political goal at this blog for the next year and a half? Helping to get a certain liberal senator from Vermont elected President. And hoping he doesn’t turn out to be a mirage like President Obama was once he’s elected. But he's the horse I'm backing-in what may be the single most important American election since 1992. I'll also be commenting about how that election signalled the end of American liberalism-and we've been sliding with steadily increasing acceleration back into pre-Great Depression Oligarchy ever since.


From The Truly FUBAR Department: By the way, did you know Americans waste 40 percent of all the food they buy while 13 percent of our children are starving? And that’s just one of the food issues this country-and frankly, the rest of the planet is facing in the 21st century. I knew Americans were complete spoiled assholes when it came to food. I’ve always been disgusted by seeing entire plates of food thrown out by people because they can. So that we waste too much food is no surprise to me.

But I have to admit, I was taken aback by that number. 

I had no idea it was that bad.

We’re definitely going to be discussing that and related issues at some point.
I’ll also be indulging the more superficial aspects of my id occasionally-not too often, don’t get nervous. For example, I’ll probably be mumbling at some point about my soft spot for statuesque women. Drink your milk so you can grow, teenage girls……………

I'll take the beautiful giantess on the end-if she'll have me...........



I’m barely 5’7 and once dated a beauty who was over 6’2’. Someone has to remind me to tell that story at some point either here or on Twitter. Remember- it takes a real man to look up to his date.

(LGBT guys not withstanding, of course. I have nothing against single sex people, but I’m happily heterosexual and just speaking for myself on this. If you've successfully surgically transitioned from the boys' to the girls' team, you're free to apply,of course. )


Anyhow-you get the idea. One day I’ll be discussing my recipe for low-fat cornbread and the perfect cup of green tea………

The next day, I’ll be reminiscing about John Byrne’s Superman and how DC’s not only buried that Superman, but brought back everything that made that drastic reboot necessary in the first place……….

Or the awesome and long lost cartoon Samurai Jack and how it deserves a better ending then a one-off comic book …………….


Or how Scott Walker is the most dangerous politician in America today that you've never heard of and the thought of him in the White House should keep you awake at night.If Ted Cruz and Donald Trump scare you more then this guy-you're not paying attention.

And that's exactly how he wants it until he gets his hands on the machinery of the American police state.


One day I’ll be complaining about an algebra problem that aggravated me and the next, I’ll be laughing about a stand up I saw on cable at 4 am the other night. One day I’ll be spitting bile and fire at the latest act of callous monstrosity by the robber baron class and later in the same day, I’ll be remembering the classic Warner Brothers cartoons I was weaned on and the hilarlous yet brilliant antics of Bugs,Daffy and all the others. One day I’ll be complaining about my gastritis and how it kept me in the bathroom all night and the next, I’ll be commenting on US foreign policy.
It’s gonna be like that. And I hope it keeps you both entertained and informed.
 
But despite the inherently capricious nature of the blog, there will be 2 recurring themes that’ll be making up the vast majority of the blog post categories: The Mathematical Sciences and it's dissemination to the masses through this and it's companion site and the ethics of progressive revolution. We seek to change American hearts and minds here-both for the better.

Whatever I choose to talk about, I can assure you, it won’t be boring. Come on by and see what I’m ranting about today. Whether you read it for political insights, my mathematical insights or how lame I think DC Comics’ New 52 has been, I think it’ll put a smile on your face.I really hope you'll choose to be along for the ride and let me do the driving for a few minutes a day.
At least, that’s what I’m hoping for.

   Lastly, it's really important you understand that like my main website, I have no intention for this to be a monologue. I'm pouring my chaotic mind and passions out here in full expection and desire of getting feedback. Whatever you feel like saying, responding, telling me, telling me where to shove it-I don't care. I want to hear from all of you, so please leave comments!

And if you have just time for a quick comment or fuck you to me too short or spontaneous, or need a dose of my wisdom but just can't get to the main blog, then follow my intermittency on Twitter @Mathemagicianme.


Until next time, I leave you with some remarkable words of wisdom from one of history's great inspirational sages:   


16
Jul 15

            Brain Droppings on Hump Day 

Hello? Hello? Is this thing on? Hello?


Well, it should be,it's my blog.


I know, I know, I promised the second part of the graduate algebra reading list tonight. But I was just too busy and too sick today and it doesn't look like I'll be finishing it before Monday. I hope I can, but sometimes life just blows for no apparent reason.


So I thought I'd share some random thoughts today. Some cognitive diaherria subsequent to the actual one I had today.


My chronic gastrisis is really beginning to piss me off. Seriously. Eating is becoming a serious problem. Sigh. Watching Food Network reminds me of when I was a normal human being. It seems an eternity ago.  


Anyone think the children of animated people in other dimensions watch live action shows on Saturday morning?


I  haven't done a formal curve-fitting regession analysis yet. But I'm becoming very sure there's an inverse relationship between my excitment over the upcoming Justice League movie and the amount of behind-the-scenes information I'm getting about it.


This is leading me to formulate the following testable hypothesis: This is going to be a piece of shit.


Seriously, it just looks like a disaster. If I'm wrong, I'll be happy to apologize. But that's how it looks to me.


And all due respect to Gal Gadot-who's lovely and not a bad actress-but she's never going to convince me she's Wonder Woman. Sorry. Princess Diana can't be built like a teenage boy in drag. 

I'm sorry,that's how Godot is built-like most modern skeletal models-turned-actresses,sadly. Just my personal.

 

Ar least they had the sense not to cast someone 5'2', so that's something.


I hope she doesn't read this and fill my blog with vitriol. Maybe I'm being unfair and barbarically alpha male here. But it has to do with my personal Platonic Form of feminine pulchritude.


 I'll blog about it in more detail at some point, but basically my Ideal of feminine beauty is a tall, curvaceous Amazon. 


5'10 and above, C to D cup bosom, a tiny waist separating the cleavage from an equally proportioned set of flaring hips and full buttocks atop long legs. In short, an Olympian goddess.


From the neck up-well, you can fill in the blank as you will. There are far more beautiful female profiles then there are physiques that fit that ideal.


And that's why Lynda Carter circa 1976 will always be Wonder Woman to me.

 

 


Those reruns of that supremely corny show made a huge impression on me as a child. How much it shaped my adult sexual ideals of women is an interesting question-one best saved for therapy sessions then a blog.


In any event-that's what turns me on and that's what I want the Supreme Superheroine to look like. Superheroes are supposed to be larger then life-and she's no exception.


That last reflection reminded me how long it's been since I went on a real date. Being broke and having sick family you're responsible for is really hard on the social life. I hope I can have one again before Grandmother Death comes for me.


If not-well, at least no man or woman ripped out my soul for kicks like a number of my friends and family had happen to them. I guess that's something.


I miss George Carlin so badly. He passed shortly after my father did-which made the yawning chasm in my life that much colder and deeper. We have so many wonderful comics now-and Bill Maher does his best every week.


But we'll never replace George. Ever. I'll never laugh again like he made me laugh.His genius glows brighter every day and every day, his insights prove more and more correct. If he had lived 3 more years, he would have seen the election of the first black president of the US and the right wing head explosion that's turned us into Tea Party Nation. I can only imagine what diamonds would have emerged from him. 


Then again, he might have ended up in Gitmo saying them given this president's track record with valid criticism. Not the crap from Fox and the other right wing propogandists-the legitimate objections a lot of us have had. 


The 4 am melancholy. Figures it would turn up and haunt my blogging now. Curse you, melancholy-I'm trying to build an audience here!


I liked Ang Lee's The Hulk much better then The Incredible Hulk. Sue me.



I also like Ballers on HBO. But it's nowhere near as good as it's predecessor, Arli$$. 


 I hope Donald Trump becomes the Republican nominee by a landslide after he calls President Obama a n***** to a 9 minute standing ovation at the Republican Convention next year-and he loses the election by the largest margin in American history.


Seriously-it would be hilarious.


Of course, the problem with that is America is stupid enough to actually elect him.................



Ok, my bed's calling me. Hopefully, the toliet won't be calling me tomorrow when I wake up.


Sleep well and  if you're Caucasian, may a cop never pull you over for speeding after you've gotten a tan at the beach.




15
Jul 15

"Monoids,R-Modules And Nonassociative Rings-These Are Some of My Favorite Things: An Updated and Expanded Suggested Reading List For Honors Undergraduate and Graduate Algebra Part I of II”

Once again, yello. Actually orange is the original font color on my Microsoft Word draft. It’s the font color I wanted to use originally at this blog-but people started sending me messages it was making them go blind. 

As Archie Bunker famously said by mangling the French in classically ignorant, working class American Conservative manner. kay sa-roo, sa-roo.

This post is an update of the last post I wrote here in 2010 before going into the twilight zone for the last few years. Many of you will notice a lot of overlap between the two posts-but you should notice a lot of differences, too. In any event, it’s my show, so pooh pooh if you don’t approve………………


For those who don’t know, I’m sort of an unofficial bibliophile for mathematical education. I inherited this love of textbooks and monographs from my inspiration, friend and unofficial mentor, Nick Metas. I was 18 years old when out of simple curiosity I called him in his office to ask him for direction in independent studies of mathematics beyond calculus-and he went on for 4 hours, naming just about every textbook and describing the subject of mathematics. That long-ago conversation is what started me on the path to becoming a mathematician.

Nowadays, the influence of Nick is very clear in my life: I have an extensive library of textbooks and monographs, people ask me all the time for references on subjects. I used to review books for the Mathematical Association Of America’s website before I couldn’t pay dues anymore. (I hope to begin doing that again at some point when I rejoin.) Like everything else, I have an opinion on most commonly used texts and monographs for all subjects-and I’m reading more every year. My hope is to begin my own small publishing company through my website by late this year. But that’s for the future.


I’ve been asked many times over the years to compose a master list of my favorite textbooks and/or monographs. On my spare time, I’ve been arrogant enough to do that in bits and pieces. Many of my posts at The Math Stack Exchange have taken this form, besides the aforementioned MAA reviews, of course.

What makes my opinions on references for mathematics different from everyone else and their mother’s lists of mathematics books is my background and life experiences. This has lead me to evaluate the quality of textbooks based on 2 criteria. Firstly, I look at mathematics textbooks from the standpoint of students, not researchers. I ask myself not which books will be the best presentation for researchers, but for talented young people aspiring not only to be researchers themselves someday, but to be educators teaching the next generation after them and presenting the material the way they wish it had been presented to them. The second concern of mine when I evaluate a book is what is the background of its intended audience? In mathematical higher education probably more than any other subject, one size most definitely does not fit all. A suitable undergraduate real analysis text at MIT will not serve well the average mathematics major at most small liberal arts colleges-just as one that will serve the average student at one of these universities will bore the hell out of their honors students who had the misfortune of going there instead of to a top flight school for any number of a hundred reasons. Every student is different and has different levels of preparation-and this does not mean they lack talent. This is a dangerous myth that tends to be perpetuated by those fortunate or wealthy enough to go to top schools. I’ll return to this point momentarily.

The list will probably undergo many revisions and additions before it reaches final form-but more importantly, I’ve decided to compose it in modular form i.e in components. This way, it’s broken into bite-sized components of manageable length that I can post here. It seems to me if I wait and try to compose it all at once-well, I’ll end up writing a 2,500 page book from the old age home I’ll be dying of cancer in. So let’s get started and hope that what little insights I can give can help neophyte students looking to broaden their knowledge base in subfields of math or are just looking for a little help in coursework they’re struggling in. Comments, input and suggestions are, of course, very welcome.

The first module here is my favorite subject in all of mathematics: algebra. (A ludicrous but sadly mandatory clarification: When a mathematics student or mathematician says ‘algebra’; it’s supposed to be understood he or she means linear and/or abstract algebra. High school algebra is, of course, the simplest special case of this wondrous arena. )

How do we define abstract algebra? Like most branches of modern mathematics, attempting a simple nonmathematical definition for non-mathematicians is a nearly impossible Catch-22 since it requires mathematical concepts to even attempt a meaningful definition. Entire philosophical treatises could probably be written attempting to answer the question and would probably fail. But I think we can try for a reasonable working definition here.

I think the best way to define algebra is that it is the general study of structures in mathematics. By a structure, we mean some kind of set -by which we mean naively a collection of objects-and a function f closed on S (the range of f is a subset of S) with a specified list of properties that characterizes that structure. For example, a group is a nonempty set S with a binary operation f such that f is associative, there is a unique element e in S such that for all elements a in S, f(e,a) = f(a,e)= a and for every a in S, there’s a unique a* such that f(a, a*)= f(a*,a)=e. Algebra deals specifically with these kinds of objects.

The pervasiveness of algebra in modern mathematics in the 21st century is astonishing. It’s more than the sheer scope of algebra itself, but the fact that most of the active areas of mathematics would not even exist without it. And I’m not talking about high-tech fields where algebra’s role is obvious-like deformation theory and higher category theory. I’m referring to the fact that most areas of mathematics are formulated in the 21st century in terms of algebraic structures. To give just one possible example of a legion, modern differential geometry would be unthinkable without the language of vector spaces and R-modules. Without tangent spaces and their associated local isomorphisms, it would be impossible to generalize calculus beyond Euclidean space. It would also be impossible to precisely define differential forms, without which most of the most interesting developments of manifold theory fall to dust. As a result, a student that’s weak in algebra needs to seriously reassess a career in mathematics.

So the least I can do is give my 2 cents on the current crop of books available.

The actual direct impetus for me writing up and posting this list was Melvyn Nathanson teaching the first semester of the year-long graduate algebra sequence at the City University Of New York Graduate Center in 2010. I began that semester sitting in on his lectures in order to begin preparations for the algebra half of my oral qualifiers for the Master’s Degree in pure mathematics at Queens College. Unfortunately, a combination of personal and financial issues prevented me from attending regularly. So that was the end of that. ( Dr. Nathanson’s lectures-and my occasional private conversations with him-are 2 of the things I miss the most about hanging out at the Graduate Center. I don’t know if he’s still active there. I’ll find out soon enough upon my return. )

I found Dr. Nathanson’s (he never told me it’s ok to call him Melvyn , so I’m going to be extra cautious as not to offend him) comments on the subject very interesting, as he has his own unique take on just about any subject. As proof, I offer this excerpt from the course’s syllabus:


In 1931, B. L. van der Waerden published the first edition of Moderne Algebra, two classic volumes, written in German, that were based in part on lectures by Emil Artin and Emmy Noether and that became the canonical work in abstract
algebra." The second edition appeared in 1937, and an English version, Modern Algebra, translated by Fred Blum and Theodore J. Benac, was published in the United States in 1949 and 1950. I and many other American mathematicians
learned algebra from the original English edition of van der Waerden. It is still a great work and I strongly recommend it for intensive study. The first volume of the seventh German edition of van der Waerden is also available in English translation, but I prefer the original. Van der Waerden's algebra begins with introductions to different algebraic structures. The first seven chapters are “Numbers and Sets," “Groups," “Rings and Fields, "Polynomials"“Theory of Fields," Continuation of Group Theory," and The Galois Theory." As proof of van der Waerden's influence, this continues to be the starting sequence of topics in most algebra courses and most algebra books, including the contemporary classic, Serge Lang's Algebra, which I also recommend. This course is different, not just in the sequence of topics, but in its philosophy. It emphasizes themes in algebra: Divisibility, dimension, decomposition, and duality,
and the course enables algebraic understanding and technique by developing these themes. The book includes all of the theorems expected in a graduate algebra course, but in a nontraditional order. The book also includes some important
topics that do not appear in van der Waerden or Lang.”

My perceived implication from the preface and his subsequent remarks was that Professor Nathanson hoped to eventually expand these notes into a textbook for a graduate algebra course. I don’t know if he ever followed through on this or what stage the book is at if he did.

But his comments got me thinking about the current state of algebra courses and the textbooks that form the basis of them. Nathanson’s experiences are not unlike those of most mathematicians of his generation: van der Waerden’s classic was the source from which he learned his algebra. Later mathematicians; particularly algebracists-such as my undergraduate algebra teacher, Kenneth Kramer-learned algebra from the earlier editions of Lang’s tome. (In fact, it was more personal for Kramer. As an honors undergraduate at Columbia in the late 1960’s, he was a student in the graduate algebra course taught by Lang himself-whose resulting lecture notes ultimately evolved into the classic text.) Most of the better universities’ graduate programs adopted Lang as the gold standard of first year graduate algebra, for better or worse, after the 1960’s. With a very few exceptions, this was the story until after the turn of the 21st century, when a host of graduate algebra texts came onto the market within a 5 year period. What was once a very sparse set of choices for this course is now a wide field of markedly diverse texts, many authored by very eminent mathematicians.

What follows is my attempt to form an amateur’s guide to these texts and my corresponding brief commentary to each. As a reviewer of textbooks, it seemed under the circumstances, that providing such a list to my erstwhile classmates in Nathanson’s course-as well as the mathematical world in general-would be a very positive undertaking. I don’t know if it would be wise, merely positive. I must add the disclaimer that I am by no means an expert; I’m merely a serious graduate student. Therefore, this reading list must be taken with a salt lick of caution as coming from an amateur and as such, it is seriously subject to revision as my knowledge grows and my mathematical style tastes change.

A major motivation in the evaluation of each of these books has been student-friendliness. Let me clarify greatly what I mean by that. A lot of top-notch mathematicians and students have an elitist, almost snobbish reaction to a textbook when you say its’ friendly. “Oh,you mean it spoon feeds the material to the brainless monkeys that pass for mathematics majors at your pathetic university? How amusing. Here at Superior U, we use only the authentic mathematics texts. Rudin.Artin Hoffman and Kunze. Alfhors. We propagate the True Word. Math is supposed to a struggle for those truly gifted enough to be worthy of it.“

Or something equally narcassistically pretentious.

I have a lot to say on this and related issues-but if I started going in depth about it here, I’d write an online book here. In future installments, I’ll begin to outline them in detail.

But in plain English, this is a bunch of crap.

The reason a lot of those “classic” texts are difficult to read isn’t because their authors were first-rate mathematicians and as such, their lessons are beyond the reach of mere mortals. In a lot of cases, it was simply because most of them never really thought about teaching; of being able to organize their deep understanding of their chosen fields -and as a result, they were very poor communicators. This lack of communication skill is reflected not only in their poor reputations as teachers, so often inversely proportional to their reps as researchers-but also in the resulting textbooks. Why don’t they? Well, again, it’s too complicated to fully go into here. But I will say that part of the reason, as any research mathematician of any prominence will tell you-is that they don’t get paid the big bucks and get the fancy titles based on how well students learn from them.

The sad part is that this myth has been perpetuated by the canonization of certain textbooks as The Books for certain classes, despite the fact that most students almost overwhelmingly despise them. And the reason why is simple: They just aren’t clear and well-organized. That makes the very act of reading them unpleasant, let alone actually learning from them. For the serious mathematics major or graduate student, this makes studying from such books virtually an act of psychic self mutilation.

To the elitists, I only have the following to say: Charles Chapman Pugh's Real Mathematical AnalysisJoseph Rotman's An Introduction to Algebraic Topology , anything by John Milnor, J.P.Serre or Jurgen Jost, Loring Tu's An Introduction to Manifolds, John McCleary's A First Course in Topology: Continuity and Dimension George F.Simmons' Differential Equations with Applications and Historical Notes,Charles Curtis' Linear Algebra: An Introductory Approach and John and Barbara Hubbard's Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach

I challenge them to consider any of these wonderful books to be spoon feeding students-and yet, they are eminently readable and wonderfully written books. In short, they are books students enjoy reading and therefore will not only learn from them-but will want to learn from them.



But an interesting trend has resulted from this myth. The students who are talented enough to learn from these texts who go on their careers to become mathematicians- and who care enough about teaching- recall their experiences as students. They don’t want to subject their students-or anyone's students-to the same torture. As a result, they try and write alternative books for students that do what they wish those texts had. The Computer Age has magnified this effect hundredfold as such books have become ridiculously easy to produce. As a result, we’ve gotten “backlash waves” of texts as alternatives to those classic tomes that created the large diversity of texts that currently exist in the various subfields of advanced mathematics. Where once there was a bare handful of such texts to choose from, a generation later, the “backlash” creates a myriad of them.

Some examples in the recent generations of math students will illustrate. Once, Alfhors’ ridiculously difficult Complex Analysis was the standard text in functions of a complex variable at U.S. graduate programs after the early 1960’s. There were a few alternatives available in English-such as Titchmarsh or Carathedory-but not a lot. This lead to an explosion of complex analysis texts in the 1970’s onward: Saks/Zygmund, Rudin, Bak/Newman, Conway, Heins, Greene/ Krantz, Jones/Singerman, Gamelin,- well, that list goes on and on. A similar backlash occurred in the 1960’s and 1970’s in general topology after an entire generation had suffered through John Kelley’s General Topology wrote a legion of such texts, including the classics by Willard and Munkres. This effect has further been enhanced by progress in those fields at the research level-which results in the presentations of the standard texts of a generation becoming outmoded. The result is the “backlash” presentations can also be “upgraded” to current language. A good example is the incorporation of category theory into advanced algebra texts post-1950’s.

I strongly believe the current large crop of graduate algebra texts is the result of a similar backlash against Lang.

I’ve gone on to some length about this because I think it’s important to keep these 2 ideas in mind- the elitist conception of Great Books and the backlash against it-when considering my readability criteria for judging such texts.

The list will be in 2 parts. The first part will focus on “warmup” texts i.e. texts that are generally too difficult to be considered first algebra texts for a standard undergraduate mathematics major, but too basic to be used as a text for a graduate course. I hope to write up a list of “basic” algebra texts for the usual students at some point. But for now, these are the books for the top students-those who have just finished honors calculus and are ready for a serious abstract mathematics course. The second part will be the heart of the list and will focus on first year graduate texts in strong programs.


So without further ado, my reading algebra list.

Enjoy.

And remember-comments and suggestions are not only welcomed, but encouraged.


Part I- Graduate Warmup: These are texts that are a little too difficult for the average undergraduate in mathematics, but aren’t quite comprehensive or rigorous enough for a strong graduate course. Of course, a lot of this is totally subjective. But it’ll make good suggestions for those struggling in graduate algebra because their backgrounds weren’t quite as strong as they thought.

Topics in Algebra, 2nd Edition by I.M.Herstein: This is the book I first learned algebra from under the sure hand of Kenneth Kramer at Queens College in his Math 337 course. It’s also the book that made me fall in love with the subject. Herstien’s style is concise yet awesomely clear at every step. His problem sets are legendarily difficult yet doable (mostly). If anyone asks me if they’re ready to take their algebra qualifier and how to prepare-I give them very simple advice: Get this book. If you can do 95 percent of the exercises, you’re ready for anything they throw at you. They’re THAT good. Warning: In true old European algebracist fashion, Herstein writes his functions in the very un-Calculus like manner on the right in composition i.e. fg= gof. This confused the author of this blog initially and no one corrected him until several weeks into the course-which lead to difficulties later on. A couple of quibbles with it-the field theory chapter is really lacking. The presentation, by today’s algebraists, may be considered somewhat old-fashioned. For example, Herstein doesn’t mention group actions and there are no commutative diagrams. This really hinders the presentation in some places. Also. Herstein tends to present even the examples-which are considerable- in their fullest generality. This makes the book harder for the beginner then it really needs to be. For example, he gives the dihedral group of rigid motions in the plane for the general n gon where n is an integer. he could start with the n=4 case and write out the full 8 member group table for the motions of the quadrilateral and then generalize. Still-I fell in love with this book. Many teachers of strong algebra courses today prefer either the more geometric approach of Artin or the similar but more modern and comprehensive approach of Dummit and Foote. Still, the book will always have a special place in my heart and I recommend it wholeheartedly for the talented beginner.

Algebra by Micheal Artin : The second edition of this book finally came out in Fall of 2010. For awhile, it looked like it might emerge posthumously-it was so long in gestation. But fortunately, this wasn’t the case. I must say in this revised review, the second edition is vastly improved over the first. The lack of exercises in the first edition has been greatly repaired with a host of new problems of varying levels of difficulty. He’s also reorganized and rewritten the book in many subtle ways that makes the writing and proofs much clearer then the first edition. Overall, the best qualities of the first edition have been preserved and improved upon. Its primary positive qualities are the heavily geometric bent and high level of presentation. The shift in emphasis from the permutation groups to matrix groups is an extremely smart one by Artin since it gives one a tool of much greater generality and simplicity while still preserving all the important properties of finite groups. (Indeed, permutations are usually explicitly represented as 2 x n matrices with integer valued bases-so the result is just a slight generalization. ) This also allows Artin to unify many different applications of algebraic structures to many different areas of mathematics-from classical geometry to Lie groups to basic topology and even some algebraic geometry (!) The major addition to the book’s presentation are many commutative diagrams allowing him to state most of the material in a completely modern manner. All through it, Artin brings an infectious love for algebra that comes through very sharply in his writing. Unfortunately, a lot of the flaws from the first edition still remain. Firstly, Artin assumes an awful lot of background in his prospective students-primarily linear algebra and basic Euclidean geometry. It might have been reasonable to assume this much background in the superhuman undergraduates at MIT in the early 1990’s, but I think that’s a stretch for most other students-even honors students. Especially nowadays. Secondly, the book is organized in a very idiosyncratic fashion that doesn’t always make sense even to people who know algebra. Nearly half the book is spent on linear algebra and group theory- rings, modules, fields are developed in a very rushed fashion. While Artin successfully expands a lot of these sections somewhat for the second edition, the book is still too unbalanced. And the discussion of modules is still too curt. Reading the second edition carefully also made me realize one of the major flaws of the overall style of the book-Artin can be painfully informal sometimes. For example, his discussion of cosets is actually confusing because he’s not as formal as he should be. It reminded me in a lot of ways of Allen Hatcher’s algebraic topology book, which suffers from a lot of the same informality. Lastly-his choice of topics for even good undergraduates is bizarre sometimes. He writes a chapter on group representations, but leaves tensor algebra and dual spaces “on the cutting room floor”? It’s a very strange choice. That being said, for all its flaws, a text of this level of daring and geometric focus by an expert of Artin’s stature is not to be ignored. I wouldn’t use it by itself, but I’d definitely keep a copy on my desk or on reserve for my students to browse.

