Jun 15
  1. Undergraduate Differential Geometry (i.e. Curves And Surfaces in Rn)

When I was an undergraduate, differential geometry appeared to me to be a study of curvatures of curves and surfaces in R3. As a graduate student I learned that it is the study of a connection on a principal bundle. I wondered what had become of the curves and surfaces, and I studied topology instead.-R.W. Sharpe's Differential Geometry: Cartan's Generalization of Klein's Erlangen Program

  1. Curves and Surfaces Lecture Notes for Geometry 1 Henrik Schlichtkrull Department of Mathematics University of Copenhagen   (PG-13) An excellent and very comprehensive set of lecture notes on classical differential geometry in low dimensions using both fine visuals and rigorous discussions and proofs. The notes require a strong knowledge of both calculus and linear algebra to be understood fully. In the United States, this would mean students with either a year of honors calculus or a semester of real analysis/advanced calculus as they will need to understand the ideas of careful limits and open and closed sets in the plane and 3-space. The notes cover all the standard topics of an undergraduate differential geometry course, with emphasis on curvature on surfaces in 3-space, curvature, the first and second fundamental forms and the Gauss formula. The real joy of these notes is how the author strikes a terrific balance between the raw geometry and the careful proofs, using the former to motivate the latter in tandem. A lot of geometers try and do this, but few actually succeed. Schlichtkrull does and very well indeed. Highly recommended.
  2. Differential Geometry Emma Carberry University of Sydney  Semester 2 2009 (PG-13))  Another impressive set of notes on classical differential geometry from a modern point of view.  These notes actually go a bit further and describe the essentials of differentiable manifold and forms to be able to conclude with a global version and proof of the Gauss-Bonnet theorem. Very well written and again, quite visual and careful, with lots of examples. The page has 2 versions of the notes-the official notes and PDF versions of the original overhead slides made with PowerPoint. Make sure you download the "slides" version of the notes, which show the actual overheads from which the written notes are distilled-there's a lot more detail in these and they look a lot nicer, too. Sadly, the exercises and their solutions aren't freely available for download, so the notes aren't as helpful as they could be. Still, a very good source for differential geometry courses at both the undergraduate and graduate levels. Highly recommended.
  3. Differential Geometry of Curves and Surfaces,Thomas Banchoff, Shiing-Shen Chern,and William Pohl SPIN Course long version 2002(PG)
  4. Differential Geometry of Curves and Surfaces Thomas Banchoff, Shiing-Shen Chern, and William Pohl  short polished version January 29, 2003  (PG)This is a quite interesting pair of notes. They are early versions of the textbook on classical differential geometry Differential Geometry of Curves And Surfaces co-authored by Banchoff and Steven Lovett. People familiar with that fine book will recognize very little of it in these early versions except the choice of  contents-indeed, in many ways they are different works. These versions were written for the 2002 and 2003 SPIN courses ( and if someone can let me know what the hell SPIN was, I'd greatly appreciate it since I can't seem to find it online) at Brown University nearly 10 years ago-and co-written by one the most eminent geometers in history in Chern: both Banchoff and Pohl were doctoral students of his.The earlier, fuller version of the notes are more or less what the title says they are: a course in differential geometry in Euclidean space using linear algebra and calculus as prerequisites. There are 3 things that stand out about the longer, less polished version of the notes. First of all, they are very formal with virtually no pictures, despite remaining very elementary in level. Secondly, although they cover the standard topics of such a course, they also include many more topics and references to the literature then the usual course presents.  The literature references are in the form of remarks, such as the attribution of the first statement of the four-vertex theorem to Mukhopadhyaya  in 1909 and the proof given to G.Herglotz. Lastly, they have an enormous number of exercises, many challenging. The main drawback of the notes besides the lack of pictures, are the many typographical errors and blank omissions. It's clear the notes needed to be drastically corrected and revised, which is probably why the authors never completed them in this form. The shorter version from 2003 only runs about one fourth the length of the longer version, but are much more polished with many fewer errors.They cover far less,though. Despite all the errors, the longer version is well worth the effort working through for students that are serious about learning differential geometry as they have a depth few sources at this level can match. Recommended for strong and patient students.
