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.

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.


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: 

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 

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.    

Jul 15


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

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.