A Course In Algebra by E.B. Vinberg This is very rapidly becoming my favorite reference for algebra. Translated from the Russian by Alexander Retakh, this book by one of the world’s preeminent algebracists is one of the best written, most comprehensive sources for undergraduate/graduate algebra that currently exists. Vinberg, like Artin, takes a very geometric approach to algebra and emphasizes the connections between it and other areas of mathematics. But Vinberg‘s book begins at a much more elementary level and gradually builds to a very high level indeed. It also eventually considers many topics not covered in Artin-including applications to physics such as the crystallographic groups and the role of Lie groups in differential geometry and mechanics! The most amazing thing about this book is how it manages to teach students such an enormous amount of algebra-from basic polynomial and linear algebra through Galois theory, multilinear algebra and concluding with the elements of representation theory and Lie groups, with an enormous number of examples and exercises that cannot be readily found in most other sources. All of it is done incredibly gently despite the steadily increasing sophistication of the material. The book has a very “Russian” style-by which I mean the author does not hesitate to both prove theorems and give applications to both geometry and physics (!) throughout. Those who know me personally know this is a position I am very sympathetic to-and for there to be a major recent abstract algebra text that takes this tack is very exciting to me.For anyone interested in writing a textbook on advanced mathematics, this is a terrific book to study for style. It is one of the most readable texts I have ever read. An absolutely first rate work that needs to be owned by any student learning algebra and any professor considering teaching it.

Abstract Algebra, 3rd edition by David S. Dummit and Richard M. Foote: Ever seen a movie or read a book where based on your tastes, everything you think and what you see in it, you should love it-but just the opposite? You don’t like it one bit and you couldn’t explain on pain of death why? That's how I feel about this book, one of the most popular and commonly used books for algebra courses-both undergraduate and graduate. It’s really frustrating that I feel that way because the book is really daring in its comprehensiveness and is surprisingly readable in many sections. It also has good exercises and more nice examples for the serious student then any book I've seen since Vinberg. So given all that, you'd think I'd be in love with the book, right? So do I, but I'm not. So what’s my problem with it? Well, first of all, it’s way too expensive. You could get both Vinberg AND a used copy of Artin for the same price as this book. (Yes, the price has come down considerably since I wrote this original review-but the book’s still ridiculously expensive brand new.) Second of all-it’s pretty dry and matter-of-fact. It just doesn’t excite me about algebra. Everything’s presented nicely and clearly-but it comes off almost like a dictionary. Lastly-the level the book is pitched at. It has pretty comprehensive coverage of the standard topics: groups, rings, field, and modules. I'm frustrated with my disappointment with D&F because the book has lots to offer.The group and field theory chapters in particular are outstanding and-I think- the highlights of the book. The book is also completely modern in outlook, it presents many commutative and functional diagrams also with many geometric examples a student will find very clarifying. It also contains some topics that are better suited for graduate courses- homological algebra and group representations, for example. The big problem is the book tries to cover all these topics at the same level and breadth as the basic material. As a result, it doesn’t succeed in developing these more sophisticated topics in enough depth for a graduate course. It also ends up covering way too much for any one-year undergraduate course. It’s certainly more comprehensive and modern then its ancestor text Herstein. So as a result, it ends up stuck in a weird level between undergraduate and graduate courses. I think this is probably what annoys me the most about this book-it comes off as a modernized and expanded,but bloated and watered down version of Herstien. Only about half the exercises are anywhere near as interesting as the ones there. I’d chop off section V altogether and expand it into a follow-up graduate text a la Knapp. I think that would a long way in improving the later sections-and the resulting 2 volume text has the potential to become the hands-down choice for the top colleges for their algebra courses. As is,it’s probably the best one book reference for algebra that currently exists and it’s nice to have handy for looking stuff up that you’ve forgotten or getting ready for exams. But it’s a problematic book to use for a course and needs to be used selectively.

OK, that ends part I. We get to the meat of the list in part II. Until then, my friends, may the forces of evil become confused on their way to your house. Ciao.




Some of the other books discussed in this post you definitely should check out: 




9
Jul 15

                                            Sad Prophecy

" Here's how I describe the American social order. The upper class has all the money, pays none of the taxes. The middle class, pays all the taxes, does all the work. The poor are there-just to scare the shit out of the middle class. Keep them showin' up at them jobs........" -the late great George Carlin, circa 1990 



8
Jul 15

                  Flies: A Brief Remembrance of Better Days 

This'll be a short one, just to keep the blog active. I don't want another 5 years to go by before I post again. I'm serious about building an audience.
 Just some random thoughts to share. 
I was going through my old,dusty books today and I came across my old dog-eared Del Rey paperback of The Best of Robert Silverberg.  In my younger days, when life was better, simpler, girls actually noticed me and I could actually eat like a normal human being without pain, I was a huge fan of the old master. Whenever we science fiction geeks got together in high school, a name we all revered was Silverberg. And going through that old book made me remember why. 
Silverberg was one of the guys aspiring science fiction writers always wanted to emulate, like aspiring rock guitarists of a previous generation always pretended to be Eric Clapton.  
Someone asked me once what my favorite Silverberg moment was. I honestly couldn't answer him. Not because there weren't any, but because there were so many.  Whenever we want to convince someone who's been trained on Joyce, Steinbeck and the stories in The New Yorker , we almost inevitably hand the unsuspecting skeptic one of the 2 people to begin with: the late great Ray Bradbury or Silverberg. 
And dammit, if Silverberg doesn't make the case for us far better than any half-assed lecture or essay would. 
"Nightflyers" "Passengers" "This is The Road", The Masks of Time, "After the Myths Went Home",
"Hawksbill Station", "At The End of Days", "Sailing to Byzantium"  "The Silent Invaders", Up The Line, Son of Man, Star Of Gypsies - you literally get out of breath trying to all the great stories the man's written. And as incredible as it sounds-at least, as incredible as it was for me to discover- the man's still writing at the tender age of 79. 
But with the wages of wisdom that are my salt and pepper temples-which occurred far younger than they should have in the course of nature-I think I now have an answer to that long-ago query, I know the answer because I reread it today and it slammed me in the head just as hard as it did when I first read it nearly 2 decades ago. It still raised goosebumps on my skin like it did when I was 14. 
It's "Flies". 
"Flies" is one of Silverberg's older works, from the period a lot of older fans consider his peak years. It was Silverberg's contribution to Harlan Ellison's legendary Dangerous Visions. And it richly deserved to be in that tome. Hell, if Silverberg had scrawled it on the back of a napkin while he was getting laid in the bathroom of a bar, any editor that had seen it and tossed it aside should not only have been fired, he (definitely he, this was 1967, remember?)  should have been blackballed so hard, the dandruff fell out of his scalp. 
I thought about giving a full review dissecting the work here. I was even eager to do it. But you know what?

The words simply eluded me.

The story is so incredibly charged, lush and existential in tone, that any words I used would just dilute it's impact.

That and I'm too damn tired right now....................lol

I'll simply say this- please go and read the story. It's been republished half a dozen times since DV. It'll be very easy to find. Invest 20 minutes of your life and read it slowly. I know people don't read anymore. As shameful as that is.

But please read this. You'll be glad you did. Trust me.

Until next time, my friends.    

6
Jul 15

RETURNING TO THE SCENE OF THE BLOG-WITH A VENGEANCE............

Well, this hasn’t worked out like I planned it to.

What else is new………….?

Nearly 5 years have passed since began this blog. Much has changed and not for the better.
I’ve barely posted at this blog since its founding. The last post dates to November, 2010, as depressing as that is. The proof lies in the rather maudlin reaction to the 2010-midterm US elections.

  Good thing I wasn’t posting during the 2012 midterms-my drunken rants might be grounds for Homeland to put me in Gitmo. After waterboarding my fat ass for kicks, of course………

 By now, if I’d been posting regularly and daringly, I might have become a runaway juggernaut blogger sensation in both academic and liberal political blogging. Yes, that sounds so smug and arrogant of me, I know. In the vast ocean of cyberspace, any of us are but a passionately screaming droplet in a roaring riptide-riddled, complex virtual electroliquid n-dimensional manifold medium spanning the whole of the human technologically embedded psychosocial ID reality. It takes a truly potent, original and dynamically and diversely informed presence to make more than a nanosecond impression on the hummingbird-like attention span of the average web surfer and the search engines they traverse.      

That and some good SEO know-how.

I think I can be one such voice and I should have tried harder to be it. I wanted to be that voice. But once again, I ended up tripping on my life.
What happened?

None of your fucking business.  

Well, that got your attention, didn’t it? Like I was demonically possessed by Chris Christie for a second…………LOL 

All kidding aside, it’s a long depressing story best told in a biography for dozens of pages to those curious enough to expend the energy and cost getting and reading such a tome. There’ll be plenty of time for self-pity later, when I can suitably manifest it for interesting reading. So I’ll table the Tale of Woe for now.  But while I won’t get into the details here and now- I will have to go into the broad brushstrokes of the picture in order to explain the new mission of this blog and its’ companion website. Ironically, if all goes as planned, it’ll be the last openly political post at this blog for several weeks.
(Of course, if an unexpected disaster of global political or economic implications occurs, I can’t promise I won’t crack and explode about it. I also make not promises not to link the articles and blog posts of other flaming liberals whose opinions I agree with. So you’ve been warned.)
Suffice to say the Great Economic Heist of 2008 finally claimed me as a victim and it’s taken me this long to climb back from suicidal despair to ambition in a new phase of my life. I realized several important things during that most recent personal Dark Age-most of which I’ll be getting to in one form or another at this blog over the next few months.
One of the most important things I finally fully realized was that America as it’s currently organized-socially, culturally, economically and politically-is a country for and by the wealthy. Sure, occasionally they throw us a crumb to prevent an annoying rebellion which would force them to spend extra money on the politicians they already own so they have the courage to send out the militia police to crush it-like the Affordable Care Act or gay marriage. But those crumbs are never done in compassion and they’re getting smaller and less frequent as our capitalist system slowly reverts to an oligarchic, feudalist ensemble of  increasingly disparate nation-states.  The rest of us are merely pungently perfumed livestock to them. Livestock they can slowly starve and torture to death before carving up to stock their pantries because they can.

They enjoy making the cow squeal in agony for the longest possible time before slitting its throat. If they’re really in a playful mood-they’ll carve the bovine up while it’s still alive and make it’s calves watch.
Worst of all-they love making the cows blame each other for it while they drink their blood in front of them-and a large number of the cows thank them for it.   
Animal Farm On Derivative Steroids.
This kind of ruthless cannibalism of the classes is nothing new in Western culture, of course. But what made America different for 2 generations in the mid to late 20th century was that we were trying to do it differently and we were succeeding. Unions protected poor and middle class workers, the government provided a floor of social safety net programs beneath which people couldn’t fall and there was a strong and effective education system that was available in principle to all. We had all but eliminated childhood hunger in this country until 20 years ago. And the wealthy was doing just fine- indeed, they made more income from the end of World War II until the early 1990’s then at any other time in our history. The only things we demanded in return was 1) a true democracy where our vote counted-hell, we even let them lobby with their wealth so they had more influence in matters that mattered to them as long as we got what little we wanted (which in retrospect, may have been a tragic error) and 2) they pay a little more so that the rest of us don’t live as medieval serfs and our children had a fighting chance at what they have. That’s all we wanted in return for not cutting their throats in a communist, Russian style revolution. And make no mistake- this country came very close to such a revolution in the 1930’s at the height of the Great Depression.
They grumbled and bitched at first. As I said, we let them get their way most of the time, so what they were complaining about is beyond most of us. But we all shared in the prosperity of the decades that followed. And most of us-regardless of class and personal wealth-were proud to be Americans.

No sane American wanted to go back to the world of Hoovervilles, soup kitchens, tommy-gun gangs  and child slave labor the climax of 160 years of the self-absorbed greed of the aristocrats had brought us-where organized crime or being a sex slave by marriage to an aristocrat was the only way for the peasants to avoid that fate.

 It was far, far, far from perfect. Ask any black, Latino or gay person over the age of 60 or any working woman of any ethnicity over the age of 50. I’m sure they have very different perceptions of those generations.

But for the most part, it was working. There was a bare minimum standard of living most of us could count on, our children had a chance at an upper class life with a little luck and a lot of work and best of all, they didn’t have to live in fear of a slow death by starvation, rape, murder or disease as their ancestors did before the New Deal.

And somewhere along the way-that changed. It changed for too many reasons to sum up in a few sentences.

 But basically what happened was that the upper class decided at some point that wasn’t good enough anymore. Why should they share?
They looked at the third world nations with slavery, poisoned food and water where only the wealthy ate without illness, where the elite bought the children of the workers as sex slaves in exchange for 5 bucks and a television, 10 cent wages and elderly poor working to death-with terrible, vicious envy.
Why couldn’t they have such power and control? Why did they have to treat the peasants as human beings? In fact, most of them were offended by the idea of the lower classes getting an  education, being protected at work, actually having a future to look forward to. “They didn’t deserve these things! They weren’t born to them! They aren’t special, like we are!”  They decided a prospering middle class was morally offensive and downright unnatural-like cats and dogs having sex.
So they decided to restore what they perceived as the natural order and make America like the Third World.

They decided they didn’t want to pay taxes anymore.

They decided national loyality is a game for the peasants to play, a good tool for encouraging servile behavior in the face of abuse.
Worst of all, they realized that for this to work, they had to ensure the lower classes had as few choices as possible. The new middles class was created primarily through providing choices that didn’t exist before to working people. They realized that had to be completely eliminated and the working class once again prisoners of circumstance for their dreams to be fulfilled.
    So they moved all our jobs overseas and began a 30 year campaign to create an American oligarchy with globally outsourced wealth. They incrementally over 40 years destroyed organized labor, public education and limited funding of elections-and then turned the same people who were suffering without these essential elements of a middle class against them. It’s an amazingly effective and tragic brainwashing campaign whose complex history and sadistic genius I’ll be discussing in detail in the future.  They created a media machine to brainwash the lower classes with the American dream, the ideas of “making it big” and “every man for himself” They took to demonizing labor and how great people take and suckers give. How personal responsibility applies to selfish teachers and union workers-and only the wealthy have the Divine Right to avoid responsibility or ethics.
 The result is a country on the verge of a 2 class system where working poor replaces the middle class and the children of adults without wealth can no longer expect to read, let alone get a quality education. The non-wealthy students of my generation are now debt vassals to the federal government which renders their educations worthless.  Despite the vast improvement of the American health care system of the Affordable Care Act-and it is a vast improvement over the sadistic Wild West of Healthcare we had before it-we still live in fear of illness bankruptsy and we likely will remain the only developed nation where this is true.
And don’t get me started on the environment and the slow death of the Earth.

All this and more I’ve seen, lived and watched. I myself was forced to abandon my graduate studies in mathematics near the finish line due to the depletion of my resources and my chronic illnesses.

And you know what?

I’ve decided I’m done just watching them strangle this world quietly.

They still may succeed in enslaving our children for centuries and leaving us here to devour our own children and die by age 30 while they live in isolated hi-tech castles above the sewers and prisons the rabble live in. It may be beyond any human power to stop that future from coming to fruition at this point.
But it’s not beyond my power to comment, criticize, inform and provide the scientific education to the masses that they dread us having. It’s not beyond my power to fight the future with knowledge, conscience and honesty.  

With the reinvigoration and renewed commitment of myself to this blog, I think it’s appropriate to reintroduce myself, the blog and its general purpose-as well as its new role as part of my larger website: T.U.L.O.O.MATH.  Allow me to introduce myself, as I’ve said

Who is.....The Mathemagician?

A man with no plan, but a freight car full of ideas and a heart of flame and frost,  fire and ice. (With apologies to George R.R. Martin for the last part, of course……….)

He is a superb tutor in mathematics and the hard sciences with over 15 years of tutoring experience at all levels from high school to beginning graduate studies.

After dabbling as a premed, then majoring in chemistry, physical chemistry and biochemistry, he was seduced into pure and applied mathematics and never looked back. He was a graduate student in pure mathematics when personal tragedy and financial ruin struck -and stopped his studies just before his oral exams for the M.S. degree. He plans to return triumphantly to complete his MS and to ultimately obtain a PHD before dying. He has 3 bachelors’ degrees, in philosophy, physical chemistry and mathematics as well as minors in biochemistry and psychology. He also has opinions on everything, both mathematical and otherwise.

He is a superb researcher and paper author who authors an online blog and more recently, is a regular poster/contributor to both The Math Stack Exchange and Math Overflow . He is also a past online reviewer of textbooks for the Mathematical Association Of America and hopes to return to this labor of love when time permits.
He is a diverse scholar and lecturer who has studied with such experts as Saul Kripke , Melvyn Nathanson and Dennis Sullivan.


He is a professional research paper/ essay/fiction writer and tutor who prides himself on a nearly perfect record of A grades as an undergraduate in written work and specializes in both paper writing for hire and tutoring students in the fine arts of academic research and paper drafting.
His personal statement: Mathematicians are sorcerers. But instead of tea leaves, toadstools, phoenix feathers, polyjuice potions and eye of newt, we conjure with Lebesgue measures, noncommutative rings, differentiable manifolds, Fourier transforms, deformation algebras and power series. We read the Book of Nature and if we ask it the right questions.


He currently is hoping to change the world with his finally completed website, TULOOMATH. This is A Free Complete Online Self Study Library And Advisory Center of Pre-University/ University Level Mathematics Lecture Notes, Inexpensive In-Print And Online Texts And Other Resources, Textbook Reviews, Student Advice, Affordable Tutoring, Counseling Services, Blog Commentary on Matters Both Mathematical and Otherwise and Much, Much More! His hope is that will become a major online resource for students of mathematics and others.

This blog was originally a Google Blog. I’m still learning WordPress, so to be honest, I still haven’t found the best way to design the blog at the website. I originally had imported the blog to the website-but I hated the way it looked and could never get it to look perfect. So here I am, back where I started. Sometimes, less truly is more.   

 I recently began a major new enterprise-the major website T.U.L.O.O.MATH. I plan for this blog to become the beating heart of the website, so I really should begin the blog’s rebirth by describing exactly what the associated website is and why I hope it’ll become a major force online for higher education.  

 This website began 2 years ago on a rainy January night. Depressed beyond finite measure by the bleak future I can now see too well, my chronic illnesses that had frustrated me both  academically and for job prospects and the financial ruin without health insurance had that all but prevented my completing my much-delayed masters’ degree,  I wanted badly to change my world in a tangible way. I wanted to make a difference, as hackneyed and clichéd as that sounds-before I grew old and died of whatever fate awaited me.

I surfed the web to try and raise my spirits. After pouring over blogs, The Huffington Post, my Facebook page and some scantily clad goddesses (hey, I’m human, sue me)-I began searching for online lecture notes in higher mathematics, physics or chemistry from universities I’d never visited. I’d been fascinated by the growth of online lecture notes for some time. Not only where they of increasing quality and originality in the subject matter they portrayed, they were free in most cases. With the rising tide of textbook prices-they’ve become a great source of self-education for poor students. I wasn’t the first to notice them, of course. There were already some sites dedicated to popularizing their owners’ favorite notes or the authors of the notes themselves pushing them. An enormous diversity of lecture notes, from grade school to PHD level, from all manner of authors, from lowly community college adjuncts to Distinguished Professors at the top Ivy League schools. And all available to anyone who wants them if one’s willing to put the time in to look for them.

And that’s when the idea hit me like Godzilla on a rampage.

Since higher education is again becoming beyond the reach of all but the aristocracy’s children in America, we can look to previous generations of peasants for alternatives to formal education.

What did they do if they really wanted an education?

 They educated themselves in the sciences and arts.

They read second hand textbooks, they took time in libraries, both personal and public, they
attended lectures where they were allowed to without getting tossed out on their ass as non-
gentry filth. History is filled with self-educated men and women who changed the world,
something they never would have had a chance to do if they followed the available
academic paths of their elitist societies. Just to name some of the more famous ones: Abraham
Lincoln , Ernest Hemingway, William Blake, Jorge Luis Borges, Howard Lovecraft, Leonardo
Da Vinci, James Watt and Malcom X. 

The language of nature is mathematics and I’m convinced that without it, even the best scientists lack the capacity for truly original breakthroughs in conceptual theory. As difficult as it is to believe, there have actually been autodidacts in mathematics in history: Nathaniel
Bowditch, George Green, Gottfried Wilhelm LeibnizSrinivasa Ramanujan, and Oliver
Heaviside

In the 21st century, the post-democratic corporatocracy autodidact peasant has a tool of
undreamt of power by previous generations for the purpose of a university-level self-
education-the Internet.  The internet is a constantly morphing treasure trove of free or
inexpensive materials, particularly online lecture notes and textbooks by mathematicians and mathematics teachers who refuse to put them behind firewalls.

There’s also a relatively large number of “old fashioned” paper textbooks  on advanced mathematics that are available extremely cheaply, particularly in used copies.  Dover Books famously was the major publisher that printed such poor man’s textbooks. Relative to the exorbitant cost of the major publishers of the “standard” university textbooks, such as Springer-Verlag, CRC Press and Cambridge University Press, brand new Dover paperbacks are indeed cheap. But they’ve risen alarmingly in price over the last 20 years and the publisher has taken recently to publishing some   texts as expensive “Phoenix” hardcover editions-whicb is rather distressing to those of us who, as students, came to identify Dover as a source of textbooks that could be had for less then the cost of an express bus trip from Queens to Manhattan. There also are now a handful of self- published textbooks and small publishing houses, which are making either new editions or reprints available very cheaply. 
  
My goal with this site is to create a single comprehensive online guide dedicated to effectively leading all aspiring students of mathematics and the sciences to all available free and/or inexpensive resources, both on the internet and in print in the new Gilded Age who want an education, but can’t afford the joke we now call higher education in our Hunger
Games-lite 1 % world. The goal is an oasis for self-education in the mathematical sciences on the web.


As I said earlier, websites with lists of links to free lecture notes or online textbooks are
nothing new of course-neither are websites with mathematicians or mathematics students
listing their favorite textbooks with commentary to guide future students looking to either self
learn or for self-help with their classes. There are plenty of such sites, a number of which I’ll be
listing in the links section. So what makes my site different?

My site is different from the others in at least 5 ways.

 First of all, I’ve systematically-over the last 2 years-searched, read, and evaluated
both the Internet and published sources. That’s right, a large percentage of the linked sources here have been read and reviewed by Yours Truly. How qualified I was to really evaluate the
more advanced source material, I can’t say for certain. What I can say is that I can-and did-evaluate how useful the materials were to me as a graduate student in mathematics looking to continue my own studies without access to formal coursework-which, of course, are exactly the kinds of students this website is designed to help! I sincerely hope my Herculean efforts pay off for the users of the site. I hope to eventually have reviews posted for ALL the source material-including reviews from other regular users of the site! More on that momentarily.

  A second difference from my website to many of the other “booklist/noteslist” sites from
across the Web is that all the sources-regardless of whether or not detailed reviews yet
exist for them-have been rated for difficulty level so that users can easily determine if they have the background to benefit from them before downloading or buying them. To my knowledge, there’s no other site that does this. My rating system is based on the one in Guillemin and Pollack’s classic textbook Differential Topology, with some minor adjustments for mere mortal students who aren’t at MIT like their students were! I’m very proud of this rating system-I hope it spares the users a lot of hassle finding what they need and at the correct level given their backgrounds.
  
   A third difference was touched upon above- I fully intend for this website to become an online community for students and  teachers of various levels-from high school to professional
level-from all over the world to not only use and share these free resources and commentary, but to contribute to the site via email and the message board. Have a review or commentary for a source or suggestions for it’s use based on your own experience using it? Post it! Got suggestions to improve the site?  Drop me an email or post it! Want to tell me to go to hell for my communist ideas and eating into the bottom line of the American corporations you either own or worship? Go ahead and post it-although I doubt you’ll like my reply…… Anyway, you get the idea. I want to hear from all of you!
  
    A fourth and critical difference is the comprehensiveness of my website. To my knowledge,
there is no site that collects links to freely available online lecture notes and/or online textbooks and course web pages that’s anywhere near as complete in content, as detailed and up to date as mine. There are several other websites that have very extensive listings of links to online lecture notes such as mine aims to have-there are quite a few listed in the Links section below. But as far as I know, none of them-none-have the degree of evaluation and guidance that my site provides for students.  There are also quite a number of sites that have reviews of mathematics textbooks and advice for their use. Most examples of this kind of site actually have many more reviews then will even be contained in my ebooks available here. 
  