  5. Differential  Geometry Lecture Notes Ruxandra Moraru University of Waterloo 2011 TeXed by David Kotik  (PG-13)  Well written and organized undergraduate  course in differential geometry that seeks to unify the classical and modern perspective by presenting curves and surfaces as submanifolds of abstract manifolds embedded in Euclidean space and defined by parametrized coordinate frames. As a result, the notes as somewhat more sophisticated then usual, requiring the elements of basic set theory and topology as well as linear algebra and calculus. Still, this approach has the advantage of unifying the concepts of curvature and torsion for curves in the plane and surfaces in 3 space. Also, there are many good examples and exercises-although surprisingly few pictures. Still, a good set of notes for strong undergraduates and will help prepare them for more advanced courses. Recommended.
  6. DIFFERENTIAL GEOMETRY: A First Course in Curves and Surfaces Theodore Shifrin University of Georgia Spring, 2015   (PG) These notes have been floating around the internet for several years now as Shifrin continues to revise and polish them for his differential geometry courses. They've become extremely popular and it's not hard to see why.They cover all the basics of classical differential geometry in the plane and 3-space, with many wonderful pictures in color, careful proofs, wonderfully literate and insightful discussions and many good exercises. I could go into more detail, but I'd prefer to let the notes speak for themselves. The notes are dedicated to the memory of S.S. Chern -Shifrin was one of Professor Chern's final doctoral students before retiring. I like to think he'd approve, given what we know about the similar lectures he gave to students decades ago that inspired an entire generation of geometers. I know I certainly do. This is one of the very best free sources for elementary differential geometry available and I'd suggest you download it soon. The notes have stabilized over the last year or so-which means Professor Shifrin may soon be preparing it for publication. Given how expensive the finished versions of his textbooks are, such as Multivariable Mathematics and Linear Algebra: A Geometric Approach, I strongly advise you take advantage of them while they're still freely available. The strongest possible recommendation.
  7. Differential Geometry second semester J. Brendan Quigley University College Dublin Spring 2009     (PG)This is an interesting set of notes that has a rather unusual content: it combines a full treatment of the calculus of several variables in Euclidean space with a treatment of local curve theory. The emphasis is on computation rather then proving and there are many beautiful computer generated pictures of surfaces and curves. Unfortunately, there is no actual surface theory apart from graphs of functions of 2 variables. As a result, I'm not sure you can really call this a differential geometry class in the usual sense-it's only half of such a class. It's more like a practical advanced calculus class for engineers based on linear algebra that uses curve theory to illustrate many of the concepts and proofs, such as the implicit function theorem and the Hessian matrix to solve extremum problems in 2 dimensions.  Still all the material, particularly that on local curve theory, is very clearly presented and will be very helpful for applied students in either kind of courses and could act as a visual supplement for a purely rigorous treatment.
  8. Elementary Differential Geometry: Curves and Surfaces  Martin Raussen AALBORG UNIVERSITY Edition 2008 (PG)Very thorough, careful and visual treatment of classical differential geometry assuming only basic linear algebra and calculus of function of several variables. Begins with a detailed review of vector algebra and geometry in Euclidean space with emphasis on the aspects needed for differential geometry, such as the determinant form of the cross product,orthogonal projections and parametrization of vector valued maps. Contains many beautiful computer and hand drawn diagrams, detailed proofs and applications to physics and geometry. There are many examples of depth and insight. Unfortunately, there are no exercises.This one quibble is a big one. Still, it's a very good set of notes and well worth using. Highly recommended.
  9. Manifolds And Differential Forms For Undergraduates Reyer Sjamaar Cornell  University 2011    (PG/PG-13)  These terrific notes have been circulating since Sjammaar posted the first draft 13 years ago and since 2006 they've more or less remained stable. I added "for undergraduates" to the title to make clear what's unique about them. Sadly,I think at most American universities,they'd probably be too difficult for undergraduate math majors unless they were especially strong students.Still, everyone seriously interested in calculus, analysis or geometry should read these notes as they are one of the most accessible introductions to the subject that currently exists. The only background needed is a strong background in linear algebra and a careful course in vector calculus in Euclidean space. A good working knowledge of Euclidean geometry and classical transformations would be helpful, but isn't necessary. Sjammaar very carefully and visually develops first differential forms in Euclidean space as multilinear functions on product vector spaces and then proceeds to develop the machinery of differential manifolds embedded in Rn .Manifolds, forms, configuration spaces and parametrizations, the exterior derivative, the real Hodge operator, the classical integration operators in R2 and R3 (Div, Grad and Curl, of course) ,pullbacks, integration of forms and the general Stokes' theorum and more. There are a ton of examples,many pictures and very good exercises. Also, although he tends to be informal at times, it never lacks clarity. This would not only be a great text for an undergraduate course, it would make terrific supplementary reading for a more advanced and abstract treatment such as Warner's classic text.  Lastly but certainly not least, Sjammaar gives many applications to both physics and geometry in a low dimensional setting which will make these notes useful to not only  mathematics students, but serious physics and engineering students. For example, he gives wonderful discussions of both the global angle functions on manifolds and a modern presentation of electromagnetic flux via forms. I cannot recommend these notes highly enough and here's hoping the author continues to make them freely available for many years to come .
  10. Geometry of Curves and Surfaces Alexander C. R. Belton Oxford University2012      (PG-13)A very rigorous,readable and careful, but terse presentation of classical differential geometry using both linear algebra and tensor notation in Rn . There are many good examples, but Belton leaves more then half the proofs as exercises and strangely,. all the pictures for the examples are missing. He leaves gaps in the notes specifically for students to fill in the proofs and pictures with computer drawings.Some solutions are given at the end, but not many. Compounding this, more difficult results are often quoted from references and not proven at all-such as the Jordan curve theorem. Very strong students presumably would like this approach and it would be helpful for those trying to prepare for prelims or course exams.But I think most students looking for a study aid or cheap course text is just going to find it frustrating and annoying. An instructor or tutor who can fill in the gaps might find it a good course text, but I'd suggest Belton fill in these gaps if he wants students to actually find them useful for independent study.   
  11. Elementary Differential Geometry: Lecture Notes Gilbert Weinstein Monash University  (PG-13) Very concise and terse notes covering all the basics of differential geometry from local curve theory through local surface theory to the the elements of the intrinsic geometry of abstract surfaces in R3 .There are very few examples and virtually no pictures, which is strange for a course that takes place entirely in low dimensions.Tensor notation is used freely alongside matrix computations, which makes the notes rather dense and dry. They're well written, but I think there's a lot better sources to learn this material from. I'd pass.
  12. Differential Geometry Class Notes Richard Koch University of Oregon March 24, 2005 (PG) Extensive, beautifully written notes on classical differential geometry of curves and surfaces with many equally lovely pictures. Requires only a good working knowledge of calculus and linear algebra. Many entertaining and fascinating historical notes, which always enrich the presentation of mathematics (well, of any hard science, really). Many careful proofs and Koch takes great pains to motivate all results, sometimes in unusual ways. To give 2 examples, the statement of  Frenet-Serret formulas connects local curve theory to transformational Euclidean geometry by stating curves with the same curvature and torsion are determined up to a Euclidean rigid motion. Also, many applications of the Gauss-Bonnet theorem are given to non-Euclidean geometry as well as deriving the classification theorem of
  13. compact surfaces in the case of constant curvature. He also gives many good references at the end, including the excellent Geometry From A Differentiable Viewpoint by John McCleary, whose historical perspective clearly heavily influenced these notes.  Unfortunately, there are no exercises and that really hinders what otherwise would have been one of the best free sources available for an introductory differential geometry course. Still, an excellent source and I would still recommend it wholeheartedly when supplemented with exercises. Highly recommended.