The last and most important difference-and the one that I think makes my site incredibly timely in this particular moment in American history-the site has a very specific and focused theme, namely being a one stop center online for the availability, expert recommending and
guidance to inexpensive self-education and references in mathematics, from the  high school level to graduate school.
 The site will also provide inexpensive personal tutoring and counseling services online. For impoverished students worldwide who cannot realistically enter college because of cost , the lecture notes and textbooks reviewed here will be of immense help. In my native America, where higher education is a)  systematically and quickly reverting to a luxury for the wealthy and b) is extorting an entire generation of former college students who are unable to complete their studies because of infinite debt that will haunt them their
entire lives, this is a necessity to avoid a life of serfdom.

From the day we're born to the day we die, we all seek to change the world in some small way. To leave a legacy, some sign we existed for future generations. For better or worse, this is mine. I hope it becomes a permanent fixture in one form or another for students worldwide to open the vast vistas of mathematics to them and future generations. 


At this blog, I hope to provide not only content related to the website-i.e. reviews of textbooks, updates and addendum to the site, mathematical observations and innuendo-but in addition, I’ll provide a wealth of other commentary and literature on other things meaningful to me. Movies, comic books, popular culture-you’ll find it all here depending on what mood I’m in. The title of the blog is very deliberately chosen. This blog will range very widely in what you find posted here on any given day, depending on my mood and thoughts. But I do particularly hope to post regularly on my liberal politics and philosophy that underlies the website and it wouldn’t be appropriate to rant on there. At first glance, these 2 aspects seem as unrelated as insects and dinosaurs. But a closer inspection shows the website was the academic child of the passions that underlie my politics and ethics-I believe education is a right to be had my all. My hope is that as education becomes increasingly inaccessible, not only will my site fill the void left by public universities, but it will inspire many more such sites on the internet.  

Many will find my views inflammatory- some will find them downright offensive. But many will agree with my bleak assessments-and I just hope some will find aspiration for solutions contained in my diatribes.

Which is why in closing, it’ll seem very counter to the call to arms I just spent pages writing to announce I plan to lay off politics for the first few weeks. At which a lot of people are probably scratching their temples.

Simply put, I want to establish my voice and personality here before beginning to tackle weighty matters. I want to establish an audience. And frankly, I’d like to lighten the mood here at first-for my benefit as much as all of you.

Think of it as Nolan Ryan in his hayday soft tossing to warm up in spring training. (I would have said Roger Clemens, but he’s effectively crapped on that image, along with all the rest of the ‘roid heads in baseball. I’ll probably bitch about that at some point, too.)

In any event, please join me in the revolution. You’ll be glad you did. At least, I hope you will.
Well, it’s nearly 4 am as I write this and my brain is shutting down. Until next time, true believers.


7
Apr 15

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15
Nov 10

"Monoids,R-Modules And Nonassociative Rings-These Are Some of My Favorite Things: A Suggested Reading List For Graduate Algebra"

Yello.

Brief political comment first.

November 2nd, 2010 may go down as one of the blackest in American history in coming decades and centuries. People who have been predicting a corporate takeover leading to a plutocracy in America -a “third world” complete with utopian gated communities defended by mercenary private security interspersed with regions of abject poverty where starving mobs of disease riddled peasants try and stay alive into their 30’s if possible-may have seen the first step towards that future on that day.

Not that our precious President and his party have been anything even remotely resembling heroic figures-between half-ass health care reform, continuing to allow the slaughter of an entire generation of young Americans, as well the slow bankruptcy of out nation to satisfy whatever mysterious powers now profit from it-they've been as cowardly and subservient to the special interests as the other side on most days. In many ways, that’s partly what led to his party’s downfall.
But with the election shenanigans with The Chamber Of Commerce-after the worst Supreme Court decision for the country at large since Plessy Vrs. Ferguson - effectively rendered the Republican party a wholly owned subsidiary of the corporations. Unless a dramatic opposition to this systematic subversion of the government occurs in the next 2 years, the very best we have to look forward to from this decision is eternal gridlock until the Commerce can buy the Presidency in 2012 and put the best candidate money can buy in there.

The poor ignorant slobs who marched for the Tea Party are in for a very rude awakening indeed-that is, if any of them have the intelligence to realize they’ve been used and discarded.

And then the future I predicted in my opening preamble-coupled with environmental collapse with only the wealthy having food and water fit for human consumption, let alone other resources-will come to pass within a generation.

That’s all I wanted to say on this for now. There’ll be much more to come when I can steal myself to a full analysis and discussion.

But now onto a promised, much more academic matter.

For those who don’t know, I’m sort of an unofficial bibliophile for mathematical education. I inherited this love of textbooks and monographs from my inspiration, friend and unofficial mentor, Nick Metas. I was 18 years old when out of simple curiosity I called him in his office to ask him for direction in independent studies of mathematics beyond calculus-and he went on for 4 hours, naming just about every textbook and describing the subject of mathematics. That long-ago conversation is what started me on the path to becoming a mathematician.

Nowadays, the influence of Nick is very clear in my life: I have an extensive library of textbooks and monographs, people ask me all the time for references on subjects and I review books for the Mathematical Association Of America’s website.(which can be found here). I have an opinion on most commonly used texts and monographs for all subjects-and I’m reading more every year. In fact, I have a private dream of beginning my own small publishing company someday.
(Of course, that’ll depend on the prediction above being dead wrong. We can all hope the country comes to it’s senses before its too late. They’re angry-that’s a good beginning Now they just have to develop enough intelligence to channel the anger constructively. Hope it happens in time. So far it doesn‘t look promising……..)

I’ve been asked many times to compose a master list of my favorite textbooks and/or monographs. The list will probably undergo many revisions and additions before it reaches final form-but more importantly, I’ve decided to compose it in modular form i.e in components. This way, it’s broken into bite-sized components of manageable length that I can post here. It seems to me if I wait and try to compose it all at once-well, I’ll end up writing a 2,500 page book from the old age home I’ll be dying of cancer in. So let’s get started and hope that what little insights I can give can help neophyte students looking to broaden their knowledge base in subfields of math or are just looking for a little help in coursework they’re struggling in. Comments, input and suggestions are, of course, very welcome.

The first module here is my favorite subject in all of mathematics: algebra. (A ludicrous but sadly mandatory clarification: When a mathematics student or mathematician says ‘algebra’; it’s supposed to be understood he or she means abstract algebra. High school algebra is, of course, the simplest special case of this wondrous arena. )

How do we define abstract algebra? Like most branches of modern mathematics, attempting a simple nonmathematical definition for non-mathematicians is a nearly impossible Catch-22 since it requires mathematical concepts to even attempt a meaningful definition. Entire philosophical treatises could probably be written attempting to answer the question and would probably fail. But I think we can try for a reasonable working definition here.

I think the best way to define algebra is that it is the general study of structures in mathematics. By a structure, we mean some kind of set -by which we mean naively a collection of objects-and a function f closed on S (the range of f is a subset of S) with a specified list of properties that characterizes that structure. For example, a group is a nonempty set S with a binary operation f such that f is associative, there is a unique element e in S such that for all elements a in S, f(e,a) = f(a,e)= a and for every a in S, there’s a unique a* such that f(a, a*)= f(a*,a)=e. Algebra deals specifically with these kinds of objects.

The pervasiveness of algebra in modern mathematics in the 21st century is astonishing. It’s more then the sheer scope of algebra itself, but the fact that most of the active areas of mathematics would not even exist without it. And I’m not talking about high-tech fields where algebra’s role is obvious-like deformation theory and higher category theory. I’m referring to the fact that most areas of mathematics are formulated in the 21st century in terms of algebraic structures. To give just one possible example of a legion, modern differential geometry would be unthinkable without the language of vector spaces and R-modules. Without tangent spaces and their associated local isomorphisms, it would be impossible to generalize calculus beyond Euclidean space. It would also be impossible to precisely define differential forms, without which most of the most interesting developments of manifold theory fall to dust. As a result, a student that’s weak in algebra needs to seriously reassess a career in mathematics.

So the least I can do is give my 2 cents on the current crop of books available.

The actual direct impetus for me writing up and posting this list was Melvyn Nathanson at the City University Of New York Graduate Center. This semester, the eminent number theorist is teaching the first semester of the year-long graduate algebra sequence there. I began this semester sitting in on his lectures in order to begin preparations for the algebra half of my oral qualifiers for the Master’s Degree in pure mathematics at Queens College. Unfortunately, a combination of personal and financial issues prevented me from attending regularly. So that was the end of that.

I found Dr. Nathanson’s (he hasn’t told me it’s ok to call him Melvyn yet, so I’m going to be extra cautious as not to offend him) comments on the subject very interesting, as he has his own unique take on just about any subject. As proof, I offer this excerpt from the course’s syllabus:

In 1931, B. L. van der Waerden published the first edition of Moderne Algebra,
two classic volumes, written in German, that were based in part on lectures by
Emil Artin and Emmy Noether and that became the canonical work in abstract
algebra." The second edition appeared in 1937, and an English version, Modern
Algebra, translated by Fred Blum and Theodore J. Benac, was published in the
United States in 1949 and 1950. I and many other American mathematicians
learned algebra from the original English edition of van der Waerden. It is still a
great work and I strongly recommend it for intensive study. The first volume of the
seventh German edition of van der Waerden is also available in English translation,
but I prefer the original. Van der Waerden's algebra begins with introductions to different
algebraic structures. The first seven chapters are “Numbers and Sets," “Groups," “Rings and
Fields, "Polynomials"“Theory of Fields," Continuation of Group Theory," and
The Galois Theory." As proof of van der Waerden's influence, this continues to
be the starting sequence of topics in most algebra courses and most algebra books,
including the contemporary classic, Serge Lang's Algebra, which I also recommend.
This course is different, not just in the sequence of topics, but in its philosophy. It
emphasizes themes in algebra: Divisibility, dimension, decomposition, and duality,
and the course enables algebraic understanding and technique by developing these
themes. The book includes all of the theorems expected in a graduate algebra
course, but in a nontraditional order. The book also includes some important
topics that do not appear in van der Waerden or Lang.”

The affirmed originality of the course, I don’t doubt. I’m still hoping to obtain a complete set of handwritten notes from some of my friends in the course, which is the official text for the course. The clear implication from the preface and his subsequent remarks is that Professor Nathanson hopes to eventually expand these notes into a textbook for a graduate algebra course.

Privately, I’m hoping to work with Dr. Nathanson as a PHD student eventually and if he follows through on this, perhaps I can be involved in the book’s drafting process. But that’s for the future.

His comments got me thinking about the current state of algebra courses and the textbooks that form the basis of them. Nathanson’s experiences are not unlike those of most mathematicians of his generation: van der Waerden’s classic was the source from which he learned his algebra. Later mathematicians; particularly algebracists-such as my undergraduate algebra teacher, Kenneth Kramer-learned algebra from the earlier editions of Lang’s tome. (In fact, it was more personal for Kramer. As an honors undergraduate at Columbia in the late 1960’s, he was a student in the graduate algebra course taught by Lang himself-whose resulting lecture notes ultimately evolved into the classic text.) Most of the better universities’ graduate programs adopted Lang as the gold standard of first year graduate algebra, for better or worse, after the 1960’s. With a very few exceptions, this was the story until after the turn of the 21st century, when a host of graduate algebra texts came onto the market within a 5 year period. What was once a very sparse set of choices for this course is now a wide field of markedly diverse texts, many authored by very eminent mathematicians.

What follows is my attempt to form an amateur’s guide to these texts and my corresponding brief commentary to each. As a reviewer of textbooks, it seemed under the circumstances, that providing such a list to my erstwhile classmates in Nathanson’s course-as well as the mathematical world in general-would be a very positive undertaking. I don’t know if it would be WISE, merely positive. I must add the disclaimer that I am by no means an expert; I’m merely a serious graduate student. Therefore, this reading list must be taken with a salt lick of caution as coming from an amateur and as such, it is seriously subject to revision as my knowledge grows and my mathematical style tastes change.

A major motivation in the evaluation of each of these books has been student-friendliness. Let me clarify greatly what I mean by that. A lot of top-notch mathematicians and students have an elitist, almost snobbish reaction to a textbook when you say its’ friendly. “Oh,you mean it spoon feeds the material to the brainless monkeys that pass for mathematics majors at your pathetic university? How amusing. Here at Superior U, we use only the authentic mathematics texts. Rudin.Artin Hoffman and Kunze. Alfhors. We propagate the True Word. Math is SUPPOSED to a struggle for those truly gifted enough to be worthy of it. “

Or something equally narcassistically pretentious.

I have a LOT to say on this and related issues-but if I started going in depth about it here, I’d write an online book here. In future installments, I’ll begin to outline them in detail.
But in plain English, this is a bunch of crap.

The reason a lot of those “classic” texts are difficult to read isn’t because their authors were first-rate mathematicians and as such, their lessons are beyond the reach of mere mortals. In a lot of cases, it was simply because most of them never really thought about teaching; of being able to organize their deep understanding of their chosen fields -and as a result, they were very poor communicators. This lack of communication skill is reflected not only in their poor reputations as teachers, so often inversely proportional to their reps as researchers-but also in the resulting textbooks. Why don’t they? Well, again, it’s too complicated to fully go into here. But I WILL say that PART of the reason, as any research mathematician of any prominence will tell you-is that they don’t get paid the big bucks and get the fancy titles based on how well students learn from them.

The sad part is that this myth has been perpetuated by the canonization of certain textbooks as The Books for certain classes, despite the fact that most students almost overwhelmingly despise them. And the reason why is simple: They just aren’t clear and well-organized. That makes the very act of reading them unpleasant, let alone actually learning from them. For the serious mathematics major or graduate student, this makes studying from such books virtually an act of psychic self mutilation.

To the elitists, I only have the following to say: Charles Chapman Pugh’s Real Mathematical Analysis, Joseph Rotman’s An Introduction To Algebraic Topology, ANYTHING by John Milnor, J.P.Serre or Jurgen Jost, Loring Tu’s An Introduction to Manifolds, John McCleary’s A First Course In Topology: Continuity And Dimension, George Simmons’ An Introduction To Differential Equations With Historical Notes,2nd edition. Charles Curtis’ Linear Algebra, 4th edition. and John And Barbara Hubbard’s Vector Calculus, Linear Algebra And Differential Forms: A Unified Approach ,3rd edition.

I challenge them to consider any of these wonderful books to be spoon feeding students-and yet, they are eminently readable and wonderfully written books. In short, they are books students ENJOY reading and therefore will not only learn from them-but will WANT to learn from them.

But an interesting trend has resulted from this myth. The students who are talented enough to learn from these texts who go on their careers to become mathematicians- and who care enough about teaching- recall their experiences as students. They don’t want to subject their students-or ANYONE’S student-to the same torture. As a result, they try and write alternative books for students that do what they wish those texts had. The Computer Age has magnified this effect hundredfold as such books have become ridiculously easy to produce. As a result, we’ve gotten “backlash waves” of texts as alternatives to those classic tomes that created the large diversity of texts that currently exist in the various subfields of advanced mathematics. Where once there was a bare handful of such texts to choose from, a generation later, the “backlash” creates a myriad of them.

Some examples in the recent generations of math students will illustrate. Once, Alfhors’ ridiculously difficult Complex Analysis was the standard text in functions of a complex variable at U.S. graduate programs after the early 1960’s. There were a few alternatives available in English-such as Titchmarsh or Carathedory-but not a lot. This lead to an explosion of complex analysis texts in the 1970’s onward: Saks/Zygmund, Rudin, Bak/Newman, Conway, Heins, Greene/ Krantz, Jones/Singerman, Gamelin,- well, that list goes on and on. A similar backlash occurred in the 1960’s and 1970’s in general topology after an entire generation had suffered through John Kelley’s General Topology wrote a legion of such texts, including the classics by Willard and Munkres.
This effect has further been enhanced by progress in those fields at the research level-which results in the presentations of the standard texts of a generation becoming outmoded. The result is the “backlash” presentations can also be “upgraded” to current language. A good example is the incorporation of category theory into advanced algebra texts post-1950’s.

I strongly believe the current large crop of graduate algebra texts is the result of a similar backlash against Lang.

I’ve gone on to some length about this because I think it’s important to keep these 2 ideas in mind- the elitist conception of Great Books and the backlash against it-when considering my readability criteria for judging such texts. So without further ado, my reading list. Enjoy.

And remember-comments and suggestions are not only welcomed, but encouraged.

Part I- Graduate Warmup: These are texts that are a little too difficult for the average undergraduate in mathematics, but aren’t quite comprehensive or rigorous enough for a strong graduate course. Of course, a lot of this is totally subjective. But it’ll make good suggestions for those struggling in graduate algebra because their backgrounds weren’t quite as strong as they thought.

Topics in Algebra by I.R. Herstien, 2nd edition: This is the book I first learned algebra from under the sure hand of Kenneth Kramer at Queens College in his Math 337 course. It’s also the book that made me fall in love with the subject. Herstien’s style is concise yet awesomely clear at every step. His problem sets are legendarily difficult yet doable (mostly). If anyone asks me if they’re ready to take their algebra qualifier and how to prepare-I give them very simple advice: Get this book. If you can do 95 percent of the exercises, you’re ready for anything they throw at you. They’re THAT good. Warning: In true old European algebraicist fashion, Herstien writes his functions in the very un-Calculus like manner on the RIGHT in composition i.e. fg= gof. This confused the author of this blog initially and no one corrected him until several weeks into the course-which lead to difficulties later on. A couple of quibbles with it-the field theory chapter is really lacking. Also. Herstein tends to present even the examples-which are considerable-in their fullest generality. This makes the book harder for the beginner then it really needs to be. For example, he gives the dihedral group of rigid motions in the plane for the general n gon where n is an integer. he could start with the n=4 case and write out the full 8 member group table for the motions of the quadrilateral and THEN generalize. Still-I fell in love with this book. The presentation is considered outdated by most mathematicians now, who prefer the more geometric approach of Artin. Still, the book will always have a special place in my heart and I recommend it wholeheartedly for the talented beginner.

Algebra by Micheal Artin : The second edition of this book finally came out in September. For awhile, it looked like it might emerge posthumously-it was so long in gestation. But fortunately, this wasn’t the case. I haven’t read it carefully yet, but from what I’ve seen of it, it looks very similar to the first edition. As for the first edition- well, I got really mixed feelings about it. Artin’s book has many, many good qualities. It’s primary positive qualities are the heavily geometric bent and high level of presentation. The shift in emphasis from the permutation groups to matrix groups is an extremely smart one by Artin since it gives one a tool of much greater generality and simplicity while still preserving all the important properties of finite groups. (Indeed, permutations are usually explicitly represented as 2 x n matrices with integer valued bases-so the result is just a slight generalization. ) This also allows Artin to unify many different applications of algebraic structures to many different areas of mathematics-from classical geometry to Lie groups to basic topology and even some algebraic geometry (!) All through it, Artin brings an infectious love for algebra that comes through very sharply in his writing. So why the hesitant recommendation?
Because it’s easy to like the book when you’ve already learned the material from other sources.
Would even a talented beginner find the book so appealing? I don’t think so. Firstly, Artin assumes an awful lot of background in his prospective students-primarily linear algebra and basic Euclidean geometry. It might have been reasonable to assume this much background in the superhuman undergraduates at MIT in the early 1990’s, but I think that’s a stretch for most other students-even honors students. Especially nowadays. Secondly, the book is organized in a very idiosyncratic fashion that doesn’t always make sense even to people who know algebra. Nearly half the book is spent on linear algebra and group theory and rings, modules, fields are developed in a very rushed fashion. Some of these sections really needed more fleshing out. Also, a lot of the group theory chapters are confusing. His discussion of both cosets and tilings in the plane are particularly discombobulated. Lastly-his choice of topics for even good undergraduates is bizarre sometimes. He writes a chapter on group representations, but leaves tensor algebra and dual spaces “on the cutting room floor”? It’s a very strange choice. The book’s main flaw is there are too few exercises and most of these are ridiculously difficult. (In all fairness, I understand this was the main problem Artin is trying to rectify in the new edition. ) That being said, for all it’s flaws, a text of this level of daring by an expert of Artin’s stature is not to be ignored. I wouldn’t use it by itself, but I’d definitely keep a copy on my desk. Apparently,though, in the 2 decades since this book was written, Artin has rethought the course and it’s structure-it remains to be seen if the new version has gotten most of the bugs out. If it has, the book will be a must-have, hands down, for students of algebra.

A Course In Algebra by E.B.Vinberg This is very rapidly becoming my favorite reference for algebra. Translated from the Russian by Alexander Retakh, this book by one of the world’s preeminent algebracists is one of the best written, most comprehensive sources for undergraduate/graduate algebra that currently exists. Vinberg, like Artin, takes a very geometric approach to algebra and emphasizes the connections between it and other areas of mathematics. But Vinberg‘s book begins at a much more elementary level and gradually builds to a very high level indeed. It also eventually considers many topics not covered in Artin-including applications to physics such as the crystallographic groups and the role of Lie groups in differential geometry and mechanics! The most amazing thing about this book is how it manages to teach students such an enormous amount of algebra-from basic polynomial and linear algebra through Galois theory, multilinear algebra and concluding with the elements of representation theory and Lie groups, with an enormous number of examples and exercises that cannot be readily found in most other sources. All of it is done incredibly gently despite the steadily increasing sophistication of the material. The book has a very “Russian” style-by which I mean the author does not hesitate to both prove theorems and give applications to both geometry and physics (!) throughout. Those who know me personally know this is a position I am very sympathetic to-and for there to be a major recent abstract algebra text that takes this tack is very exciting to me.
For anyone interested in writing a textbook on advanced mathematics, this is a terrific book to study for style. It is one of the most readable texts I have ever read. An absolutely first rate work that needs to be owned by any student learning algebra and any professor considering teaching it.

Abstract Algebra, 3rd edition by David S. Dummit and Richard M. Foote : Ever seen a movie or read a book where based on your tastes, everything you think and what you see in it, you should love it-but just the opposite? You don’t like it one bit and you couldn’t explain on pain of death why? THAT’S how I feel about this book, one of the most popular and commonly used books for algebra courses-both undergraduate and graduate. It has a good, very comprehensive selection of material, good exercises and lots of nice examples for the serious student. So what’s my problem with it? Well, first of all, it’s WAY too expensive. You could get both Vinberg AND a used copy of Artin for the same price as this book. Second of all-it’s pretty dry and matter-of-fact. It just doesn’t excite me about algebra. Everything’s presented nicely and clearly-but it comes off almost like a dictionary. Lastly-the level the book is pitched at. It has pretty comprehensive coverage of the standard topics: groups, rings, field, and modules. It also contains some topics that are better suited for graduate courses- homological algebra and group representations, for example. The problem is the book tries to cover all these topics equally. As a result, it doesn’t succeed in developing all of them in enough depth for a graduate course and it ends up covering way too much for any one-year undergraduate course. And to be frank, a lot of the presentation of the undergraduate material is very similar to that of Herstein- except the only about half the exercises are anywhere near as interesting as the ones there. I think this is probably what annoys me the most about this book-it comes off as a bloated, watered down version of Herstien. It’s nice to have handy for looking stuff up that you’ve forgotten -but for its price, see if you can borrow a copy instead.

The Big Three: These are the 3 textbooks that up until about 10 years ago, were the standard texts at the top graduate programs in the U.S. to use for first year graduate algebra courses and for qualifying exams at PHD programs in algebra. Of course, at such programs, the line between graduate and undergraduate coursework is somewhat ambiguous. But I think most mathematicians would agree with me on this assessment.

Algebra, 3rd edition by Serge Lang:

Ok, let’s get the elephant in the room out of the way first.

Lang is a good example of the kind of strange “canonization’ of textbooks in academia which I’ve mentioned before at this blog and other places. It’s funny how some mathematicians-particularly algebracists at the more prestigious programs- that get very self-righteous and uppity when you question whether or not Lang should be used as a first-year graduate text anymore with all the new choices. I can’t help but use some of the remarks of a frequent poster at Math Overflow in this regards.
Let’s call him Mr. G.
Mr. G is a talented undergraduate at one of the more prominent universities to study mathematics in the Midwestern United States. Like the author of this blog, he also has been occasionally slammed for shooting off his big mouth on MO by the moderators.
He and I have had several heated exchanged about his Bourbaki-worship: G believes that the Bourbaki texts are sacred tomes that are the only “real” texts for mathematicians and applications are for nonmathematicans. But I’ll let his own words state his position far better then I can. Here is a recent exchange between Mr. G and 2 mathematicians who are frequent posters at MO: let’s call them Dr. H and Dr. L. This was a question regarding the presentation of graduate algebra. (I obviously can’t be more specific then that-to do so would identify the participants.)