  14. Geometry and Topology I Differential Geometry and Topology of Curves and Surfaces Mohammad Ghomi Georgia Tech 2007(PG-13) These are the first set of Ghomi's notes for his geometry and topology courses which he's given on and off at Penn State and Georgia Tech since 2004. These are for the undergraduate level course-there's a sequel for a graduate level course we discuss here. A look at the full contents is instructive to show how this course differs from most others: Basics of Euclidean geometry, Cauchy-Schwarz inequality,curves, reparametrizations, length, Cauchy's integral formula, curves of constant width, isometries of Euclidean space, formulas for curvature of smooth regular curves, Fox-Milnor theorem, curves of constant curvature,  the principal normal, signed curvature, turning angle, Hopf's theorem on winding number, fundamental theorem for planar curves,osculating circle,
  15. Kneser's Nesting Theorem, total curvature, convex curves,four vertex theorem, Shur's arm lemma, isoperimetric inequality,torsion, Frenet-Seret frame, helices, spherical curves, abstract definition of surface, differential maps,Gaussian curvature, Gauss map, shape operator, coefficients of the first and second fundamental forms, curvature of graphs,Gaussian curvature as a measure of local convexity, ratio of areas, and products of principal curvatures,intrinsic metric and isometries of surfaces, Gauss's Theorema Egregium, Brioschi's formula for Gaussian curvature,Gauss's formulas, Christoffel symbols, Gauss and Codazzi-Mainardi equations, Riemann curvature tensor and a  second proof of Gauss's Theorema Egregium,covariant derivative and Lie bracket,Riemann curvature tensor,induced Lie bracket on surfaces,self adjointness of the shape operator, Riemann curvature tensor of surfaces, Gauss and Codazzi Mainardi equations,geodesic curvature and applications of the Gauss-Bonnet theorem. These notes are very well written and lucid, but despite the minimal prerequisites, quite a bit more sophisticated then most notes at this level . Ghomi strongly emphasizes the connection-no pun intended-between curve and surface theory, basic  topology and the classical Euclidean and non-Euclidean geometries, particularly spherical geometry. He also covers a number of results that are usually considered too difficult for a first course, such as the Fox-Milnor theorem and  Brioschi's formula for Gaussian  curvature.They're concise yet extensive and also quite a bit more challenging then most courses at this level.  The assumed background isn't explicitly stated, but reading them, it's hard to imagine any student being able to get far with them without at least a strong background in calculus, linear algebra,basic topology and Euclidean geometry.  The scope of the notes is largely expanded by the many, many exercises-and while most of them are straightforward, none of them are softball questions. They all require some thought and a number are very difficult. They also have surprisingly many pictures for such a formal course. They're not for the casual reader, but if you want to teach or take a serious course in this very important material, this is a very good place to begin. It would also make a fine review or supplement for students about to embark on a graduate course. Highly recommended.
  16. Elementary Differential Geometry Hovhannes M. Khudaverdian University of Manchester Lecture Notes 2010  (PG)This is is a course in differential geometry at Manchester that assumes a good command of both calculus and linear algebra. Again, this is a course in the Cambridge style: rather concise and challenging, covers a lot of ground efficiently but very clearly. However, it has many good examples and the author is very good at emphasizing the points that lead to better understanding of other points. For example, the discussions on curves in the plane are unified by the central example of the distance, velocity  and acceleration vector traces.Khudaverdian informally motivates the idea of parametrization and reparametrization as follows: Two curves are equivalent curves (belong to the same equivalence class) if  these parameterised curves ( paths) have the same images. We come to equivalent curves if we consider the movement along the same trajectory with different speeds. Non-parameterised curve|it is trajectory of point. Or in other words: two equivalent curves have the same image.  This intertwining of rigorous definition and informal description is a hallmark of the notes and is very effective. All the standard topics are covered in this manner, with a number of exercises. Sadly, there are no pictures and the homework exercises are missing online. Still, a strong set of notes for a first course, quite readable and informative. Highly recommended.