@Dr. H: The first graduate algebra course is often going to be the student's first introduction to algebra. It's supposed to be abstract and intense! If you muddy the waters with applications, your students will never get to that level of Zen you achieve after stumbling around in an algebra course. It's like point-set topology, except the rabbit-hole called algebra goes much deeper and is much more important. –Mr.G
@Mr.G: After my first algebra course I still didn't understand why I should actually much about Galois theory from a practical point of view until I saw $GF(2^n)$ in all sorts of applications. My experience has been that most students--even graduate students studying algebra--are not going to be interested in abstraction for its own sake. Mechanics can help to motivate calculus. The same can be true of information theory and algebra. –Dr.H
@Dr.H: Graduate math students shouldn't be taught things "from a practical point of view". This isn't a gen. ed. class, and the abstract perspective one gains by really engaging algebra "as it is practiced) is completely worth the "journey in the desert", as it were. This is the "Zen" I was talking about. Also, I think that characterizing algebra as "abstraction for abstraction's sake" is really missing the point tremendously. – Mr.G.
@Mr.G. The journey through algebra does not necessarily have to go through the desert, nor is that necessarily the best or most ideal path. It might be so for you, but it is certainly not the best path for everybody. There are numerous other paths to take, most of which can lead and have lead people to mathematical understanding and success. Please take a moment to consider, for instance, Richard Borcherds' recent algebraic geometry examples post. – Dr. L.

Speaking for myself, I firmly believe in heeding Lebesgue’s warning about the state of the art in mathematics: “ Reduced to general theories, mathematics would become a beautiful form without content: It would quickly die.” Generality in mathematics is certainly important, but it can and often is, overly done. But I digress. My point is that Mr.G’s attitude is typical of the Lang-worshipper: That if you can’t deal with Lang, you’re not good enough to be a graduate student in mathematics. Or to use Mr.G’s own words on another thread on the teaching of graduate algebra: ”Lang or bust.” Many feel the “journey in the desert” of Lang is a rite of passage for graduate students, much as Walter Rudin’s Principles of Mathematical Analysis is for undergraduates.
Well, there’s no denying Lang’s book is one of a kind and it’s very good in many respects. People ask me a lot how I feel about Lang’s remarkable career as a textbook author. It’s important to note I never met the man, sadly-and everything I know about him is second hand.
Reading Lang’s books brings to mind a quote from Paul Halmos in his classic autobiography I Want To Be A Mathematician .which was said in reference to another famous Hungarian mathematician, Paul Erdos: “I don’t like the kind of arithmetic-geometric-combinatorial problems Erdos likes, but he’s so good at them, you can’t help but be impressed. “
I don’t like the kind of ultra-abstract, application-devoid, Bourbakian, minimalist presentations Lang was famous for-but he was SO good at writing them, you can’t help but be impressed.
And Algebra is his tour de force.
The sheer scope of the book is stunning. The book more or less covers everything that’s covered in the later editions of van der Waerden-all from a completely categorical, commutative diagram with functors point of view. There’s also a generous helping of algebraic number theory and algebraic geometry from this point of view as well. His proofs are incredibly concise and with zero fat, but quite clear if you take the effort to follow them and fill in the blanks. The chapters on groups and fields are particularly good. Lang also is an amazingly thorough and responsible scholar; each chapter is brimming with references to original proofs and their source papers. This is a book by one of the giants in the field and it’s clearly a field he had enough respect for to know his way around the literature remarkably well-and he believed in giving credit where credit is due. Quite a few results and proofs-such as localization of rings, applications of representation theory to functional analysis and the homology of derivations, simply don’t appear in other texts. The last point is one I think Lang doesn’t get a lot of credit for, without which the book would be all but unreadable: He gives many, many examples for each concept-many nonstandard and very difficult to ferret out of the literature. Frankly, the book would be worth having just for this reason alone.

So fine, why not go with the party line then of “Lang or bust”?

Because the book is absurdly difficult, that’s why.

First of all, it’s ridiculously terse. It takes 2-3 pages of scrap paper sometimes to fill in the details in Lang’s proofs. Imagine doing that for OVER 911 PAGES. And worse, the terseness increases as one progresses in the book. For the easier topics, like basic group theory and Galois theory, it’s not so bad. But the final sections on homological algebra and free resolutions are almost unreadable. You actually get exhausted working through them.

AND WE HAVEN’T EVEN TALKED ABOUT THE EXERCISES YET.

“Yeah, I’ve heard the horror stories about Lang’s exercises in the grad algebra book. C’mon, they’re not THAT bad are they?”

You’re right, they’re not.

They’re WORSE.

I mean, it’s just ludicrous how hard some of these exercises are.

I’ll just describe 2 of the more ridiculous exercises and it pretty much will give you an idea what I’m talking about. Exercise 30 on page 256 asks for the solution of an unsolved conjecture by Emil Artin. That’s right, you read correctly. Lang puts in parentheses before it: “The solution to the following exercise is not known.” No shit? And you expect first year graduate students-even at Yale-to have a chance? I’m sorry, that’s not a reasonable thing to do!

Then there’s the famous-or more accurately, infamous-exercise in the chapter on homological algebra: “Take any textbook on homological algebra and try and prove all the results without looking up the proofs.”

I know in principle, that’s what we’re all supposed to do with any mathematical subject we’re learning. But HOMOLOGICAL ALGEBRA?!?

(An aside: I actually had a rather spirited discussion via email with Joseph Rotman, Professor Emeritus of the University of Illinois at Urbana-Champaign, over this matter-a guy that knows a thing or 3 about algebra. Rotman felt I was too hard on Lang for assigning this problem. He thought Lang was trying to make a point with the exercise, namely that homological algebra just looks harder then any other subject, it really isn’t. Well, firstly, that’s a debatable point Lang was trying to make if so. Secondly, I seriously doubt any graduate student who’s given this as part of his or her final grade is going to be as understanding as Rotman was. Actually, it’s kind of ironic Rotman thinks that since I know many a graduate student who would have failed the homology part of their Lang-based algebra course without Rotman’s book on the subject! )

My point is I don’t care how good your students are, it’s educational malpractice to assign problems like that for mandatory credit. And even if you don’t and just leave them as challenges for the best students-isn’t that rubbing salt in the wounds inflicted by this already Draconian textbook?

These exercises are why so many mathematicians have bitter memories of Lang from their student days.

A lot of you may be whining now that I just don’t like hard books. That’s just not true. Herstien is plenty difficult for any student and it’s one of my favorites. In fact, quite the contrary. You’re really supposed to labor over good mathematics texts anyway-math isn’t supposed to be EASY. An easy math textbook is like a workout where you’re not even winded at the end-it’s doubtful you’re going to get any benefits from it.

But Lang isn’t just hard; I don’t just mean students have to labor over the sections before getting them.

The average graduate student learning algebra from Lang’s book is like a fat guy trying to get in shape by undergoing a 3 month U.S. Marine Corp boot camp and having a steady diet of nothing but vitamins, rice cakes and water. Assuming he doesn’t drop dead of a heart attack halfway through, such a regimen will certainly have the desired effect-but it will be inhumanely arduous and unpleasant.

And there are far less Draconian methods of obtaining the same results.

So unless one is a masochist, why in God’s name would you use Lang for a first year graduate course in algebra?

Is it a TERRIBLE book? No-as I said above, it has many good qualities and sections. As a reference for all the algebra one will need in graduate school unless becoming an algebraicist, the book is second to none.

Would I use it as a text for a first year graduate course or qualifying exam in algebra?

HELL NO.

Algebra by Thomas Hungerford : This has become a favorite of a lot of graduate students for their algebra courses and it’s pretty easy to see why-at least at first glance. It’s nearly as demanding as Lang-but it’s much shorter and more selective, has a lot more examples of elementary difficulty and the exercises are tough but manageable.
The main problem with this book occurs in the chapter on rings and it boils down to a simple choice. Hungerford-for some strange reason-decides to define rings without a multiplicative identity.

I know in some ring theoretic cases, this is quite useful. But for most of the important results in basic ring and module theory, this results in proofs that are much more complicated since this condition needs to be “compensated” for by considering left and right R-modules as separate cases. Hungerford could alleviate this considerably by giving complete, if concise, proofs as Lang does in most cases.

But he doesn’t. He only sketches proofs in more then half the cases.

The result is that every section on rings and modules is very confusing. In particular, the parts on modules over commutative rings and homological algebra-which I really need for my upcoming exam-are all over the place.

Still, the book has a lot of really nicely presented material from a totally modern, categorical point of view. The first chapter on category theory is probably the best short introduction there is in the textbook literature and the section on group theory is very nice indeed.

Basic Algebra, 2nd edition by Nathan Jacobson, volumes I and II: I say we should nominate Dover Books for a Nobel Peace Prize for their recent reissue of this classic. The late Nathan Jacobson, of course, was one of the giants of non-commutative ring theory in the 20th century.

He was also a remarkable teacher with an awesome record of producing PHDs at Yale, including Charles Curtis, Kevin Mc Crimmon, Louis H.Rowen, George Seligman, David Saltman and Jerome Katz. His lectures at Yale on abstract algebra were world famous and had 2 incarnations in book form: The first, the 3 volume Lectures In Abstract Algebra, was for a generation the main competition for van der Waerden as the text for graduate algebra courses. Basic Algebra is the second major incarnation- the first edition came out in the 1970’s and was intended as an upgraded course in algebra for the extremely strong mathematics students entering Yale from high school during the Space Age. The first volume-covering classical topics like groups, rings, modules, fields and geometric constructions-was intended as a challenging undergraduate course for such students. The second volume-covering an overview of categorical and homological algebra as well as the state-of-the-art (circa 1985) of non-commutative ring theory-was intended as a graduate course for first year students. The complete collapse of the American educational system in the 1990’s has rendered both volumes useless as anything but graduate algebra texts. Still, given that the second volume was going for nearly 400 dollars at one point online in good condition, it’s reissue by Dover in wonderfully cheap editions is a serious cause for celebration.

Both books are beautifully and authoritatively written with a lot of material that isn’t easily found in other sources, such as sections on non-associative rings , Jordan and Lie algebras, metric vector spaces and an integrated introduction to both universal algebra and category theory. They are rather sparse in examples compared with other books, but the examples they DO have are very well chosen and described thoroughly. There are also many fascinating, detailed historical notes introducing each chapter, particularly in the first volume.

The main problem with both books is that Jacobson’s program here absolutely splits in half algebra into undergraduate and graduate level topics; i.e. without and with categorical and homological structures. This leads to several topics being presented in a somewhat disjointed and inefficient manner because Jacobson refuses to combine them in a modern presentation-module theory in particular suffers from this organization. Personally, I didn’t find it THAT big an issue with a little effort-but a lot of other students have complained about it. Also, some of the exercises are quite difficult, rivaling Lang’s. Even so, the sheer richness of these books make them true classics. If graduate students are willing to work a little to unify the various pieces of the vast puzzle that Jacobson presents here with astonishing clarity, he or she will be greatly rewarded by a master’s presentation and depth of understanding.

The New Kids On The Block: As I said earlier, the adoption of Lang worldwide as the canonical graduate algebra text had a backlash effect that’s been felt with a slew of new graduate texts. I haven’t seen them all, but I’ve seen quite a few. Here’s my commentary on the ones I’m most familiar with.

Basic Algebra/Advanced Algebra by Anthony W. Knapp: This is probably the single most complete reference for abstract algebra that currently exists. It is also paradoxically, the single most beautiful, comprehensive textbook on it. Knapp taught both undergraduate and graduate algebra at SUNY Stonybrook for nearly 3 decades-and these volumes are the finished product of the tons of lecture notes that resulted. The purpose of these books, according to Knapp, is to provide the basis for all the algebra a mathematician needs to know to be able to attend a conference on algebra and understand it. If so, he’s succeeded beyond all expectations. The main themes of both books are group theory and linear algebra (construed generally i.e. module theory and tensor algebra) . The first volume corresponds roughly to what could possibly be covered at the undergraduate level from reviews basic number theory and linear algebra up to an honors undergraduate course in abstract algebra (groups, rings, fields, Galois theory, multilinear algebra, module theory over commutative rings). The second corresponds to graduate syllabus focusing on topics in noncom mutative rings, algebraic number theory and algebraic geometry( adeles and ideles, homological algebra, Wedderburn-Artin ring theory, schemes and varieties, Grobner bases, etc.) This is the dream of what an advanced textbook should be-beautifully written, completely modern and loaded with both examples and challenging exercises that are both creative and not too difficult. In fact, the exercises are really extensions of the text where many topics and applications are in fact derived-such as Jordan algebras, Fourier analysis and Haar groups, Grothendieck groups and schemes, computer algebra and much, much more. The group actions on sets are stressed throughout. Also, categorical arguments are given implicitly before categories are explicitly covered by giving many commutative diagram arguments as universal properties. (This avoids the trap Jacobson fell into.) Best of all-there are hints and solutions to ALL the exercises in the back of each volume. I would LOVE to use this set to teach algebra one day-either as the main texts, as supplements or just references-but if you enjoy algebra, you HAVE to have a copy. Hopefully, there will be many editions to come.

Abstract Algebra, 2nd edition by Pierre Grillet: The first edition of this book was simply called “Algebra” and it came out in 1999. To me, this book is what Lang should be. Grillet is an algebriacist and award-winning teacher at Tulane University. Interestingly, he apparently carried out the revision in the aftermath of Katrina. The book covers all the standard and more modern topics in a concise, very modern manner-much like Lang. Unlike Lang, though, Grillet is extremely readable, selective in his content and highly structured with many digressions and historical notes. The sheer depth of the book is amazing. Unlike Lang, which focuses entirely on what-or Hungerford, which explains a great deal but also is very terse-Grillet focuses mainly on why things are defined this way in algebra and how the myriad results are interconnected. It also has the best one chapter introduction to category theory and universal algebra I’ve ever seen-and it occurs in Chapter 17 after the previous 16 chapters where commutative diagrams are constructed on virtually every other page. So by the time the student gets to category theory, he or she has already worked a great deal with the concepts implicitly in the previous chapters. This is very typical of the presentation. Also, Grillet doesn’t overload the book with certain topics and give the short shriff to others-many texts are half group theory and half everything else, for instance. Grillet gives relatively short chapters on very specific topics-which makes the book very easy to absorb. The exercises run the gamut from routine calculations to proofs of major theorems. The resulting text is a clinic in how to write “Bourbaki” style texts and it would be a great alternative to either Lang or Hungerford.

Algebra: A Graduate Course by I.Martin Issacs: This is a strange book. After being out of print for over a decade, it was recently reissued by the AMS. Isaacs claims this course was inspired by his teacher at Harvard, Lynn Loomis, whose first-year graduate course Issacs took in 1960 there. Like Loomis’ course, Issacs emphasizes noncom mutative aspects first, focusing mainly on group theory. He then goes on to commutative theory-discussing ring and ideal theory, Galois theory and cyclotomy. The book is one of the most beautifully written texts I’ve ever seen, with most theorems proved and most example constructions left as exercises. Unfortunately, Issacs’ material choice seems to follow his memories of his graduate course in 1960 far too closely-this choice of topics would be a first year graduate course at a top university ONLY before the 1960’s. Issacs omits completely multilinear algebra, category theory and homological algebra. How can you call such a book in 2010 a graduate course? That being said-it is wonderfully written and if supplemented by a text on homological algebra, it could certainly serve as half of such a course-either Osborne or Joseph Rotman’s books on the subject fill in the omissions very nicely.

And now-my very favorite algebra book of all time. Drum roll,pleeeeeeeeease................

Advanced Modern Algebra by Joseph J. Rotman: I haven’t seen the second edition, but I’m very familiar with the first. Rotman may be the best writer of algebra textbooks alive. Hell, he may be the best writer of university-level mathematics textbooks PERIOD. Serious. So when his graduate textbook came out, I begged, borrowed and cajoled until I could buy it. And it was one of the best textbooks I ever bought. The contents of the book are, as the AMS’ blurb discusses:

“This book is designed as a text for the first year of graduate algebra, but it can also serve as a reference since it contains more advanced topics as well. This second edition has a different organization than the first. It begins with a discussion of the cubic and quartic equations, which leads into permutations, group theory, and Galois theory (for finite extensions; infinite Galois theory is discussed later in the book). The study of groups continues with finite abelian groups (finitely generated groups are discussed later, in the context of module theory), Sylow theorems, simplicity of projective unimodular groups, free groups and presentations, and the Nielsen-Schreier theorem (subgroups of free groups are free). The study of commutative rings continues with prime and maximal ideals, unique factorization, noetherian rings, Zorn's lemma and applications, varieties, and Grobner bases. Next, noncommutative rings and modules are discussed, treating tensor product, projective, injective, and flat modules, categories, functors, and natural transformations, categorical constructions (including direct and inverse limits), and adjoint functors. Then follow group representations: Wedderburn-Artin theorems, character theory, theorems of Burnside and Frobenius, division rings, Brauer groups, and abelian categories. Advanced linear algebra treats canonical forms for matrices and the structure of modules over PIDs, followed by multilinear algebra. Homology is introduced, first for simplicial complexes, then as derived functors, with applications to Ext, Tor, and cohomology of groups, crossed products, and an introduction to algebraic K-theory. Finally, the author treats localization, Dedekind rings and algebraic number theory, and homological dimensions. The book ends with the proof that regular local rings have unique factorization.”

That says what’s in the book. What it doesn’t tell you is what makes this incredible book so special and why it deserves a second edition so quickly with the AMS: Rotman’s gifted style as a teacher, lecturer and writer. The book is completely modern, amazingly thorough and contains discussions of deep algebraic matters completely unmatched in clarity. As proof, read the following excerpt from the first edition, how Rotman explains the basic idea of category theory and it’s importance in algebra:

Imagine a set theory whose primitive terms, instead of set and element, are set and function.
How could we define bijection, cartesian product, union, and intersection? Category theory
will force us to think in this way. Now categories are the context for discussing general
properties of systems such as groups, rings, vector spaces, modules, sets, and topological
spaces, in tandem with their respective transformations: homomorphisms, functions, and
continuous maps. There are two basic reasons for studying categories: The first is that they
are needed to define functors and natural transformations (which we will do in the next
sections); the other is that categories will force us to regard a module, for example, not in
isolation, but in a context serving to relate it to all other modules (for example, we will
define certain modules as solutions to universal mapping problems).

I dare you to find a description of category theory that would serve a novice better. The book is filled with passages like that-as well as hundreds of commutative diagrams, examples, calculations and proofs of astounding completeness and clarity. Rotman presents algebra as a huge, beautiful puzzle of interlocking pieces-one he knows as well as anyone in the field. The one minor complaint is the book’s exercises-they’re a little soft compared to the ones in Hungerford or Lang. And the sheer size of the book-1008 HARDBACK BOUND PAGES!- is a bit daunting. ( Rotman joked with me via email that more then a few times, he mistakenly carried it to his calculus class and had to go back to his office to switch books.) But these are very minor quibbles in a book destined to become a classic. If I had to choose one textbook for graduate algebra and it’s qualifier and couldn’t pick any others-THIS is the one I’d pick, hands down.
Word from the AMS and those who have seen it that the second edition is even better-the index has been greatly improved and entire sections have been rewritten to emphasize noncom mutative algebra-which is appropriate for a graduate course.

I suggest you all place your orders now. You’ll thank me later, I promise.

I now return you to your regularly scheduled lives.

Thank you for your attention.

Peace.

Student Of Fortune

15
Nov 10

"Monoids,R-Modules And Nonassociative Rings-These Are Some of My Favorite Things: A Suggested Reading List For Graduate Algebra"


Yello.

Brief political comment first.

November 2nd, 2010 may go down as one of the blackest in American history in coming decades and centuries. People who have been predicting a corporate takeover leading to a plutocracy in America -a “third world” complete with utopian gated communities defended by mercenary private security interspersed with regions of abject poverty where starving mobs of disease riddled peasants try and stay alive into their 30’s if possible-may have seen the first step towards that future on that day.


Not that our precious President and his party have been anything even remotely resembling heroic figures-between half-ass health care reform, continuing to allow the slaughter of an entire generation of young Americans, as well the slow bankruptcy of out nation to satisfy whatever mysterious powers now profit from it-they've been as cowardly and subservient to the special interests as the other side on most days. In many ways, that’s partly what led to his party’s downfall.


But with the election shenanigans with The Chamber Of Commerce-after the worst Supreme Court decision for the country at large since Plessy Vs. Ferguson - effectively rendered the Republican party a wholly owned subsidiary of the corporations. Unless a dramatic opposition to this systematic subversion of the government occurs in the next 2 years, the very best we have to look forward to from this decision is eternal gridlock until the Commerce can buy the Presidency in 2012 and put the best candidate money can buy in there.

The poor ignorant slobs who marched for the Tea Party are in for a very rude awakening indeed-that is, if any of them have the intelligence to realize they’ve been used and discarded.

And then the future I predicted in my opening preamble-coupled with environmental collapse with only the wealthy having food and water fit for human consumption, let alone other resources-will come to pass within a generation.

That’s all I wanted to say on this for now. There’ll be much more to come when I can steal myself to a full analysis and discussion.

But now onto a promised, much more academic matter.

For those who don’t know, I’m sort of an unofficial bibliophile for mathematical education. I inherited this love of textbooks and monographs from my inspiration, friend and unofficial mentor, Nick Metas. I was 18 years old when out of simple curiosity I called him in his office to ask him for direction in independent studies of mathematics beyond calculus-and he went on for 4 hours, naming just about every textbook and describing the subject of mathematics.

That long-ago conversation is what started me on the path to becoming a mathematician.

Nowadays, the influence of Nick is very clear in my life: I have an extensive library of textbooks and monographs, people ask me all the time for references on subjects and I review books for the Mathematical Association Of America’s website. I have an opinion on most commonly used texts and monographs for all subjects-and I’m reading more every year. In fact, I have a private dream of beginning my own small publishing company someday.


(Of course, that’ll depend on the prediction above being dead wrong. We can all hope the country comes to it’s senses before its too late. They’re angry-that’s a good beginning Now they just have to develop enough intelligence to channel the anger constructively. Hope it happens in time. So far it doesn‘t look promising……..)

I’ve been asked many times to compose a master list of my favorite textbooks and/or monographs. The list will probably undergo many revisions and additions before it reaches final form-but more importantly, I’ve decided to compose it in modular form i.e in components. This way, it’s broken into bite-sized components of manageable length that I can post here.

It seems to me if I wait and try to compose it all at once-well, I’ll end up writing a 2,500 page book from the old age home I’ll be dying of cancer in. So let’s get started and hope that what little insights I can give can help neophyte students looking to broaden their knowledge base in subfields of math or are just looking for a little help in coursework they’re struggling in. Comments, input and suggestions are, of course, very welcome.

The first module here is my favorite subject in all of mathematics: algebra. (A ludicrous but sadly mandatory clarification: When a mathematics student or mathematician says ‘algebra’; it’s supposed to be understood he or she means abstract algebra. High school algebra is, of course, the simplest special case of this wondrous arena. )

How do we define abstract algebra? Like most branches of modern mathematics, attempting a simple nonmathematical definition for non-mathematicians is a nearly impossible Catch-22 since it requires mathematical concepts to even attempt a meaningful definition. Entire philosophical treatises could probably be written attempting to answer the question and would probably fail. But I think we can try for a reasonable working definition here.

I think the best way to define algebra is that it is the general study of structures in mathematics. By a structure, we mean some kind of set -by which we mean naively a collection of objects-and a function f closed on S (the range of f is a subset of S) with a specified list of properties that characterizes that structure. For example, a group is a nonempty set S with a binary operation f such that f is associative, there is a unique element e in S such that for all elements a in S, f(e,a) = f(a,e)= a and for every a in S, there’s a unique a* such that f(a, a*)= f(a*,a)=e. Algebra deals specifically with these kinds of objects.

The pervasiveness of algebra in modern mathematics in the 21st century is astonishing. It’s more then the sheer scope of algebra itself, but the fact that most of the active areas of mathematics would not even exist without it.


And I’m not talking about high-tech fields where algebra’s role is obvious-like deformation theory and higher category theory. I’m referring to the fact that most areas of mathematics are formulated in the 21st century in terms of algebraic structures. To give just one possible example of a legion, modern differential geometry would be unthinkable without the language of vector spaces and R-modules. Without tangent spaces and their associated local isomorphisms, it would be impossible to generalize calculus beyond Euclidean space. It would also be impossible to precisely define differential forms, without which most of the most interesting developments of manifold theory fall to dust. As a result, a student that’s weak in algebra needs to seriously reassess a career in mathematics.

So the least I can do is give my 2 cents on the current crop of books available.

The actual direct impetus for me writing up and posting this list was Melvyn Nathanson at the City University Of New York Graduate Center. This semester, the eminent number theorist is teaching the first semester of the year-long graduate algebra sequence there. I began this semester sitting in on his lectures in order to begin preparations for the algebra half of my oral qualifiers for the Master’s Degree in pure mathematics at Queens College. Unfortunately, a combination of personal and financial issues prevented me from attending regularly. So that was the end of that.

I found Dr. Nathanson’s (he hasn’t told me it’s ok to call him Melvyn yet, so I’m going to be extra cautious as not to offend him) comments on the subject very interesting, as he has his own unique take on just about any subject. As proof, I offer this excerpt from the course’s syllabus:


In 1931, B. L. van der Waerden published the first edition of Moderne Algebra,two classic volumes, written in German, that were based in part on lectures by
Emil Artin and Emmy Noether and that became the canonical work in abstract algebra." The second edition appeared in 1937, and an English version, Modern Algebra, translated by Fred Blum and Theodore J. Benac, was published in the United States in 1949 and 1950. I and many other American mathematicians learned algebra from the original English edition of van der Waerden. It is still a great work and I strongly recommend it for intensive study. The first volume of the seventh German edition of van der Waerden is also available in English translation, but I prefer the original. Van der Waerden's algebra begins with introductions to different algebraic structures. The first seven chapters are “Numbers and Sets," “Groups," “Rings and Fields, "Polyn-omials"“Theory of Fields," Continuation of Group Theory," and The Galois Theory." As proof of van der Waerden's influence, this continues to be the starting sequence of topics in most algebra courses and most algebra books, including the contemporary classic, Serge Lang's Algebra, which I also recommend. This course is different, not just in the sequence of topics, but in its philosophy. It emphasizes themes in algebra: Divisibility, dimension, decomposition, and duality,
and the course enables algebraic understanding and technique by developing these themes. The book includes all of the theorems expected in a graduate algebra course, but in a nontraditional order. The book also includes some important topics that do not appear in van der Waerden or Lang.”


The affirmed originality of the course, I don’t doubt. I’m still hoping to obtain a complete set of handwritten notes from some of my friends in the course, which is the official text for the course. The clear implication from the preface and his subsequent remarks is that Professor Nathanson hopes to eventually expand these notes into a textbook for a graduate algebra course.

Privately, I’m hoping to work with Dr. Nathanson as a PHD student eventually and if he follows through on this, perhaps I can be involved in the book’s drafting process. But that’s for the future.

His comments got me thinking about the current state of algebra courses and the textbooks that form the basis of them. Nathanson’s experiences are not unlike those of most mathematicians of his generation: van der Waerden’s classic was the source from which he learned his algebra. Later mathematicians; particularly algebracists-such as my undergraduate algebra teacher, Kenneth Kramer-learned algebra from the earlier editions of Lang’s tome. (In fact, it was more personal for Kramer. As an honors undergraduate at Columbia in the late 1960’s, he was a student in the graduate algebra course taught by Lang himself-whose resulting lecture notes ultimately evolved into the classic text.) Most of the better universities’ graduate programs adopted Lang as the gold standard of first year graduate algebra, for better or worse, after the 1960’s. With a very few exceptions, this was the story until after the turn of the 21st century, when a host of graduate algebra texts came onto the market within a 5 year period. What was once a very sparse set of choices for this course is now a wide field of markedly diverse texts, many authored by very eminent mathematicians.

What follows is my attempt to form an amateur’s guide to these texts and my corresponding brief commentary to each. As a reviewer of textbooks, it seemed under the circumstances, that providing such a list to my erstwhile classmates in Nathanson’s course-as well as the mathematical world in general-would be a very positive undertaking. I don’t know if it would be WISE, merely positive. I must add the disclaimer that I am by no means an expert; I’m merely a serious graduate student. Therefore, this reading list must be taken with a salt lick of caution as coming from an amateur and as such, it is seriously subject to revision as my knowledge grows and my mathematical style tastes change.

A major motivation in the evaluation of each of these books has been student-friendliness. Let me clarify greatly what I mean by that. A lot of top-notch mathematicians and students have an elitist, almost snobbish reaction to a textbook when you say its’ friendly. “Oh,you mean it spoon feeds the material to the brainless monkeys that pass for mathematics majors at your pathetic university? How amusing. Here at Superior U, we use only the authentic mathematics texts. Rudin.Artin Hoffman and Kunze. Alfhors. We propagate the True Word. Math is SUPPOSED to a struggle for those truly gifted enough to be worthy of it. “

Or something equally pretentiously narcissistic.


I have a LOT to say on this and related issues-but if I started going in depth about it here, I’d write an online book here. In future installments, I’ll begin to outline them in detail.
But in plain English, this is a bunch of crap.


The reason a lot of those “classic” texts are difficult to read isn’t because their authors were first-rate mathematicians and as such, their lessons are beyond the reach of mere mortals. In a lot of cases, it was simply because most of them never really thought about teaching; of being able to organize their deep understanding of their chosen fields -and as a result, they were very poor communicators.

This lack of communication skill is reflected not only in their poor reputations as teachers, so often inversely proportional to their reps as researchers-but also in the resulting textbooks. Why don’t they? Well, again, it’s too complicated to fully go into here. But I will say that part of the reason, as any research mathematician of any prominence will tell you-is that they don’t get paid the big bucks and get the fancy titles based on how well students learn from them.

The sad part is that this myth has been perpetuated by the canonization of certain textbooks as The Books for certain classes, despite the fact that most students almost overwhelmingly despise them. And the reason why is simple: They just aren’t clear and well-organized. That makes the very act of reading them unpleasant, let alone actually learning from them. For the serious mathematics major or graduate student, this makes studying from such books virtually an act of psychic self mutilation.

To the elitists, I only have the following to say: Charles

Chapman Pugh's Real Mathematical Analysis
, Joseph

Rotman's An Introduction to Algebraic Topology
, anything by John Milnor, J.P.Serre or Jurgen Jost, Loring Tu's

An Introduction to Manifolds
, John

McCleary's A First Course in Topology: Continuity and Dimension
, George

F.Simmons' Differential Equations with Applications and Historical Notes,
Charles

Curtis' Linear Algebra: An Introductory Approach
andJohn and

Barbara Hubbard's Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach


I challenge them to consider any of these wonderful books to be spoon feeding students-and yet, they are eminently readable and wonderfully written books. In short, they are books students enjoy reading and therefore will not only learn from them-but will want to learn from them.

But an interesting trend has resulted from this myth. The students who are talented enough to learn from these texts who go on their careers to become mathematicians- and who care enough about teaching- recall their experiences as students. They don’t want to subject their students-or anyone's students-to the same torture. As a result, they try and write alternative books for students that do what they wish those texts had. The Computer Age has magnified this effect hundredfold as such books have become ridiculously easy to produce. As a result, we’ve gotten “backlash waves” of texts as alternatives to those classic tomes that created the large diversity of texts that currently exist in the various subfields of advanced mathematics. Where once there was a bare handful of such texts to choose from, a generation later, the “backlash” creates a myriad of them.

Some examples in the recent generations of math students will illustrate. Once, Alfhors’ ridiculously difficult Complex Analysis was the standard text in functions of a complex variable at U.S. graduate programs after the early 1960’s. There were a few alternatives available in English-such as Hille's treatise or Carathedory's-but not a lot. This lead to an explosion of complex analysis texts in the 1970’s onward: Saks/Zygmund, Rudin, Bak/Newman, Conway, Heins, Greene/ Krantz, Jones/Singerman, Gamelin,- well, that list goes on and on. A similar backlash occurred in the 1960’s and 1970’s in general topology after an entire generation had suffered through John Kelley’s General Topology wrote a legion of such texts, including the classics by Willard and Munkres.

This effect has further been enhanced by progress in those fields at the research level-which results in the presentations of the standard texts of a generation becoming outmoded. The result is the “backlash” presentations can also be “upgraded” to current language. A good example is the incorporation of category theory into advanced algebra texts post-1950’s.

I strongly believe the current large crop of graduate algebra texts is the result of a similar backlash against Lang.

I’ve gone on to some length about this because I think it’s important to keep these 2 ideas in mind- the elitist conception of Great Books and the backlash against it-when considering my readability criteria for judging such texts. So without further ado, my reading list. Enjoy.

And remember-comments and suggestions are not only welcomed, but encouraged.


Part I- Graduate Warmup: These are texts that are a little too difficult for the average undergraduate in mathematics, but aren’t quite comprehensive or rigorous enough for a strong graduate course. Of course, a lot of this is totally subjective. But it’ll make good suggestions for those struggling in graduate algebra because their backgrounds weren’t quite as strong as they thought.




Topics in Algebra, 2nd Edition by I.R.Herstein
This is the book I first learned algebra from under the sure hand of Kenneth Kramer at Queens College in his Math 337 honors course. It’s also the book that made me fall in love with the subject. Herstein’s style is concise yet awesomely clear at every step. His problem sets are legendarily difficult yet doable (mostly). If anyone asks me if they’re ready to take their algebra qualifier and how to prepare-I give them very simple advice: Get this book. If you can do 95 percent of the exercises, you’re ready for anything they throw at you. They’re that good. Warning: In true old European algebracist fashion, Herstein writes his functions in the very un-Calculus like manner on the right in composition i.e. fg= gof. This confused the author of this blog initially and no one corrected him until several weeks into the course-which lead to difficulties later on. A couple of quibbles with it-the field theory chapter is really lacking. Also. Herstein tends to present even the examples-which are considerable-in their fullest generality. This makes the book harder for the beginner then it really needs to be. For example, he gives the dihedral group of rigid motions in the plane for the general n gon where n is an integer. he could start with the n=4 case and write out the full 8 member group table for the motions of the quadrilateral and THEN generalize. Still-I fell in love with this book. The presentation is considered outdated by most mathematicians now, who prefer the more geometric approach of Artin. Still, the book will always have a special place in my heart and I recommend it wholeheartedly for the talented beginner.




Algebra 2nd Edition by Michael Artin The second edition of this book finally came out in September. For awhile, it looked like it might emerge posthumously-it was so long in gestation. But fortunately, this wasn’t the case. I haven’t read it carefully yet, but from what I’ve seen of it, it looks very similar to the first edition. As for the first edition- well, I got really mixed feelings about it. Artin’s book has many, many good qualities. It’s primary positive qualities are the heavily geometric bent and high level of presentation. The shift in emphasis from the permutation groups to matrix groups is an extremely smart one by Artin since it gives one a tool of much greater generality and simplicity while still preserving all the important properties of finite groups. (Indeed, permutations are usually explicitly represented as 2 x n matrices with integer valued bases-so the result is just a slight generalization. ) This also allows Artin to unify many different applications of algebraic structures to many different areas of mathematics-from classical geometry to Lie groups to basic topology and even some algebraic geometry (!) All through it, Artin brings an infectious love for algebra that comes through very sharply in his writing. So why the hesitant recommendation?
Because it’s easy to like the book when you’ve already learned the material from other sources.
Would even a talented beginner find the book so appealing? I don’t think so. Firstly, Artin assumes an awful lot of background in his prospective students-primarily linear algebra and basic Euclidean geometry. It might have been reasonable to assume this much background in the superhuman undergraduates at MIT in the early 1990’s, but I think that’s a stretch for most other students-even honors students. Especially nowadays. Secondly, the book is organized in a very idiosyncratic fashion that doesn’t always make sense even to people who know algebra. Nearly half the book is spent on linear algebra and group theory and rings, modules, fields are developed in a very rushed fashion. Some of these sections really needed more fleshing out. Also, a lot of the group theory chapters are confusing. His discussion of both cosets and tilings in the plane are particularly discombobulated. Lastly-his choice of topics for even good undergraduates is bizarre sometimes. He writes a chapter on group representations, but leaves tensor algebra and dual spaces “on the cutting room floor”? It’s a very strange choice. The book’s main flaw is there are too few exercises and most of these are ridiculously difficult. (In all fairness, I understand this was the main problem Artin is trying to rectify in the new edition. ) That being said, for all it’s flaws, a text of this level of daring by an expert of Artin’s stature is not to be ignored. I wouldn’t use it by itself, but I’d definitely keep a copy on my desk. Apparently,though, in the 2 decades since this book was written, Artin has rethought the course and it’s structure-it remains to be seen if the new version has gotten most of the bugs out. If it has, the book will be a must-have, hands down, for students of algebra.



A Course in Algebra by E.B.Vinberg This is very rapidly becoming my favorite reference for algebra. Translated from the Russian by Alexander Retakh, this book by one of the world’s preeminent algebracists is one of the best written, most comprehensive sources for undergraduate/graduate algebra that currently exists. Vinberg, like Artin, takes a very geometric approach to algebra and emphasizes the connections between it and other areas of mathematics. But Vinberg‘s book begins at a much more elementary level and gradually builds to a very high level indeed. It also eventually considers many topics not covered in Artin-including applications to physics such as the crystallographic groups and the role of Lie groups in differential geometry and mechanics! The most amazing thing about this book is how it manages to teach students such an enormous amount of algebra-from basic polynomial and linear algebra through Galois theory, multilinear algebra and concluding with the elements of representation theory and Lie groups, with an enormous number of examples and exercises that cannot be readily found in most other sources. All of it is done incredibly gently despite the steadily increasing sophistication of the material. The book has a very “Russian” style-by which I mean the author does not hesitate to both prove theorems and give applications to both geometry and physics (!) throughout. Those who know me personally know this is a position I am very sympathetic to-and for there to be a major recent abstract algebra text that takes this tack is very exciting to me. For anyone interested in writing a textbook on advanced mathematics, this is a terrific book to study for style. It is one of the most readable texts I have ever read. An absolutely first rate work that needs to be owned by any student learning algebra and any professor considering teaching it.








Abstract Algebra, 3rd Edition by David S Dummit and Richard M. Foote Ever seen a movie or read a book where based on your tastes, everything you think and what you see in it, you should love it-but just the opposite? You don’t like it one bit and you couldn’t explain on pain of death why? THAT’S how I feel about this book, one of the most popular and commonly used books for algebra courses-both undergraduate and graduate. It has a good, very comprehensive selection of material, good exercises and lots of nice examples for the serious student. It's also presented in a completely modern manner with commutative diagrams and lots of nice nuances.

So what’s my problem with it?

Well, first of all, it’s way too expensive. You could get both Vinberg and a used copy of Artin for the same price as this book.

Second of all-it’s pretty dry and matter-of-fact. It just doesn’t excite me about algebra the way Vinberg or Herstein do. Everything’s presented nicely and clearly-but it comes off almost like a dictionary.

Lastly-the level the book is pitched at. It has pretty comprehensive coverage of the standard topics: groups, rings, field, and modules. It also contains some topics that are better suited for graduate courses- homological algebra and group representations, for example. The big problem is the book tries to cover all these topics equally. As a result, it doesn’t succeed in developing all of them well. For example, the chapters on basic group and ring theory are excellent, but modules are presented in a very general but somewhat disjointed way.Also, the level of the book mean the more advanced topics aren't covered in enough depth for a graduate course. And to be frank, a lot of the presentation of the undergraduate material is very similar to that of Herstein- except the only about half the exercises are anywhere near as interesting as the ones there.

I think this is probably what annoys me the most about this book-it comes off as a bloated, watered down version of Herstien. I think the book would have been helped enormously by being split into 2 volumes: The first volume for undergraduate topics and the second for more advanced topics. Instead, the authors tried to cram everything into one volume that can be used for both undergraduate and graduate courses and the result is quite uneven in quality.

The verdict?

It’s probably the single best reference for undergraduate algebra that exists.As such, it's very handy for looking stuff up that you’ve forgotten or preparing for qualifying exams. But as a learning textbook,it's a mixed bag. To be honest, I'd rather use Vinberg or the first volume of Knapp as a text at this level. For its price, see if you can borrow a copy instead.


The Big Three: These are the 3 textbooks that up until about 10 years ago, were the standard texts at the top graduate programs in the U.S. to use for first year graduate algebra courses and for qualifying exams at PhD programs in algebra. Of course, at such programs, the line between graduate and undergraduate coursework is somewhat ambiguous. But I think most mathematicians would agree with me on this assessment.





Algebra 3rd edition by Serge Lang Ok, let’s get the elephant in the room out of the way first.


Lang is a good example of the kind of strange “canonization’ of textbooks in academia which I’ve mentioned before at this blog and other places. It’s funny how some mathematicians-particularly algebracists at the more prestigious programs- that get very self-righteous and uppity when you question whether or not Lang should be used as a first-year graduate text anymore with all the new choices. I can’t help but use some of the remarks of a frequent poster at Math Overflow in this regards.

Let’s call him Mr. G.


Mr. G is a talented undergraduate at one of the more prominent universities to study mathematics in the Midwestern United States. Like the author of this blog, he also has been occasionally slammed for shooting off his big mouth on MO by the moderators.


He and I have had several heated exchanged about his Bourbaki-worship. He believes that the Bourbaki texts are sacred tomes that are the only “real” texts for mathematicians and applications are for non-mathematicans. But I’ll let his own words state his position far better then I can. Here is a recent exchange between Mr. G and 2 mathematicians who are frequent posters at MO: let’s call them Dr. H and Dr. L. This was a question regarding the presentation of graduate algebra. (I obviously can’t be more specific then that-to do so would identify the participants.)

@Dr. H: The first graduate algebra course is often going to be the student's first introduction to algebra. It's supposed to be abstract and intense! If you muddy the waters with applications, your students will never get to that level of Zen you achieve after stumbling around in an algebra course. It's like point-set topology, except the rabbit-hole called algebra goes much deeper and is much more important. –Mr.G

@Mr.G: After my first algebra course I still didn't understand why I should actually much about Galois theory from a practical point of view until I saw $GF(2^n)$ in all sorts of applications. My experience has been that most students--even graduate students studying algebra--are not going to be interested in abstraction for its own sake. Mechanics can help to motivate calculus. The same can be true of information theory and algebra. –Dr.H

@Dr.H: Graduate math students shouldn't be taught things "from a practical point of view". This isn't a gen. ed. class, and the abstract perspective one gains by really engaging algebra "as it is practiced) is completely worth the "journey in the desert", as it were. This is the "Zen" I was talking about. Also, I think that characterizing algebra as "abstraction for abstraction's sake" is really missing the point tremendously. – Mr.G.

@Mr.G. The journey through algebra does not necessarily have to go through the desert, nor is that necessarily the best or most ideal path. It might be so for you, but it is certainly not the best path for everybody. There are numerous other paths to take, most of which can lead and have lead people to mathematical understanding and success. Please take a moment to consider, for instance, Richard Borcherds' recent algebraic geometry examples post. – Dr. L.



Speaking for myself, I firmly believe in heeding Lebesgue’s warning about the state of the art in mathematics: “ Reduced to general theories, mathematics would become a beautiful form without content: It would quickly die.” Generality in mathematics is certainly important, but it can and often is, overly done. But I digress. My point is that Mr.G’s attitude is typical of the Lang-worshipper: That if you can’t deal with Lang, you’re not good enough to be a graduate student in mathematics. Or to use Mr.G’s own words on another thread on the teaching of graduate algebra: ”Lang or bust.” Many feel the “journey in the desert” of Lang is a rite of passage for graduate students, much as Walter Rudin’s Principles of Mathematical Analysis is for undergraduates.

Well, there’s no denying Lang’s book is one of a kind and it’s very good in many respects. People ask me a lot how I feel about Lang’s remarkable career as a textbook author. It’s important to note I never met the man, sadly-and everything I know about him is second hand.


Reading Lang’s books brings to mind a quote from Paul Halmos in his classic autobiography, I Want to be a Mathematician: An Automathography
.which was said in reference to another famous Hungarian mathematician, Paul Erdos: “I don’t like the kind of arithmetic-geometric-combinatorial problems Erdos likes, but he’s so good at them, you can’t help but be impressed. “


I don’t like the kind of ultra-abstract, application-devoid, Bourbakian, minimalist presentations Lang was famous for-but he was SO good at writing them, you can’t help but be impressed.

And Algebra is his tour de force.


The sheer scope of the book is stunning. The book more or less covers everything that’s covered in the later editions of van der Waerden-all from a completely categorical, commutative diagram -with-functors point of view. There’s also a generous helping of algebraic number theory and algebraic geometry from this point of view as well. His proofs are incredibly concise and with zero fat, but quite clear if you take the effort to follow them and fill in the blanks. The chapters on groups and fields are particularly good.

Lang also is an amazingly thorough and responsible scholar; each chapter is brimming with references to original proofs and their source papers. This is a book by one of the giants in the field and it’s clearly a field he had enough respect for to know his way around the literature remarkably well-and he believed in giving credit where credit is due. Quite a few results and proofs-such as localization of rings, applications of representation theory to functional analysis and the homology of derivations, simply don’t appear in other texts. The last point is one I think Lang doesn’t get a lot of credit for, without which the book would be all but unreadable: He gives many, many examples for each concept-many nonstandard and very difficult to ferret out of the literature. Frankly, the book would be worth having just for this reason alone.

So fine, why not go with the line then of “Lang or bust”?

Because the book is absurdly difficult, that’s why.

First of all, it’s ridiculously terse. It takes 2-3 pages of scrap paper sometimes to fill in the details in Lang’s proofs. Imagine doing that for OVER 911 PAGES. And worse, the terseness increases as one progresses in the book. For the easier topics, like basic group theory and Galois theory, it’s not so bad. But the final sections on homological algebra and free resolutions are almost unreadable. You actually get exhausted working through them.

And we haven't even talked about the exercises yet.

“Yeah, I’ve heard the horror stories about Lang’s exercises in the grad algebra book. C’mon, they’re not that bad are they?”

You’re right, they’re not.

They’re worse.

I mean, it’s just ludicrous how hard some of these exercises are.

I’ll just describe 2 of the more ridiculous exercises and it pretty much will give you an idea what I’m talking about. Exercise 30 on page 256 asks for the solution of an unsolved conjecture by Emil Artin. That’s right, you read correctly. Lang puts in parentheses before it: “The solution to the following exercise is not known.”

No shit? And you expect first year graduate students-even at Yale-to have a chance? I’m sorry, that’s not a reasonable thing to do!

Then there’s the famous-or more accurately, infamous-exercise in the chapter on homological algebra: “Take any textbook on homological algebra and try and prove all the results without looking up the proofs.”

I know in principle, that’s what we’re all supposed to do with any mathematical subject we’re learning. But homological algebra?!?

(An aside: I actually had a rather spirited discussion via email with Joseph Rotman, Professor Emeritus of the University of Illinois at Urbana-Champaign, over this matter-a guy that knows a thing or 3 about algebra. Rotman felt I was too hard on Lang for assigning this problem. He thought Lang was trying to make a point with the exercise, namely that homological algebra just looks harder then any other subject, it really isn’t. Well, firstly, that’s a debatable point Lang was trying to make if so. Secondly, I seriously doubt any graduate student who’s given this problem as a significant part of his or her final grade is going to be as understanding as Rotman was. Actually, it’s kind of ironic Rotman thinks that since I know many a graduate student who would have failed the homology part of their Lang-based algebra course without Rotman’s book on the subject!
)

My point is I don’t care how good your students are, it’s educational malpractice to assign problems like that for mandatory credit. And even if you don’t and just leave them as challenges for the best students-isn’t that rubbing salt in the wounds inflicted by this already Draconian textbook?

These exercises are why so many mathematicians have bitter memories of Lang from their student days.

A lot of you may be whining now that I just don’t like hard books. That’s just not true. Herstien is plenty difficult for any student and it’s one of my favorites. In fact, quite the contrary. You’re really supposed to labor over good mathematics texts anyway-math isn’t supposed to be EASY. An easy math textbook is like a workout where you’re not even winded at the end-it’s doubtful you’re going to get any benefits from it.

But Lang isn’t just hard; I don’t just mean students have to labor over the sections before getting them.

The average graduate student learning algebra from Lang’s book is like a fat guy trying to get in shape by undergoing a 3 month U.S. Marine Corp boot camp and having a steady diet of nothing but vitamins, rice cakes and water. Assuming he doesn’t drop dead of a heart attack halfway through, such a regimen will certainly have the desired effect-but it will be inhumanely arduous and unpleasant.

And there are far less Draconian methods of obtaining the same results.

So unless one is a masochist, why in God’s name would you use Lang for a first year graduate course in algebra?

Is it a terrible book? No-as I said above, it has many good qualities and sections. As a reference for all the algebra one will need in graduate school unless becoming an algebracist, the book is second to none.

Would I use it as a text for a first year graduate course or qualifying exam in algebra?

Hell no.




Algebra by Thomas Hungerford This has become a favorite of a lot of graduate students for their algebra courses and it’s pretty easy to see why-at least at first glance. It’s nearly as demanding as Lang-but it’s much shorter and more selective, has a lot more examples of elementary difficulty and the exercises are tough but manageable.


The main problem with this book occurs in the chapter on rings and it boils down to a simple choice. Hungerford-for some strange reason-decides to define rings without a multiplicative identity. I know in some ring theoretic cases, this is quite useful. But for most of the important results in basic ring and module theory, this results in proofs that are much more complicated since this condition needs to be “compensated” for by considering left and right R-modules as separate cases. Hungerford could alleviate this considerably by giving complete, if concise, proofs as Lang does in most cases.

But he doesn’t. He only sketches proofs in more then half the cases.

The result is that every section on rings and modules is very confusing. In particular, the parts on modules over commutative rings and homological algebra-which a lot of students really need to learn-are all over the place.

And the field theory chapter is a train wreck, frankly.

Still, the book has a lot of really nicely presented material from a totally modern, categorical point of view. The first chapter on category theory is probably the best short introduction there is in the textbook literature and the section on group theory is very nice indeed.

But I think Grillet does a considerably better job as a one volume alternative to Lang.







Basic Algebra 2nd edition Volume 1 and Volume II by Nathan Jacobson

I say we should nominate Dover Books for a Nobel Peace Prize for their recent reissue of this classic. The late Nathan Jacobson, of course, was one of the giants of non-commutative ring theory in the 20th century.

He was also a remarkable teacher with an awesome record of producing PHDs at Yale, including Charles Curtis, Kevin Mc Crimmon, Louis H.Rowen, George Seligman, David Saltman and Jerome Katz. His lectures at Yale on abstract algebra were world famous and had 2 incarnations in book form: The first, the 3 volume Lectures In Abstract Algebra, was for a generation the main competition for van der Waerden as the text for graduate algebra courses. Basic Algebra is the second edition-the first edition came out in the 1970’s and was intended as an upgraded course in algebra for the extremely strong mathematics students entering Yale from high school during the Space Age. The first volume-covering classical topics like groups, rings, modules, fields and geometric constructions-was intended as a challenging undergraduate course for such students. The second volume-covering an overview of categorical and homological algebra as well as the state-of-the-art (circa 1985) of commutative and non-commutative ring and module theory as well as the representation theory of finite groups-was intended as a course for strong first year graduate students. The complete collapse of the American educational system in the 1990’s has rendered both volumes useless as anything but graduate algebra texts. Still, given that the second volume was going for nearly 400 dollars at one point online in good condition, it’s reissue by Dover in wonderfully cheap editions is a serious cause for celebration.

(BTW-an interesting story I got from one of Jacobson's former students in the 1980's:Apparently most of Jacobson's undergraduate students struggled mightily using the first volume. Eventually,he abandoned using it as an undergraduate text and switched to Herstien's text, which the students liked much better.However, the first year graduate students and advanced research students in algebra at Yale loved both books and those were the clintele Jacobson used the texts for for the rest of his career.)


Both books are beautifully and authoritatively written in Jacobson's unique style, with a lot of material that isn’t easily found in other sources, such as sections on non-associative rings , Jordan and Lie algebras, metric vector spaces and an integrated introduction to both universal algebra and category theory. The material in both volumes on ring and field theory-both Galois theory and general fields-are fantastic.They are concise and rather sparse in examples compared with other books, but the examples they do have are very well chosen and described thoroughly. There are also many fascinating, detailed historical notes introducing each chapter, particularly in the first volume.

The main problem with both books is that Jacobson’s program here absolutely splits in half algebra into undergraduate and graduate level topics; i.e. without and with categorical and homological structures. This leads to several topics being presented in a somewhat disjointed and inefficient manner because Jacobson refuses to combine them in a modern presentation-module theory in particular suffers from this organization. Personally, I didn’t find it that big an issue with a little effort-but a lot of other students have complained about it. Also, some of the exercises are quite difficult, rivaling Lang’s. Even so, the sheer richness of these books make them true classics. If graduate students are willing to work a little to unify the various pieces of the vast puzzle that Jacobson presents here with astonishing clarity, he or she will be greatly rewarded by a master’s presentation and depth of understanding.



The New Kids On The Block: As I said earlier, the adoption of Lang worldwide as the canonical graduate algebra text had a backlash effect that’s been felt with a slew of new graduate texts. I haven’t seen them all, but I’ve seen quite a few. Here’s my commentary on the ones I’m most familiar with.






Basic Algebra and Advanced Algebra by Anthony W.Knapp

This is probably the single most complete reference for abstract algebra that currently exists. It is also paradoxically, the single most beautiful, comprehensive textbook on it. Knapp taught both undergraduate and graduate algebra at The State University of New York at Stonybrook for nearly 3 decades.These volumes are the finished product of the tons of lecture notes that resulted. The purpose of these books, according to Knapp, is to provide the basis for all the algebra a mathematician needs to know to be able to attend a conference on algebra and understand it.

If so, he’s succeeded beyond all expectations.

The main themes of both books are group theory and linear algebra (construed generally i.e. module theory and tensor algebra).The first volume corresponds roughly to what could possibly be covered at the undergraduate level from reviews of basic number theory and linear algebra up to an honors undergraduate course in abstract algebra (groups, rings, fields, Galois theory, multilinear algebra, module theory over commutative rings). The second corresponds to a first year graduate syllabus focusing on topics in noncommutative rings, algebraic number theory and algebraic geometry( adeles and ideles, homological algebra, Wedderburn-Artin ring theory, schemes and varieties, Grobner bases, etc.)

This is the dream of what an advanced mathematics textbook should be-beautifully written, completely modern and loaded with both examples and challenging exercises that are both creative and not too difficult. In fact, the exercises are really extensions of the text where many topics and applications are in fact derived-such as Jordan algebras, Fourier analysis and Haar groups, Grothendieck groups and schemes, computer algebra and much, much more. The group actions on sets are stressed throughout. Also, categorical arguments are given implicitly before categories are explicitly covered by giving many commutative diagram arguments as universal properties. (This avoids the trap Jacobson fell into.) Best of all-there are hints and solutions to ALL the exercises in the back of each volume. I would love to use this set to teach algebra one day-either as the main texts, as supplements or just references.

But if you enjoy algebra, you have to have a copy. Hopefully, there will be many editions to come.





Abstract Algebra 2nd edition by Pierre Grillet

The first edition of this book was simply called “Algebra” and it came out in 1999. To me, this book is what Lang should be. Grillet is an algebracist and award-winning teacher at Tulane University. Interestingly, he apparently carried out the revision in the aftermath of Katrina. The book covers all the standard and more modern topics in a concise, very modern manner-much like Lang. Unlike Lang, though, Grillet is extremely readable, selective in his content and highly structured with many digressions and historical notes. The sheer depth of the book is amazing. Unlike Lang, which focuses entirely on what-or Hungerford, which explains a great deal but also is very terse-Grillet focuses mainly on why things are defined this way in algebra and how the myriad results are interconnected. It also has the best one chapter introduction to category theory and universal algebra I’ve ever seen-and it occurs in Chapter 17 after the previous 16 chapters where commutative diagrams are constructed on virtually every other page. So by the time the student gets to category theory, he or she has already worked a great deal with the concepts implicitly in the previous chapters. This is very typical of the presentation. Also, Grillet doesn’t overload the book with certain topics and give the short shriff to others-many texts are half group theory and half everything else. Grillet gives relatively short chapters on very specific topics-which makes the book very easy to absorb. The exercises run the gamut from routine calculations to proofs of major theorems. The resulting text is a clinic in how to write “Bourbaki” style texts and it would be a great alternative to either Lang or Hungerford.




Algebra: A Graduate Course by I.Martin Issacs: This is a strange book. After being out of print for over a decade, it was recently reissued by the AMS. Isaacs claims this course was inspired by his teacher at Harvard, Lynn Loomis, whose first-year graduate course Issacs took in 1960 there. Like Loomis’ course, Issacs emphasizes noncom mutative aspects first, focusing mainly on group theory. He then goes on to commutative theory-discussing ring and ideal theory, Galois theory and cyclotomy. The book is one of the most beautifully written texts I’ve ever seen, with most theorems proved and most example constructions left as exercises. This isn't my taste in texts, I prefer the converse: Advanced books with a lot of explicit examples that leave theorem proofs for the reader.But in this case, Issacs writes so beautifully, it's really hard to complain. He also discusses some beautiful and very unusual topics, like x-groups and finite character theory. Unfortunately,for the most part,Issacs’ material choice seems to follow his memories of his graduate course in 1960 far too closely.This choice of topics would be a first year graduate course at a top university only before the 1960’s. Issacs omits completely multilinear algebra, category theory and homological algebra-which is of course, a huge problem. How can you call such a book in 2010 a graduate course?It would be very hard to use such a text by itself as a graduate course in algebra-it would need to be extensively supplemented to provide modern topics any student would need for a qualifying exam. That being said-it is wonderfully written.If supplemented by a text on homological algebra, it could certainly serve as half of such a course. Either Osborne or Joseph Rotman’s books on the subject would fill in the omissions very nicely.The final chapter of Ash would also serve this need well and inexpensively.

Of course, that begs the question-if Issacs' book needs this much supplementing, then why both spend the considerable cost of the book? While I like the book immensely,I suggest borrowing a copy rather then emptying your bank account on it. A used copy for gotten online on the cheap would be a great asset, too.

And now-my very favorite algebra book of all time. Drum roll,pleeeeeeeeease................





Advanced Modern Algebra 1st edition by Joseph J. Rotman
I haven’t seen the second edition, but I’m very familiar with the first. Rotman may be the best writer of algebra textbooks alive. Hell, he may be the best writer of university-level mathematics textbooks PERIOD. Serious. So when his graduate textbook came out, I begged, borrowed and cajoled until I could buy it. And it was one of the best textbooks I ever bought. The contents of the book are, as the AMS’ blurb discusses:“This book is designed as a text for the first year of graduate algebra, but it can also serve as a reference since it contains more advanced topics as well. This second edition has a different organization than the first. It begins with a discussion of the cubic and quartic equations, which leads into permutations, group theory, and Galois theory (for finite extensions; infinite Galois theory is discussed later in the book). The study of groups continues with finite abelian groups (finitely generated groups are discussed later, in the context of module theory), Sylow theorems, simplicity of projective unimodular groups, free groups and presentations, and the Nielsen-Schreier theorem (subgroups of free groups are free). The study of commutative rings continues with prime and maximal ideals, unique factorization, noetherian rings, Zorn's lemma and applications, varieties, and Grobner bases. Next, noncommutative rings and modules are discussed, treating tensor product, projective, injective, and flat modules, categories, functors, and natural transformations, categorical constructions (including direct and inverse limits), and adjoint functors. Then follow group representations: Wedderburn-Artin theorems, character theory, theorems of Burnside and Frobenius, division rings, Brauer groups, and abelian categories. Advanced linear algebra treats canonical forms for matrices and the structure of modules over PIDs, followed by multilinear algebra. Homology is introduced, first for simplicial complexes, then as derived functors, with applications to Ext, Tor, and cohomology of groups, crossed products, and an introduction to algebraic K-theory. Finally, the author treats localization, Dedekind rings and algebraic number theory, and homological dimensions. The book ends with the proof that regular local rings have unique factorization.”
That says what’s in the book. What it doesn’t tell you is what makes this incredible book so special and why it deserves a second edition so quickly with the AMS: Rotman’s gifted style as a teacher, lecturer and writer. The book is completely modern, amazingly thorough and contains discussions of deep algebraic matters completely unmatched in clarity. As proof, read the following excerpt from the first edition, how Rotman explains the basic idea of category theory and it’s importance in algebra:

Imagine a set theory whose primitive terms, instead of set and element, are set and function.How could we define bijection, cartesian product, union, and intersection? Category theory will force us to think in this way. Now categories are the context for discussing general properties of systems such as groups, rings, vector spaces, modules, sets, and topological spaces, in tandem with their respective transformations: homomorphisms, functions, and continuous maps. There are two basic reasons for studying categories: The first is that they are needed to define functors and natural transformations (which we will do in the next
sections); the other is that categories will force us to regard a module, for example, not in
isolation, but in a context serving to relate it to all other modules (for example, we will
define certain modules as solutions to universal mapping problems).


I dare you to find a description of category theory that would serve a novice better. The book is filled with passages like that-as well as hundreds of commutative diagrams, examples, calculations and proofs of astounding completeness and clarity. Rotman presents algebra as a huge, beautiful puzzle of interlocking pieces-one he knows as well as anyone in the field. The one minor complaint is the book’s exercises-they’re a little soft compared to the ones in Hungerford or Lang. And the sheer size of the book-1000 hardback bound pages!- is a bit daunting. ( Rotman joked with me via email that more then a few times, he mistakenly carried it to his calculus class and had to go back to his office to switch books.) But these are very minor quibbles in a book destined to become a classic. If I had to choose one textbook for graduate algebra and it’s qualifier and couldn’t pick any others-THIS is the one I’d pick, hands down.

Word from the AMS and those who have seen it that the second edition is even better-the index has been greatly improved and entire sections have been rewritten to emphasize noncom mutative algebra-which is appropriate for a graduate course.

I suggest you all place your orders now. You’ll thank me later, I promise.

I now return you to your regularly scheduled lives.

Thank you for your attention.

Peace.


Other books mentioned in this post(worth checking out):












1
Nov 10

A Sidebar On My Willing Exile From Math Overflow

Crap, took WAY too long between posts again. And this is going to have to be a short one because of the lateness of the hour.

Need to do something about that. But between chronic insomnia and a sinus infection, it’s been all I can do to think.

To the main point of this post:The guys at Math Overflow have finally had it with my shenanigans.

After my 5th suspension from the board for…well, to be honest, I’m still not completely sure. According to moderator Ben Webster ( formerly MIT C.L.E. Moore Instructor, now at the University of Oregon-you may also recognize him from his days blogging at The Secret Blogging Seminar ) , the reason was as follows:

Andrew- No one has ever been suspended on MO for the contents of mathematical statements, even if we disagree with them. The issue is your rude comments on other answers; I would call it "bad sportmanship," but MO is not a game. For example "I can't believe this guy puts down a high school slogan and gets 13 points for it and I got downvoted for "Probability is real analysis with the concept of an expectation." " on Michael Lugo's answer.

As far as I'm concerned, this is equivalent to jumping up after a seminar and shouting "You guys are clapping for that? That was a terrible talk!" which I think we can all agree would not be socially acceptable behavior.

Note to the audience: generally the moderators have adopted a policy of not arguing with Andrew on meta, since it just seems to create more drama. In this case I thought it was important to point out that the issue was not Andrew's mathematical statements (which as I said before, we would not suspend people over), but rather his behavior in comments.

Uh, ok, Ben.

My “behavior” ,as he so puts it, was simply being myself. For those who know me, that seems to be more then enough. Despite my best efforts to tone it down for the board, things just deteriorated further and further. At one point, I was emailed and messaged by several of the members telling me-politely but in no uncertain terms-that my antics were making a bad impression on the mathematical community in general and that I was thus endangering my future career and job prospects.

I really don’t like being threatened.

And make no mistake, as nicely as it was delivered, that’s what it was. A threat.

There was a time I’d have told the whole bunch of them to go fuck themselves for openers and go on for several HTML pages about them and their mothers and wives.

But if I had-well, I would have deserved what I got.

Firstly, when you’re dealing with people like Webster, Andy Putnam at Rice University and Pete L.Clark at the University of Georgia-and this drama is all going on in front of frequent posters Terence Tao, Tom Gowers and Richard Stanley (!) - well, it’s pretty obvious you’re not going to win this one.

And it’s more then that. This isn’t MY site, it’s THIERS.

This wasn’t really about right and wrong, it was about me trying to make their site something it’s not because I wanted to be able to say these things in front of professional mathematicians and get their feedback.

But that’s not what MO is for. It’s for research level and academic questions regarding the mathematical community. They set it up, they police it, they make the rules. I was just a guest.

And they decided I was messing up their furniture and would rather I left. That’s entirely within their rights to do.

And contrary to what some people may think, I don’t enjoy offending people. I wasn’t following the rules and I was dead wrong here. No matter how morally indignant I might want to look.

So after this last incident, I left my pride on the floor and reopened shop here, where I can have no rules but my own.

In closing regarding this incident, I wanted to let everyone over at Math Overflow I never intended any offense. I’m a passionate, opinionated guy. I was the proverbial bull in a china shop.
Not that that’s anything new for me. Once again, I want to apologize to everyone there and hope one day I can return.

More importantly,I hope to be posting here at a regular basis as I prepare for my oral qualifying exams in algebra and topology-hopefully,to be taken around Christmas, no later. I want to let everyone there know they are more then welcome to respond here. They are also free to say WHATEVER THEY WANT ABOUT THE POSTS OR ME. The first amendment is very much alive here.

(For as long as the government allows it after Tuesday,of course. More on that later in the week.)

Well,it’s nearly 3 am now, so the planned posted reading list for graduate algebra courses-sadly-is going to have to wait.

But hopefully-not very long at all!

Ciao!

EDIT 11/6/2010: As Terry Tao and another poster commented after the first draft if this post went online, I wasn't fair in my blanket description of those aforementioned comments as all threats. In fact, many were by well intentioned posters who didn't want me blow my career by shooting off my stupid mouth. For those posters, I should and will apologize. They meant well and I shouldn't be lumping them in with the idiots-"trolls" they call them on MO-who emailed me and warned me "they'd fix my career when I apply for jobs." THAT was a threat.But those brave enough to countermand me were very well intentioned.As such,I apologize once again.
See,I'm not above admitting an error. Rare as it is........

24
Aug 10

I HAVE RETURNED! Hide the women and children. Well, maybe not the women…………..

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After a disappointing year of graduate school, I’ve returned at the close of summer and the beginning of my last semester (I HOPE) of schoolwork in my life. Flecks of premature grey adorn my beard as the analogous summer of my life ends and the chill of winter slowly but inevitably approaches-and a PHD still nowhere in sight due to the collapse of the American empire.

I will develop my observations on this matter in depth in future posts.
But forget the barbarians being at the gate: The guard was their all-too-willing accomplices as they walked right in and enslaved the entire kingdom under the pretexts of “free market economy“ and “lobbying“.

I fear for the future of this country. The Radical Red Staters are poised to take advantage of 2 generations of steadily degenerating education of our citizens to polarize and divide this nation, so that it’s no longer a battle of liberal and conservative ideas. Rather, it’s now between thinking, logical, scientifically objective people and obsessively tribal, dogmatic, racist, exclusionary Christian fundamentalist extremists-with the latter group being frighteningly agitated into a frenzy by a paradoxically self righteous and simultaneously self-servingly duplicitous leadership who blasts their inflammatory beacons on the internet and talk-radio.

And just think: All it took was the election of the first African-American president-and the first real attempts at social progress in over a decade as a logical response by his administration to the worst economic disaster in our history-for the populace of the US to revert socially to the 1930’s. Socialism. Illegal aliens. Doubts over evolution being forcibly imposed in school textbooks by a few concerned educators in Texas.

Somewhere in Hell, Adolf Hitler, Joseph Stalin and Joseph McCarthy are laughing their asses off. I see the cliff coming and no one seems to want to slam on the brakes before we go over………

Again-MUCH more on this in future posts. But I can’t justify or contextualize this first post’s purpose without some preamble of the desperation felt by my aging self as the world I knew as a teenager slips away-to be replaced by a darker jungle of uncertainty.

The wake of my father’s death-now 4 years distant-has financially crippled what remains of my family. My father’s 18 year battle with cancer-which he ultimately lost-has now left us with a debt that barely allows my mother to buy food. I know-we’re hardly alone in that. But this has taken it’s toll on my career. I’ve had to take 3 graduate courses in the last year just to maintain my eligibility for health insurance as well as student loans to PAY for that insurance. (Of course, this being America, you’d think this would be the primary place for free health care for all it’s citizens. Of course, we don’t live in a logical or moral reality, do we?)

That alone would make struggling through these courses hard enough. But then in a moment of desperation-I made a very foolish choice regarding my health. I really can’t get into the specifics here-I hope to do so in a future post in connection to a larger issue.
I will say this: I’ve officially learned the hard way that if you have no other available methods of controlling psychological illnesses other then drugs and someone tells you to go off them cold turkey and “you’ll feel so much better, all you need is to man up!”-then if you actually listen to them, the REAL imbecile in the conversation is YOU.

As you can imagine, the long term result was somewhat less then expected. I rectified the result, but the damage was done. And now I’m stuck with 3 subpar grades in courses I didn’t even fucking WANT. It’s entirely possible now my dreams of an Ivy League PHD programs-unless I solve the Riemann Hypothesis-are out the door and my career ends here with a Master’s degree and a future as a high school teacher.

The wonderful destiny awaits of dying of cardiac complications from prostate cancer when I’m 75-face down in puddle of cold decaf tea grading high school algebra papers from student who can barely read.

So it goes.

I haven’t given up yet. I can’t. Why not? That’s what a rational person of my intelligence would do.

Well, we’re all going to die of something. I lost my youth long ago-sacrificed it for my family. I think I’ve finally made my peace with that. It hurts like hell what I lost. But we’re all losing so much as the second decade of the 21st century opens. Parents losing their children in 2 pointless wars that have bankrupted the country the survivors are coming back to. PHDs mopping floors because those educated in other nations or of higher pedigrees are preferred for the vanishingly small number of jobs available.

Again, the cliff approaches for all but the top 1% of us.

I dunno about all of you, but I’d rather go off the cliff screaming my rage at this pointless, Godless reality and fighting with my last breath against the dying of the light.

Which is why I’m taking the initiative and beginning my own online business. (Nice segway, huh?)

I began this Ebay store on whim. I’ve been selling textbooks online for nearly 8 years on and off for pocket cash-and all told, it’s worked out pretty well. But “working out” has meant what’s basically chump change by selling second hand books and cannibalizing my own library.
So I started thinking: What kind of cash can I make with a REAL inventory of books to sell?

So one thing lead to another and viola! Parthenon Academic Books.

Actually, it wasn’t that simple. History never is.

So what’s the scoop behind it?

I’d love to tell you guys, but I’m passing out at almost 3 am. So it’ll have to wait-along with several other things-for the next post.

After all, how can you build a following to a blog without good cliffhangers?

TO BE CONTINUED!!!

In the meantime, please check it out! Yes, I’m begging. Begging is something we all should get used to-most of us will be making a living doing it for the next few decades. So it’s good practice for all of us.

Wait, wait, don’t go! At LEAST give me a chance to let you see the place before you blow it off. And yes, you can SEE what you’re buying. That’s the first main advantage of buying from me. I’ll show you EVERYTHING before you buy-you don’t have to take my word for it like Amazon and the rest of the corporate text machines.

What are the others? Well, since I have a webpage, why should I waste my time just repeating what’s there?

My About Me Page:

http://members.ebay.com/ws/eBayISAPI.dll?ViewUserPage&userid=locasciosales

And honestly-how mamy stores HAVE THIER OWN BLOG? No,not THIS one. I can't have anyone upstaging me,let alone filling this air with complaints about the dustjackets of purchased books. Nope-I got my own blog just for that. And even better,it's got more photos of the inventory and is updated by me semiweekly. Does Amazon do that or let you speak directly with thier CEO? I think not.

The blog:
http://parthenonacademicbooks.blogspot.com/2010/08/welcome-to-parthenon-blog-where.html

Come on in and browse-any questions,feel free to ask.

Be back soon,same blogsphere plane and frequency.

Beware, Citizens...........

24
Aug 10

I HAVE RETURNED! Hide the women and children. Well, maybe not thewomen…………..

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After a disappointing year of graduate school, I’ve returned at the close of summer and the beginning of my last semester (I HOPE) of schoolwork in my life. Flecks of premature grey adorn my beard as the analogous summer of my life ends and the chill of winter slowly but inevitably approaches-and a PHD still nowhere in sight due to the collapse of the American empire.

I will develop my observations on this matter in depth in future posts.
But forget the barbarians being at the gate: The guard was their all-too-willing accomplices as they walked right in and enslaved the entire kingdom under the pretexts of “free market economy“ and “lobbying“.

I fear for the future of this country. The Radical Red Staters are poised to take advantage of 2 generations of steadily degenerating education of our citizens to polarize and divide this nation, so that it’s no longer a battle of liberal and conservative ideas. Rather, it’s now between thinking, logical, scientifically objective people and obsessively tribal, dogmatic, racist, exclusionary Christian fundamentalist extremists-with the latter group being frighteningly agitated into a frenzy by a paradoxically self righteous and simultaneously self-servingly duplicitous leadership who blasts their inflammatory beacons on the internet and talk-radio.

And just think: All it took was the election of the first African-American president-and the first real attempts at social progress in over a decade as a logical response by his administration to the worst economic disaster in our history-for the populace of the US to revert socially to the 1930’s. Socialism. Illegal aliens. Doubts over evolution being forcibly imposed in school textbooks by a few concerned educators in Texas.

Somewhere in Hell, Adolf Hitler, Joseph Stalin and Joseph McCarthy are laughing their asses off. I see the cliff coming and no one seems to want to slam on the brakes before we go over………

Again-MUCH more on this in future posts. But I can’t justify or contextualize this first post’s purpose without some preamble of the desperation felt by my aging self as the world I knew as a teenager slips away-to be replaced by a darker jungle of uncertainty.

The wake of my father’s death-now 4 years distant-has financially crippled what remains of my family. My father’s 18 year battle with cancer-which he ultimately lost-has now left us with a debt that barely allows my mother to buy food. I know-we’re hardly alone in that. But this has taken it’s toll on my career. I’ve had to take 3 graduate courses in the last year just to maintain my eligibility for health insurance as well as student loans to PAY for that insurance. (Of course, this being America, you’d think this would be the primary place for free health care for all it’s citizens. Of course, we don’t live in a logical or moral reality, do we?)

That alone would make struggling through these courses hard enough. But then in a moment of desperation-I made a very foolish choice regarding my health. I really can’t get into the specifics here-I hope to do so in a future post in connection to a larger issue.
I will say this: I’ve officially learned the hard way that if you have no other available methods of controlling psychological illnesses other then drugs and someone tells you to go off them cold turkey and “you’ll feel so much better, all you need is to man up!”-then if you actually listen to them, the REAL imbecile in the conversation is YOU.

As you can imagine, the long term result was somewhat less then expected. I rectified the result, but the damage was done. And now I’m stuck with 3 subpar grades in courses I didn’t even fucking WANT. It’s entirely possible now my dreams of an Ivy League PHD programs-unless I solve the Riemann Hypothesis-are out the door and my career ends here with a Master’s degree and a future as a high school teacher.

The wonderful destiny awaits of dying of cardiac complications from prostate cancer when I’m 75-face down in puddle of cold decaf tea grading high school algebra papers from student who can barely read.

So it goes.

I haven’t given up yet. I can’t. Why not? That’s what a rational person of my intelligence would do.

Well, we’re all going to die of something. I lost my youth long ago-sacrificed it for my family. I think I’ve finally made my peace with that. It hurts like hell what I lost. But we’re all losing so much as the second decade of the 21st century opens. Parents losing their children in 2 pointless wars that have bankrupted the country the survivors are coming back to. PHDs mopping floors because those educated in other nations or of higher pedigrees are preferred for the vanishingly small number of jobs available.

Again, the cliff approaches for all but the top 1% of us.

I dunno about all of you, but I’d rather go off the cliff screaming my rage at this pointless, Godless reality and fighting with my last breath against the dying of the light.

Which is why I’m taking the initiative and beginning my own online business. (Nice segway, huh?)

I began this Ebay store on whim. I’ve been selling textbooks online for nearly 8 years on and off for pocket cash-and all told, it’s worked out pretty well. But “working out” has meant what’s basically chump change by selling second hand books and cannibalizing my own library.
So I started thinking: What kind of cash can I make with a REAL inventory of books to sell?

So one thing lead to another and viola! Parthenon Academic Books.

Actually, it wasn’t that simple. History never is.

So what’s the scoop behind it?

I’d love to tell you guys, but I’m passing out at almost 3 am. So it’ll have to wait-along with several other things-for the next post.

After all, how can you build a following to a blog without good cliffhangers?

TO BE CONTINUED!!!
Be back soon,same blogsphere plane and frequency.

Beware, Citizens...........

Update 7/4/2015 : Parthenon Academic Books was a complete disaster. It may still exist in some virtual shell sense, with the original page and listings in some remote corner of the net, like an aborted fetus left to rot by some spoiled sociopathic teenage mother (EW, nice image) , but that's all it is now. An  aborted dream. My dearest hope is that my latest and far larger effort at www.tuloomath.com succeeds in far greater fashion. Check it out and let me know.

12
Jul 09

Re: arXiv:The Modern Cure For B.A.D Mathematicans?

This'll be a short one. But hopefully something I'll comment on more in the future.

I've found a terrific article by Professor Melvin Henriksen published in The Mathematical Intelligencer in 1993 and republished online at the Topology Atlas by him.Henriksen-if you're unfamiliar with him-is currently Professor Emertis at Harvey Mudd College and he's one of the few remaining active-sort of-mathematicans in one of my favorite areas of mathematics:point set topology. He's also published quite a bit in algebra,has been very active in historical aspects of mathematics and is currently one of the major overseers of the mammoth virtual site Math Forum supported by Drexel University.

The elitism in mathematics is nothing new-nor,sadly,is it unique to mathematics among academic endevors.Henriksen-never one to keep his opinions to himself-summed up the situation-and the reasons behind it-beautifully and informatively,in this article:

http://at.yorku.ca/t/o/p/c/10.htm

If anything,the situation has WORSENED since he wrote this article.When you mention certain branches of mathematics at some unmentionable universities that believe only the half a dozen places on Earth called "Ivies" are where civilized humans exist and everywhere else is untamed jungle with blood drinking,grunting barbarians with pieces of paper masquerading as educated humans-you actually get audible laughter. They don't even try to be kind about it. Why should they? Anyone who can't see that they're right is a fool and won't get promoted anywhere. At least,not if they have anything to say about it-and sadly,they do.

I'm currently investigating the possible role of general topology in additive number theory. To be honest,if arXiv-simply called "Archive" by most of us-didn't exist,I doubt I could publish it ANYWHERE. This remarkable tool has changed publishing forever through open access and free publishing in mathematics and physics with no formal refereeing-and therefore no monkey buisness. Any attempts to shut it down or seriously regulate it should be met with savage resistance. Such attempts at regulation has already begun in the form of the vote in 2004 by the Archive board to allow only preprints-presumably because copyright issues could arise that could jeapordize corporate profits. The bottom line is that it's begun-the attempts to control it.
(The copyrighting of concepts as property is a terrifying phenomenon that I hope to tackle in depth another time.Suffice to say this is a Pandora's Box that threatens all original thinking if it's not strictly controlled.)
It's a wonderful reality we live in currently in this regard-free publishing. Let's not let the B.A.D. crew wreck it like they've wreaked whatever doesn't serve thier purposes.
In other words,buisness as usual on Planet Earth.

12
Jul 09

Re: arXiv:The Modern Cure For B.A.D Mathematicans?

This'll be a short one. But hopefully something I'll comment on more in the future.

I've found a terrific article by Professor Melvin Henriksen published in The Mathematical Intelligencer in 1993 and republished online at the Topology Atlas by him. Henriksen-if you're unfamiliar with him-is currently Professor Emeritus at Harvey Mudd College and he's one of the few remaining active-sort of-mathematicians in one of my favorite areas of mathematics:point set topology. He's also published quite a bit in algebra,has been very active in historical aspects of mathematics and is currently one of the major overseers of the mammoth virtual site Math Forum supported by Drexel University.

The elitism in mathematics is nothing new-nor,sadly,is it unique to mathematics among academic endeavors. Henriksen-never one to keep his opinions to himself-summed up the situation-and the reasons behind it-beautifully and informatively,in this article:

http://at.yorku.ca/t/o/p/c/10.htm

If anything,the situation has worsened since he wrote this article.

When you mention certain branches of mathematics at some unmentionable universities that believe only the half a dozen places on Earth called "Ivies" are where civilized humans exist and everywhere else is untamed jungle with blood drinking,grunting barbarians with pieces of paper masquerading as educated humans-you actually get audible laughter. They don't even try to be kind about it. Why should they? Anyone who can't see that they're right is a fool and won't get promoted anywhere. At least,not if they have anything to say about it-and sadly,they do.

I'm currently investigating the possible role of general topology in additive number theory. To be honest,if arXiv-simply called "Archive" by most of us-didn't exist,I doubt I could publish it anywhere. This remarkable tool has changed publishing forever through open access and free publishing in mathematics and physics with no formal refereeing-and therefore no monkey business. Any attempts to shut it down or seriously regulate it should be met with savage resistance. Such attempts at regulation has already begun in the form of the vote in 2004 by the Archive board to allow only preprints-presumably because copyright issues could arise that could jeapordize corporate profits. The bottom line is that it's begun-the attempts to control it.

(The copyrighting of concepts as property is a terrifying phenomenon that I hope to tackle in depth another time.Suffice to say this is a Pandora's Box that threatens all original thinking if it's not strictly controlled.)

It's a wonderful reality we live in currently in this regard- free publishing. Let's not let the B.A.D. crew and their wealthy allies wreck it like they've wreaked whatever doesn't serve their purposes.

In other words,business as usual on Planet Earth.

6
Jul 09

A Brief (Partial) Apology For Speaking out of Turn: Calculus, Cirricula And Constudents...............

I'm not usually one to apologize when I feel someone is being a dick. Anyone that knows me knows that. But my guilt has gotten the better of me and I think I need to amend my swipe at McMasters' University professor James Stewart.I think I was angry at the decaying civilization around me and I took it out on him.
I'm really apologizing just for one small part of the rant that I felt was beneath me. It was simply untrue and would be very unfair for me to say about someone I've never even heard lecture or speak once.I referred to Stewart as a "grotesquely overpaid hack without an ounce of mathematical talent".
Well,that was completely untrue and unfair:Professor Stewart is actually a very fine teacher and mathematician from what I know of him.(It turns out he's the mathematical grandson of the famous Oxford mathematician E.C.Titchmarsh.I didn't know that and found that kind of interesting in and of itself.)
I dug out my copy of the third edition of his textbook to act physical evidence in this trial of my conscience.I also borrowed a copy of the 6th edition.He's made a lot of improvements in the text since I used it-a lot more pictures, the exponential and logarithmic functions are introduced and discussed MUCH earlier (in the first chapter,in fact),and in general a lot more explicit focus on the overall process of problem solving,which was only indirectly stated in the edition I'm familiar with. Stewart was a graduate student of George Polya before moving on to get his PHD-the influence of the Stanford problem solving master is all over this textbook in both editions. Stewart approaches calculus as a problem solving enterprise first and foremost-such an approach is bound to be pragmatic and will intentionally sacrifice rigor where it obscures understanding.
In short, Stewart is trying to teach his students how to become intelligent problem solvers above all else. As teachers (and speaking for myself as an aspiring teacher at the college level), discouraging the good intentions behind such an approach is the LAST THING we should want to do.
It's easy to forget how confusing calculus and physics is when one first seriously tackles it as a college undergraduate-or for the more fortunate and/or talented, high school. As a result, it's easy to get on your high horse and badmouth a text like this from the viewpoint of someone who's mastered a good portion of rigorous mathematics. Stewart offers to take the student by the hand and walk him or her step by step through the fog-showing them tricks of the trade along the way and tried-and-true methods of attacking problems in ways that not only obtain solutions,but a complete understanding of the MEANING of what's being asked of them. "What do they want from you?What will satisfy the question?" This is what Stewart is trying to teach with his book.
It's really informative in this regard to read Stewart's own comments on the text from an interview done by the MAA on July 6th,comparing it to the texts he used as a student at Stanford University and The University Of Toronto in the 1960's:

IP: How have mathematics textbooks changed over the years?
JS: Compared with the textbooks that I had as a student, textbooks are so much better now. I don’t know how kids learned from these old books. There was no motivation. It was very austere. You can go too far in the other direction, but the state of the exposition of mathematics is just so much better than it was three decades ago.
As an author of the high school textbooks in the 70s, I kept my eye on trends in education. The new math had been well ensconced by then. But what I observed and decried was the waves, the extremes, the pendulum going back and forth from the new math back to basics. You still see this, especially in the U.S., especially at the high school level, where it is much more virulent. At that time, I longed to get hold of that pendulum and stop it somewhere in a sensible middle. People get too dogmatic.

Even more insightful into Stewart's thinking is his comments on teaching and what he's doing lately:

IP: Are you still teaching?
JS: Although I am Professor Emeritus at McMaster, a year ago I was appointed professor of mathematics at the University of Toronto, and I have twice taught first-year calculus. Although I don’t teach fulltime anymore, I love teaching. Being an author is a pretty solitary, sedentary occupation, so I miss the social aspect—which is teaching. I do it partly to keep in touch with kids, because it brings out the best in me, and to give me new ideas for new editions of my books.
This fall I am introducing a new course at the University of Toronto on problem solving. I introduced such a course at McMaster quite some time ago.
When I was a graduate student at Stanford I fell under the spell of George Polya, who was retired but used to come in and give these problem-solving talks. He had all of us—teachers and students alike—literally sitting on the edges of our seats with mathematical excitement, presenting data, asking us to make conjectures.
The idea is: Suppose you’re faced with a problem that you have never seen before. How do you get started? The first few lectures introduce some basic principles of problem solving. The remaining lectures start with a “problem of the day.” How would you solve it? What strategy would you use? What about trying a special case or solving a simpler problem first? It’s my favorite course to teach.
I’m doing that this fall, working with some of the faculty at the University of Toronto so that they can carry on after me. It will be a kind of capstone course. You’re drawing on everything that you’ve learned up to that point, putting it together. There’s no new content whatsoever. But once you take a problem out of the context of a specific course, it becomes harder.

Now THAT'S a book I'd love to read-a problem solving textbook by Stewart that emerges from that course!
But sadly,Stewart seems to miss that the problem with this approach to calculus which has made his book so successful is also why it's damaging to students used by itself. The result of the "practical" nature of the text is that THE FACT THAT IT'S A BOOK ON CALCULUS BECOMES COMPLETELY INCIDENTAL.He never asks the all important "why" questions that brought the real number system and the structure of real analysis into focus for mathematicans in the 19th century.Everything's given a name-Sum Rule,Product Rule,Method of Secants,etc.-which makes them tailor-made for memorization rather then learning.He gives quite good "geometric" explainations-such as a good discussion of motivating the definition of the derivative as the limit of a sequence of secant lines to a point on a curve.But such a discussion is completely independent of the definition of a derivative as a limit.It might as well appear in a book on physics or geometry. As a result, it's all completely mechanical-the fact that it's a book on calculus almost become irrelevant!
And this is the problem he fails to see:TO MOST OF TODAY'S STUDENTS, IT IS IRRELEVANT. YOU MAY AS WELL BE TEACHING THEM HOW TO PLAY CHECKERS AND THEY MEMORIZE THE RULES. Sure, a few students will really look at the very nice geometrical arguements and walk away really learning something. But most students-who make up most of today's colleges and whom the university administrators are aiming to sell calculus to-couldn't care less. I call such students constudents-a hybrid of conmen and students. They aren't interested in learning-in fact, like thieves excited about stealing and not getting caught or cheating husbands who call thier wives to tell them they're going to be late while getting oral sex from thier mistress-getting an A while never learning a damn thing is exciting to them.
I know what some of you are thinking: "Andrew,come on-that's human nature,there's always going to be students like that!" Sure,of course.But the big advantage of the rigorous calculus texts of the past was that it was almost impossible for such students to con thier way to a good grade-the fact that rigorous mathematics was an ESSENTIAL part of the structure of the course ensured they actually had to learn something to do reasonably well. And the course acted to ensure that students with impure motives who didn't even try DIDN'T get good grades.
Books like Stewart's have eliminated this failsafe altogether.
I remember as a premed sitting around with a number of students taking calculus using Stewart and the discussion of the exam was like they were talking about a football game and how they were going to "beat" the exam. They came up with codes,mnemonics,word games-not a single theorum or concept or proof. I made the idiotic mistake of asking if anyone actually learned the material and the whole table erupted with laughter. The President of the Student Medical Association smiled at me like The Grinch.
"Winning is about APPEARING to know what you're doing,not actually doing it.Don't worry,Andrew-you can always work taking out the trash in my office on 5th avenue."
Our society rewards this kind of behavior.Why?Becuase letting these monsters use Stewart and get thier A's without learning anything is good for buisness,that's why. The university gets to pack the classes with 200 PAYING students by making this a required course,the students get thier A's which the college can use to improve it's ranking standing so that administrators get promoted for making so much money and helping public relations and off they go to Ivy League medical schools thinking urea is made in the kidney-and worse,not giving a shit.
And 5 years later they're killing and crippling patients left and right and being aquitted at malpractice trials because the only one in the room who's a better liar then they are is the son of a bitch defending them. A book like Stewart's ENABLES this kind of system.
I have no problem with Stewart wanting to make the book of a problem solving nature-as I've said,this leads to the book having many positive qualities.My problem is that including mathematical rigor need not be contrarian to this intention and for someone claiming to be so devout to teaching, Stewart refuses to acknowledge this.
Sadly, I think he's too smart not to see this. I think his position is one of willful ignorance in a corrupt academic culture that's made him not only very wealthy for his occupation, but very famous. I doubt anyone outside of McMaster would have ever heard of him without this text. And I stand by my earlier criticism of Stewart of his ridiculous excess with his own concert hall. He loves music, fine. Bless him. But spending more on his hobby then 5 families spend on thier homes is nauseating and he should be ashamed of himself. Of course,he's hardly alone in that in this day and age.

But he's an academic. He should know better.

Frankly,I think he DOES and his own words betray this:

When I started writing my first book, I had no idea you could make any money writing books. That was not a motivation at all. It was a surprise, but it enabled me to build this house. And I’ve got to continue to work to pay for the house. The house’s cost [$24 million] is double the original estimates.

It sounds like he has a very strong motivation for continuing to enable the sharks. Amazing what people are able to justify to themselves.

I hope Dr.Stewart keeps making money and succeeds in paying for his house so his heirs have the proceeds from using it as a tourist trap when he passes away. I hope all his kids and grandkids go to Harvard from it and maybe follow in his footsteps as a teacher instead of becoming criminal defense attorneys and bankers as the later generations usually do when the first generation creates a fortune for them. And I hope a lot of teachers of calculus use it as a supplementary reference or secondary source for thier calculus courses and as the main text in high school courses.

I just hope one day someone has the balls to challenge the American way someday and writes the text that replaces Stewart by combining mathematical rigor with his teaching skills to give us a calculus text for students and not constudents.

And I hope I and my loved ones are never at the mercy of enabled in a hospital with thier lawyers' number constantly in thier back pocket.

Welcome To The Twilight Jungle. Abandon All Honesty And Integrity Ye That Enter Here.................

6
Jul 09

A Brief (Partial) Apology For Speaking out of Turn: Calculus, CirriculaAnd Constudents...............

I'm not usually one to apologize when I feel someone is being a dick.

Anyone that knows me knows that.

But my guilt has gotten the better of me and I think I need to make amends for my last post.
Namely,my swipe at McMasters' University professor James Stewart.

I think I was angry at the decaying civilization around me and I took it out on him.

I'm really apologizing just for one small part of the rant that I felt was beneath me. It was simply untrue and would be very unfair for me to say about someone I've never even heard lecture or speak once.I referred to Stewart as a "grotesquely overpaid hack without an ounce of mathematical talent".

Well,that was completely untrue and unfair:Professor Stewart is actually a very fine teacher and mathematician from what I know of him. (It turns out he's the mathematical grandson of the famous Oxford mathematician E.C.Titchmarsh. I didn't know that and found that kind of interesting in and of itself.)

I dug out my copy of the third edition of his textbook to act physical evidence in this trial of my conscience.I also borrowed a copy of the 6th edition.

He's made a lot of improvements in the text since I used it-a lot more pictures, the exponential and logarithmic functions are introduced and discussed MUCH earlier (in the first chapter,in fact),and in general a lot more explicit focus on the overall process of problem solving,which was only indirectly stated in the edition I'm familiar with. Stewart was a graduate student of George Polya before moving on to get his PhD-the influence of the Stanford problem solving master is all over this textbook in both editions. Stewart approaches calculus as a problem solving enterprise first and foremost-such an approach is bound to be pragmatic and will intentionally sacrifice rigor where it obscures understanding.

In short, Stewart is trying to teach his students how to become intelligent problem solvers above all else. As teachers (and speaking for myself as an aspiring teacher at the college level), discouraging the good intentions behind such an approach is the last thing we should want to do. It's easy to forget how confusing calculus and physics is when one first seriously tackles it as a college undergraduate-or for the more fortunate and/or talented, high school. As a result, it's easy to get on your high horse and badmouth a text like this from the viewpoint of someone who's mastered a good portion of rigorous mathematics. Stewart offers to take the student by the hand and walk him or her step by step through the fog-showing them tricks of the trade along the way and tried-and-true methods of attacking problems in ways that not only obtain solutions,but a complete understanding of the MEANING of what's being asked of them. "What do they want from you?What will satisfy the question?" This is what Stewart is trying to teach with his book.

It's really informative in this regard to read Stewart's own comments on the text from an interview done by the MAA on July 6th,comparing it to the texts he used as a student at Stanford University and The University Of Toronto in the 1960's:

IP: How have mathematics textbooks changed over the years?
JS: Compared with the textbooks that I had as a student, textbooks are so much better now. I don’t know how kids learned from these old books. There was no motivation. It was very austere. You can go too far in the other direction, but the state of the exposition of mathematics is just so much better than it was three decades ago.
As an author of the high school textbooks in the 70s, I kept my eye on trends in education. The new math had been well ensconced by then. But what I observed and decried was the waves, the extremes, the pendulum going back and forth from the new math back to basics. You still see this, especially in the U.S., especially at the high school level, where it is much more virulent. At that time, I longed to get hold of that pendulum and stop it somewhere in a sensible middle. People get too dogmatic.

Even more insightful into Stewart's thinking is his comments on teaching and what he's doing lately:

IP: Are you still teaching?
JS: Although I am Professor Emeritus at McMaster, a year ago I was appointed professor of mathematics at the University of Toronto, and I have twice taught first-year calculus. Although I don’t teach fulltime anymore, I love teaching. Being an author is a pretty solitary, sedentary occupation, so I miss the social aspect—which is teaching. I do it partly to keep in touch with kids, because it brings out the best in me, and to give me new ideas for new editions of my books. This fall I am introducing a new course at the University of Toronto on problem solving. I introduced such a course at McMaster quite some time ago. When I was a graduate student at Stanford I fell under the spell of George Polya, who was retired but used to come in and give these problem-solving talks. He had all of us—teachers and students alike—literally sitting on the edges of our seats with mathematical excitement, presenting data, asking us to make conjectures. The idea is: Suppose you’re faced with a problem that you have never seen before. How do you get started? The first few lectures introduce some basic principles of problem solving. The remaining lectures start with a “problem of the day.” How would you solve it? What strategy would you use? What about trying a special case or solving a simpler problem first? It’s my favorite course to teach.I’m doing that this fall, working with some of the faculty at the University of Toronto so that they can carry on after me. It will be a kind of capstone course. You’re drawing on everything that you’ve learned up to that point, putting it together. There’s no new content whatsoever. But once you take a problem out of the context of a specific course, it becomes harder.

Now that's a book I'd love to read-a problem solving textbook by Stewart that emerges from that course!

But sadly,Stewart seems to miss that the problem with this approach to calculus which has made his book so successful is also why it's damaging to students used by itself.The result of the "practical" nature of the text is that the fact that it's a book on calculus becomes completely incidental.

He never asks the all important "why" questions that brought the real number system and the structure of real analysis into focus for mathematicians in the 19th century.Everything's given a name-Sum Rule,Product Rule,Method of Secants,etc.-which makes them tailor-made for memorization rather then learning.He gives quite good "geometric" explanations-such as a good discussion of motivating the definition of the derivative as the limit of a sequence of secant lines to a point on a curve.But such a discussion is completely independent of the definition of a derivative as a limit.It might as well appear in a book on physics or geometry. As a result, it's all completely mechanical-the fact that it's a book on calculus almost become irrelevant!

And this is the problem he fails to see:To most of today's students,it is irrelevant.You may as well be teaching them how to play checkers and they memorize the rules.

Sure, a few students will really look at the very nice geometrical arguments and walk away really learning something.

But most students-who make up most of today's colleges and whom the university administrators are aiming to sell calculus to-couldn't care less.

I call such students constudents-a hybrid of conmen and students. They aren't interested in learning-in fact, like thieves excited about stealing and not getting caught or cheating husbands who call their wives to tell them they're going to be late while getting oral sex from their mistress-getting an A while never learning a damn thing is exciting to them.

I know what some of you are thinking: "Come on-that's human nature,there's always going to be students like that!"

Sure,of course.

But the big advantage of the rigorous calculus texts of the past was that it was almost impossible for such students to con their way to a good grade-the fact that rigorous mathematics was an essential part of the structure of the course ensured they actually had to learn something to do reasonably well. And the course acted to ensure that students with impure motives who didn't even try didn't get good grades.

Books like Stewart's have eliminated this fail-safe altogether.

I remember as a premed sitting around with a number of students taking calculus using Stewart and the discussion of the exam was like they were talking about a football game and how they were going to "beat" the exam. They came up with codes,mnemonics,word games-not a single theorem or concept or proof. I made the idiotic mistake of asking if anyone actually learned the material and the whole table erupted with laughter. The President of the Student Medical Association smiled at me like The Grinch.

"Winning is about APPEARING to know what you're doing,not actually doing it.Don't worry-you can always work taking out the trash in my office on 5th avenue."

Our society rewards this kind of behavior.Why?Because letting these monsters use Stewart and get thier A's without learning anything is good for business,that's why. The university gets to pack the classes with 200 paying students by making this a required course,the students get thier A's which the college can use to improve it's ranking standing so that administrators get promoted for making so much money and helping public relations and off they go to Ivy League medical schools thinking urea is made in the kidney-and worse,not giving a shit.
And 5 years later they're killing and crippling patients left and right and being acquitted at malpractice trials because the only one in the room who's a better liar then they are is the son of a bitch defending them.

A book like Stewart's ENABLES this kind of system.

I have no problem with Stewart wanting to make the book of a problem solving nature-as I've said,this leads to the book having many positive qualities.My problem is that including mathematical rigor need not be contrarian to this intention and for someone claiming to be so devout to teaching, Stewart refuses to acknowledge this.

Sadly, I think he's too smart not to see this. I think his position is one of willful ignorance in a corrupt academic culture that's made him not only very wealthy for his occupation, but very famous. I doubt anyone outside of McMaster would have ever heard of him without this text.

And I stand by my earlier criticism of Stewart of his ridiculous excess with his own concert hall.

He loves music, fine. Bless him. But spending more on his hobby then 5 families spend on their homes is nauseating and he should be ashamed of himself.

Of course,he's hardly alone in that in this day and age.

But he's an academic. He should know better.

Frankly,I think he does and his own words betray this:

When I started writing my first book, I had no idea you could make any money writing books. That was not a motivation at all. It was a surprise, but it enabled me to build this house. And I’ve got to continue to work to pay for the house. The house’s cost [$24 million] is double the original estimates.

It sounds like he has a very strong motivation for continuing to enable the sharks. Amazing what people are able to justify to themselves.

I hope Dr.Stewart keeps making money and succeeds in paying for his house so his heirs have the proceeds from using it as a tourist trap when he passes away. I hope all his kids and grandkids go to Harvard from it and maybe follow in his footsteps as a teacher instead of becoming criminal defense attorneys and bankers as the later generations usually do when the first generation creates a fortune for them. And I hope a lot of teachers of calculus use it as a supplementary reference or secondary source for their calculus courses and as the main text in high school courses.

I just hope one day someone has the balls to challenge the American way someday and writes the text that replaces Stewart by combining mathematical rigor with his teaching skills to give us a calculus text for students and not con-students.

And I hope I and my loved ones are never at the mercy of the enabled MDs in a hospital with their lawyers' number constantly in their back pocket.

Welcome To The Twilight Jungle.

Abandon All Honesty And Integrity Ye That Enter Here.................

6
Jul 09

A Brief Ode To Stewart's Calculus-NOT.........................

I just read a really funny post at Ars Mathematica and had to share it with all of you with commentary. Apparently, the question's come up with what McMaster's University's self-made gazillionare James Stewart did with all his royalites from the infamous calculus book every other university's department uses. Apparently, he built a gigantic house with his own personal concert hall in the middle of it. You don't believe me? See for yourself :

http://online.wsj.com/article_email/SB123872378357585295-lMyQjAxMDI5MzA4NDcwMjQzWj.html

This was so he-a trained violinist-could perform with his friends in the privacy and comfort of his own mansion. Talk about hubris worthy of being struck down by the Gods with the Ceres asteriod.Apparently the only way Stewart's fragile self esteem could make it as a violinist in a concert hall was to have one built for himself where he'd be the star of the show every single night...............LOL
Sigh. Only in America would that seem like a logical action and not a gigantic excess of self-centered indulgence. I pass-on my way to the bus-recently homeless families of 4 living in thier cars with thier 4 year old daughter crying to the mother, "Mommy,what happened to my bed?" Meanwhile, this grotesquely overpaid hack without an ounce of mathematical talent is spending 3 times what these poor people's former house was worth because he doesn't want to embarrass himself in public with his violin playing.................

But be that as it may-I was honestly asked:How bad IS Stewart's book and what are some of YOUR favorite texts?What would YOU use to teach calculus given the chance?
Well,sadly,since I was a complete imbecile in high school and didn't know grades mattered in life-and my parents being laborers,well,they didn't know either-I ended up at The City University Of New York instead of a REAL college.(I made many friends there and learned a lot-but let no one be decieved my lack of pedigree will give me a huge battle ahead for any degree of success.) So my first exposure to calculus WAS Stewart.
In all fairness,it's not as bad as some people make it out to be. The real positive about the book is the IMMENSE number of exercises with complete solutions. Unfortuately,that's a double edged sword and it's also the main reason it's completely unpalatable for mathematicans:It reduces calculus to a step-by-step, plug-and-chug bag of techniques without even any mathematical insight or thinking. Anything that requires more thought then a baboon is either completely omitted or shunted to a mythical "advanced calculus" course-which no longer exists,of course. The students don't have to do any real thinking at all-which is why most students love it,of course. Let's face it-THAT'S why the bottom feeding universities buy it every year-so the premeds,accounting students,actuaries,pre-law and all the rest of the master cheaters that form the vast majority of bodies filling the enormous lecture halls of the average 200 student calculus course can program the solutions of all thier exams into thier programmable calculators.
"This is AMERICA. Let the Japanese waste thier time thinking and just give me my f***ing A so I can go out and screw people over for 6 figures a year,geek."
It's also why Stewart would never have become so absurdly wealthy writing a book that is the very pinnacle of mediocrity in any other academic system BUT America's. It's why a piece of crap like CHARMED was on for 7 years while great shows like FARSCAPE vanish, why TRANSFORMERS:THE REVENGE OF THE FALLEN-with a mindless plot and racist "black" Autobots-is the #1 film in America-it's why we sold our blood won freedoms to a stupid evil Texan from a rich family we elected king for the illusion of safety while Americans lost the entire Bill of Rights for 8 years.
"Americans aren't stupid!" Really?You must be living in a different USA then I am.
So it goes.
My favorites? Well,when anyone tells you Spivak's CALCULUS is the best calculus book ever-EVER-it's really hard to argue. It's incredibly beautiful and a model of clarity. But much more then that,with every word,picture and exercise,Spivak asks the reader to THINK about the concepts before him or her before setting the task of doing it. Really THINK about it.
Is it too hard for the average student? Well,depends on what you mean by the average student. The average student cheating thier way through every homework and test and sleeping with TAs to get a 4.0 to get into Harvard medicial school,sure. But if you're talking about the average student-not necessarily a mathematics or natural science student-who reads everything with an effort and wonders and asks real questions even if they don't understand or particuarly like it because they're there to LEARN something-it would be a struggle. But with a good teacher by thier side, they could definitely get through it.
And they'd be all the better for it. For the mathematically talented, the book will become a treasured keepsake for a lifetime.The chapter on infinite series alone is worth photocopying and keeping.
I refuse to recommend soft,"applied" books.To me,the pure/applied mathematics distinction is a symptom of the problem above. There is no pure math or applied math-there is only MATHEMATICS. If you don't realize that,you're not part of the solution,you're part of the problem. That being said-the main problem with using Spivak is that he has virtually no applications-just one lame application of vector algebra to celestial mechanics late in the book. The main point of calculus is to calcul-ATE. Theory is important and all well and good, but teaching calculus as real analysis completely devoid of application is a little like teaching music students the complete mechanics of writing scores and symphonies,but never teaching them how to play!!!!
A book that fascinates me and I'd love to try to use for a basic calculus course one day is Donald Estep's PRACTICAL ANALYSIS IN ONE VARIABLE. Estep,a numerical analyst, teaches a basic real analysis course combined with a basic calculus course, using numerical methods to motivate the rigorous development of the real numbers and epsilon-delta arguements-with DOZENS of actual real-world examples from chemistry and physics!!! I'd be a little scared to use the book,though-Estep makes a couple of really strange choices. The biggest one is deciding NOT TO DISCUSS INFINITE SERIES-TO ESTEP, INFINITE SERIES IS BEST DONE WITH COMPLEX VARIABLES,SO HE DECIDES TO FORGET IT. HUH?!?
My favorite all around calculus book is a nearly forgotten one by a legendary teacher-CALCULUS,2nd edition by Edwin E.Moise-based on the course in calculus that Moise taught for many years at Harvard and won several awards for. It's completely rigorous, yet beautifully intuitive with many,many pictures and geometric insight motivated using Euclidean geometry such as lines,planes and conic sections, as well as many,many physical applications. THIS is the book I would use to teach my children calculus.Go to the library and check it out for yourself if you're disappointed with the ton of fluff the departments are trying to push on you to teach calculus with. You'll thank me later,I promise.
Stewart and his private concert hall.Yet another example we are living in the era of the barbarians at the gate. It's so frustrating-with no address,you can't even drive by and throw a firebomb through his window to burn it down...........LOL

6
Jul 09

A Brief Ode To Stewart's Calculus-NOT.........................

I just read a really funny post at Ars Mathematica and had to share it with all of you with commentary. Apparently, the question's come up with what McMaster's University's self-made gazillionare James Stewart did with all his royalites from the famous/infamous-depending on how much you care about mathematics-calculus book every other university's department uses.
Apparently, he built a gigantic house with his own personal concert hall in the middle of it. You don't believe me? See for yourself :

http://online.wsj.com/article_email/SB123872378357585295-lMyQjAxMDI5MzA4NDcwMjQzWj.html

This was so he-a trained violinist-could perform with his friends in the privacy and comfort of his own mansion. Talk about hubris worthy of being struck down by the Gods with the Ceres asteroid.

Apparently the only way Stewart's fragile self esteem could make it as a violinist in a concert hall was to have one built for himself where he'd be the star of the show every single night.

    Sigh.

Only in America would that seem like a logical action and not a gigantic excess of self-centered indulgence. I recently passed-on my way to the bus-a recently homeless family of 4 living in their car with their 4 year old daughter crying to the mother, "Mommy,what happened to my bed?"

Meanwhile, this grotesquely overpaid hack without an ounce of mathematical talent is spending 3 times what these poor people's former house was worth because he doesn't want to embarrass himself in public with his violin playing.................

     But be that as it may-I was honestly asked:How bad is Stewart's book and what are some of your favorite texts?What would you use to teach calculus given the chance?

Well,sadly,since I was a complete imbecile in high school and didn't know grades mattered in life-and my parents being laborers,well,they didn't know either-I ended up at The City University Of New York instead of a real college.(I made many friends there and learned a lot-but let no one be deceived my lack of pedigree and relatively advanced age will give me a huge battle ahead for any degree of success.)

So my first exposure to calculus was Stewart.

In all fairness,I was being a bit disingenuous up to this point. It's not as bad as many people make it out to be. The real positive about the book is the immense number of exercises with complete solutions.

Unfortunately,that's a double edged sword and it's the main reason it's completely unpalatable for mathematicians:It reduces calculus to a step-by-step, plug-and-chug bag of techniques without any real mathematical insight or thinking. Anything that requires more thought then a baboon is either completely omitted,put in rushed optional sections or shunted to a mythical "advanced calculus" course.

The students don't have to do any real thinking at all. Which is why most students-particularly the non-science majors-love it,of course.

Let's face it-that's why the bottom feeding universities buy it every year-so the premeds,accounting students,actuaries,pre-law and all the rest of the master cheaters that form the vast majority of bodies filling the enormous lecture halls of the average 200 student calculus course can program the solutions of all their exams into their programmable calculators.

"This is Anerica. Let the Japanese waste their time thinking and just give me my f***ing A so I can go out and screw people over for 6 figures a year working for Goldman-Sachs,geek."

It's also why Stewart would never have become so absurdly wealthy writing a book that is the very pinnacle of mediocrity in any other academic system but America's.

It's why a piece of crap like Charmed was on for 7 years while great shows like Farscape vanish, why Transformers:The Revenge Of The Fallen-with a mindless plot and racist "black" Autobots-is the #1 film in America at this writing.

It's why we sold our blood won freedoms to a stupid evil Texan from a rich family we elected king for the illusion of safety while Americans lost the entire Bill of Rights for 8 years.

"Americans aren't stupid!"

Really? You must be living in a different USA then I am.

So it goes.

My favorites? Well,when anyone tells you Micheal Spivak's Calculus
is the best calculus book ever-ever-it's really hard to argue. It's incredibly beautiful and a model of clarity. But much more then that,with every word,picture and exercise, Spivak asks the reader to think about the concepts before him or her before setting the task of doing it. Really THINK about it.

Is it too hard for the average student?

Well,depends on what you mean by the average student.

The average student cheating their way through every homework and test and sleeping with TAs to get a 4.0 to get into Harvard medical school,sure.

But if you're talking about a typical smart and curious undergraduate student-not necessarily a mathematics or natural science student-who reads everything with a real effort,wonders and asks real questions even if they don't understand or particularly like it because they're there to learn something?

I think they can.

Yes,it would be a struggle-particularly in learning how to calculate with epsilonic limit arguments for the first time. But with a good,patient teacher by their side, they could definitely get through it.In the process,they'd learn an enormous amount-not only about calculus, but logical thinking and problem solving. Which is useful in all walks of life, not just the sciences-and they'd be all the better for it.

For the mathematically talented, the book will become a treasured keepsake for a lifetime.The chapter on infinite series alone is worth photocopying and keeping.

I refuse to recommend soft,"applied" books.To me,the pure/applied mathematics distinction is a symptom of the problem above. There is no pure math or applied math-there is only mathematics. If you don't realize that,you're not part of the solution,you're part of the problem. That being said-the main problem with using Spivak is that he has virtually no applications-just one lame application of vector algebra to celestial mechanics late in the book. The main point of calculus is to calcul-ATE. Theory is important and all well and good, but teaching calculus as real analysis completely devoid of application is a little like teaching music students the complete mechanics of writing scores and symphonies,but never teaching them how to play!!!!

A book that fascinates me and I'd love to try using for a basic calculus course one day is Donald Estep's Practical Analysis In One Variable
. Estep,a numerical analyst, teaches a basic real analysis course combined with a basic calculus course, using numerical methods to motivate the rigorous development of the real numbers via Cauchy sequences of rationals and epsilon-delta arguments-with dozens of actual real-world examples from chemistry and physics!!! It's precisely the kind of book I wish there were more of-a book that combines application and fully careful mathematical development. I'd be a little scared to use the book,though-Estep makes a couple of really strange choices. The biggest one is deciding not to discuss infinite series. To Estep, infinite series is best done with complex variables,so he decides to omit them altogether. Huh?!? I hope there's a second edition where he adds a chapter on infinite series. Still, it's a relatively minor flaw in an altogether marvelous book that should be in everyone's library who loves calculus.

My favorite all around calculus book is a nearly forgotten one by a legendary teacher-Calculus by Edwin E.Moise.
It's based on both the regular and honors versions of the course in calculus that Moise taught for many years at Harvard and won several awards for. It's completely rigorous, yet beautifully intuitive with many,many pictures and geometric insight motivated using Euclidean geometry such as lines,planes and conic sections, as well as many physical applications. It's not quite as rigorous as Estep or Spivak,but it is considerably more careful then the average calculus book.

It breaks my heart this book is out of print and I'd love to republish it myself one day. This is the book I would use to teach my children calculus.Go to the library and check it out for yourself if you're disappointed with the ton of fluff the departments are trying to push on you to teach calculus with.

You'll thank me later,I promise.

Stewart and his private concert hall.Yet another example we are living in the era of the barbarians at the gate. It's so frustrating-with no address,you can't even drive by and throw a firebomb through his window to burn it down...........LOL

26
Jan 09

Re:The Vampires Of American Medicine And WTF Does "Well Defined" Mean?!?

So much for posting regularly at this blog.I may as well just shut it down and start again.

But I won't. I WILL keep trying to post on a regular basis for the rest of the summer until the blog catches on. Or it doesn't. A blog is for the author,no one else.Anyone else reads it,that's a plus.

I AM hoping it does catch on,though. I have a lot of thoughts on many things ongoing-but now's not the time. If anything, small posts will begin appearing regularly.

This summer-my last one before applying to PHD programs has not gone well. Sleep has eluded me for the better part of a month-stolen by gut pain combined with frequent urination. And the wonderful health care system of America has assured my internest can't see me.

I don't have a right to live according to the AMA, you see-not enough money to buy good health.

That's why they let my father die of agonizing prostate cancer at the end-they crunched the numbers and thier profits simply outwieghed my dad's treatment. So they gave us the bullshit story that "There's nothing more we can do." The cancer metastisized througout his bones over his last few weeks, giving him a death you wouldn't wish on Bernie Madoff.

Meanwhile, if he was a drug kingpin who dropped off 5 million in CASH,I wonder if a miraculous treatment they suddenly remembered about would have appeared and extended his life by 5-10 years. Since corporations now control the publication of most medical research as well as the mass media, we'd never know if one existed no matter how much you researched.

I can get fully into this here, but I WILL say this: The fact that Yale Medical School considered seriously adding ACTING CLASSES to it's required cirriculia for the M.D. for all students entering after 2011 to "improve maximally productive patient-practitioner interaction"(translation:to make the doctors the best con-artists possible) speaks volumes of the age of medicine we live in-and why I turned my back on that world years ago. I consider myself VERY lucky to have good and trustworthy doctors-but I can't tell you how hard my family searched to find them.
100 monsters for every one like them.

"We're coming for your money and we'll GET it all. We're the only real winners.The players don't stand a chance." -from the screenplay of Martin Scorsese's CASINO

Changing the subject to something mathematical, something on the web caught my eye yesterday and I just need to share it with the house. Ever wonder what well-defined means? It's amazing how many graduate students-particularly those working in category theory and the higher altitudes of algebra,where the phrase probably comes up most-never ask what that means. It's kind of accepted everyone "sorta" knows what it means. And for most people,that's good enough.
I remember the first time I ever wondered about it-it was in Kenneth Kramer's honors abstract algebra course a few years ago at Queens College. He was sketching the proof of Cayley's theorum on the fact that every group is the same as some group of permutations on a set (i.e. they're isomorphic). ( Actually, he wasn't proving it,he just wanted to sketch the proof because he'd rather spend the classes' time developing the theory of group actions on a set, of which Cayley's theorum is a special case-i.e. a group acting on itself. But I digress.............)
He was constructing the composition map which is the isomorphism of a group G onto it's corresponding permutation group acting on it's underlying set S -I forget what he denoted it as,call it P(S). He commented the map was clearly well-defined. I raised my hand in frustration since I'd asked the question before and never gotten a straight answer from any professor (some of them actually got annoyed with it and made unkind remarks about my age as a student)
What followed was one of the most impressionable moments of my student career as Dr.Kramer and I exchanged comments on what exactly it meant to be well-defined. "It means it's not ambiguous what the value assigned is, Andrew-that we don't get 2 values for the same arguement." "Oh, you mean the relation actually specifies a function?" "Well, not exactly-if the formula IS a function, you're absolutely right. But this may not be a function and still be a well defined mathematical object." I didn't get it. After a few minutes of him giving a few examples, no progress was made. He ultimately asked me to table the question so we don't waste any more of the classes' time.
I did so,but ultimately,it disturbed me. Dr.Kramer is a gifted teacher on all matters mathematical-an early student of John Tate's at Harvard-and usually the most pleasant and patient of people with even the stupidest of students' questions. In fact,I'll be taking a course on elliptic functions with him at the City University Of New York Graduate Center this fall. The sheer wieght of the subsequent coursework-the first 6 chapters of Herstein's classic Topics In Algebra in a VERY intensive 2 semesters,plus his own notes-prevented us from broaching the subject further. All that really got settled was that it was pretty clear what "well-defined" meant if the object under consideration was a function-in fact, it's almost redundant. But how would you describle a general mathematical object as being "well defined"?

Leave it to Tom Gowers to make everybody happy.

There are several blogs online I try so hard not to miss. Peter Woit's Not Even Wrong, Terrance Tao's, John Baez's The N-Category Cafe' , The Secret Blogging Seminar and a few others. But nothing matches Gower's blog for sheer beauty of writing and thinking about mathematics. A lot of people can do mathematics, a lot more people can teach mathematics, and even more people can talk about mathematics .(Sadly, this is whether or not they know what the fuck they're talking about or not...........)

There are so few who can do all of the above.

Elias Stien can do it (sometimes).

Melvyn Nathanson can do it.

James Stasheff can do it. Better then anyone I've ever heard.

William Thurston can do it.

But for my money, no one does it better currently and consistently then Tom Gowers. His blog should be required reading for all mathematicans and serious math students. (By the way-his old teacher at Cambridge, Bela Bollabos-is also great at all of the above. I doubt that's an accident. )
Anywho, I was reading Gowers' blog and low and behold, Gowers also wanted to know, after grading the exams for the year at Cambridge and discovering NONE of his students understood it,either-what's it mean for something to be well defined?

People who know me know I'm Socratic to a fault, to the point of making people violent. I almost NEVER agree with EVERYTHING someone says.

But this is rare occasion when I'm speechless with complete conviction and agreement with someone else's analysis. As I said, leave it to Gowers to give the perfect answer to a great question.

I'll simply let the beauty,depth and simplicity of Gowers' blogpost speak for itself-I simply have nothing to add to it. Nothing at all. Anyone asks me this question in the future, I'll simply give them a copy of Gowers' post. For all basic mathematical discussions that may come up in the future, I seriously doubt anyone can debunk this discussion.

It's THAT good.

Oh,screw the self-engratiating pontification,here's Gowers. And if you don't bookmark his blog, shame on you.

Good night to all,fellow travelers. Until next time.


http://gowers.wordpress.com/2009/06/08/why-arent-all-functions-well-defined/#more-605

26
Jan 09

Re:The Vampires Of American Medicine And WTF Does "Well Defined" Mean?!?

So much for posting regularly at this blog.I may as well just shut it down and start again.

But I won't. I WILL keep trying to post on a regular basis for the rest of the summer until the blog catches on. Or it doesn't. A blog is for the author,no one else.Anyone else reads it,that's a plus.

I AM hoping it does catch on,though. I have a lot of thoughts on many things ongoing-but now's not the time. If anything, small posts will begin appearing regularly.

This summer-my last one before applying to PHD programs has not gone well. Sleep has eluded me for the better part of a month-stolen by gut pain combined with frequent urination. And the wonderful health care system of America has assured my internest can't see me.

I don't have a right to live according to the AMA, you see-not enough money to buy good health.

That's why they let my father die of agonizing prostate cancer at the end-they crunched the numbers and thier profits simply outwieghed my dad's treatment. So they gave us the bullshit story that "There's nothing more we can do." The cancer metastisized througout his bones over his last few weeks, giving him a death you wouldn't wish on Bernie Madoff.

Meanwhile, if he was a drug kingpin who dropped off 5 million in CASH,I wonder if a miraculous treatment they suddenly remembered about would have appeared and extended his life by 5-10 years. Since corporations now control the publication of most medical research as well as the mass media, we'd never know if one existed no matter how much you researched.

I can get fully into this here, but I WILL say this: The fact that Yale Medical School considered seriously adding ACTING CLASSES to it's required cirriculia for the M.D. for all students entering after 2011 to "improve maximally productive patient-practitioner interaction"(translation:to make the doctors the best con-artists possible) speaks volumes of the age of medicine we live in-and why I turned my back on that world years ago. I consider myself VERY lucky to have good and trustworthy doctors-but I can't tell you how hard my family searched to find them.
100 monsters for every one like them.

"We're coming for your money and we'll GET it all. We're the only real winners.The players don't stand a chance." -from the screenplay of Martin Scorsese's CASINO

Changing the subject to something mathematical, something on the web caught my eye yesterday and I just need to share it with the house. Ever wonder what well-defined means? It's amazing how many graduate students-particularly those working in category theory and the higher altitudes of algebra,where the phrase probably comes up most-never ask what that means. It's kind of accepted everyone "sorta" knows what it means. And for most people,that's good enough.
I remember the first time I ever wondered about it-it was in Kenneth Kramer's honors abstract algebra course a few years ago at Queens College. He was sketching the proof of Cayley's theorum on the fact that every group is the same as some group of permutations on a set (i.e. they're isomorphic). ( Actually, he wasn't proving it,he just wanted to sketch the proof because he'd rather spend the classes' time developing the theory of group actions on a set, of which Cayley's theorum is a special case-i.e. a group acting on itself. But I digress.............)
He was constructing the composition map which is the isomorphism of a group G onto it's corresponding permutation group acting on it's underlying set S -I forget what he denoted it as,call it P(S). He commented the map was clearly well-defined. I raised my hand in frustration since I'd asked the question before and never gotten a straight answer from any professor (some of them actually got annoyed with it and made unkind remarks about my age as a student)
What followed was one of the most impressionable moments of my student career as Dr.Kramer and I exchanged comments on what exactly it meant to be well-defined. "It means it's not ambiguous what the value assigned is, Andrew-that we don't get 2 values for the same arguement." "Oh, you mean the relation actually specifies a function?" "Well, not exactly-if the formula IS a function, you're absolutely right. But this may not be a function and still be a well defined mathematical object." I didn't get it. After a few minutes of him giving a few examples, no progress was made. He ultimately asked me to table the question so we don't waste any more of the classes' time.
I did so,but ultimately,it disturbed me. Dr.Kramer is a gifted teacher on all matters mathematical-an early student of John Tate's at Harvard-and usually the most pleasant and patient of people with even the stupidest of students' questions. In fact,I'll be taking a course on elliptic functions with him at the City University Of New York Graduate Center this fall. The sheer wieght of the subsequent coursework-the first 6 chapters of Herstein's classic Topics In Algebra in a VERY intensive 2 semesters,plus his own notes-prevented us from broaching the subject further. All that really got settled was that it was pretty clear what "well-defined" meant if the object under consideration was a function-in fact, it's almost redundant. But how would you describle a general mathematical object as being "well defined"?

Leave it to Tom Gowers to make everybody happy.

There are several blogs online I try so hard not to miss. Peter Woit's Not Even Wrong, Terrance Tao's, John Baez's The N-Category Cafe' , The Secret Blogging Seminar and a few others. But nothing matches Gower's blog for sheer beauty of writing and thinking about mathematics. A lot of people can do mathematics, a lot more people can teach mathematics, and even more people can talk about mathematics .(Sadly, this is whether or not they know what the fuck they're talking about or not...........)

There are so few who can do all of the above.

Elias Stien can do it (sometimes).

Melvyn Nathanson can do it.

James Stasheff can do it. Better then anyone I've ever heard.

William Thurston can do it.

But for my money, no one does it better currently and consistently then Tom Gowers. His blog should be required reading for all mathematicans and serious math students. (By the way-his old teacher at Cambridge, Bela Bollabos-is also great at all of the above. I doubt that's an accident. )
Anywho, I was reading Gowers' blog and low and behold, Gowers also wanted to know, after grading the exams for the year at Cambridge and discovering NONE of his students understood it,either-what's it mean for something to be well defined?

People who know me know I'm Socratic to a fault, to the point of making people violent. I almost NEVER agree with EVERYTHING someone says.

But this is rare occasion when I'm speechless with complete conviction and agreement with someone else's analysis. As I said, leave it to Gowers to give the perfect answer to a great question.

I'll simply let the beauty,depth and simplicity of Gowers' blogpost speak for itself-I simply have nothing to add to it. Nothing at all. Anyone asks me this question in the future, I'll simply give them a copy of Gowers' post. For all basic mathematical discussions that may come up in the future, I seriously doubt anyone can debunk this discussion.

It's THAT good.

Oh,screw the self-engratiating pontification,here's Gowers. And if you don't bookmark his blog, shame on you.

Good night to all,fellow travelers. Until next time.


http://gowers.wordpress.com/2009/06/08/why-arent-all-functions-well-defined/#more-605