  17. Introduction to Geometry H.M. Khudaverdian University of Manchester 2013 (PG) This is a 2nd year U.K. university level course  on geometry by Khudaverdian that contains a large intersection with the differential geometry notes above. The prerequisites are about the ame, except these notes require in addition a good grounding in basic Euclidean geometry, like a good high school course would give. The notes are interestingly structured-some parts are written in standard prose like a textbook and some are much curter and telegraphic, almost in "bullet point" presentation style.(I only bring it up because most substantial lecture notes are written in either of these 2 styles, but this is the first I've seen that does both. ) There is a great emphasis on linear and multilinear algebra in these notes-differential forms in low dimensional Euclidean space are developed in detail, in particular 0 and 1-forms. Classical transformation geometry is developed entirely as linear operators in Euclidean space- which isn't unique to these notes, of course, but it's done very nicely and neatly here. Much of the material on curves and surfaces in the earlier notes forms the last part of the course- again, with emphasis on the central example of the velocity operator and it's related maps. There are many examples, many clear and  quite illuminating. Except for the organization of the classical differential geometry material, which has already been commented on above, there's really nothing strikingly original here. But there's nothing wrong with presenting standard and important material well and the author certainly does that here. Recommended.
  18.  Differential Geometry: CURVES AND SURFACES IN R3 CHUU-LIAN TERNG Prelininary Draft University of Peking /University of California Irvine 2003 (PG-13)
  19. Differential Geometry:Curves and Surfaces, Part I by Chuu-Lian Terng, Winter quarter 2005 Department of Mathematics, University of California at Irvine (PG)
  20. Differential Geometry:Curves and Surfaces in R3 , Part II by Chuu-Lian  Terng, Spring quarter 2005 Department of Mathematics, University of California at Irvine (PG) These are 2 versions of essentially the same notes. The first is an earlier version that differs from the later version only in being somewhat briefer with somewhat more emphasis on analysis. The second 2 links are to 2 halves of a year-long undergraduate course in differential geometry on curves and surfaces at UCI. This is another very formal, "algebraic" presentation of classical curve and surface theory with no pictures and lots of good exercises. Very similar in approach and content to Khudaverdian's or Ghomi's notes. It's all presented very clearly and nicely, albiet in a dry,matter-of-fact manner. If you like that approach, you'll like these notes. Personally, I'd like either Shifrin's, Koch's or Hajlasz's notes a lot better. But worth a look.
  21. Undergraduate Differential Geometry Piotz Hajlasz University of Pittsburgh 2012 (PG)  This is the first of the marvelous lecture notes I’ve discovered of the author-I plan on using them myself extensively when prelims come. The author is a well-known analyst at The University of Pittsburgh who does something that sadly, very few professors in the United States do anymore who  are engaged in serious research: He writes his own “textbooks” for the courses he teaches i.e. complete sets of lecture notes for either handing out or for his students to download in scanned form from the internet. These are his notes for the undergraduate differential geometry course which presumes just basic calculus and a good background in linear algebra. They form a subset of  the author's much more extensive graduate notes on the subject. The content of the totality of these notes is astonishing. While very little in any of these notes is original, they are some of the clearest, most detailed notes on classical differential geometry I’ve ever seen. They give the impression of a master teacher at work, who not only inspires his students with such wonderful lectures, but takes pride in doing so. I hope my inclusion of his notes will make them more well known to people engaging in serious study in analysis and geometry.When I begin my publishing company someday, he’ll be one of the first people I contact to put out inexpensive editions of his lectures. The highest possible recommendation.
  22. Differential Geometry of Cuves and Surfaces B. Shultz University of California Riverside 2012 (PG) An old fashioned but well written set of lecture notes for an undergraduate differential geometry course. Requires quite minimal background-just computational calculus of several variables and some very basic linear algebra up to and including determinants. The author develops everything else he needs, such as open sets in the plane and R3 . The course actually uses Schaum's Outline of Differential Geometry  as the textbook, which I find interesting since although cheap, it's way too old fashioned for a modern course on differential geometry. To me, it looks like a set of notes written in the 1920's! But Shultz insists, so these notes are really just for supplementation.  As a result, there's a lot of holes in them, no pictures and nowhere near as many examples as one would like. But many theorems are given very careful proofs with good insights. The real value of these notes are the enormous number of historical notes and references in them-Shultz surveys virtually the entire undergraduate literature in the course of these notes with good commentary. So to be honest, although they're well written and contain a lot of good material, I think the real value of these notes is as a study guide and comprehensive overview of the literature for students and teachers. When used that way with a more standard textbook, I think Shultz' notes will be immensely helpful for students learning differential geometry.
  23. Notes on Differential Geometry for Undergraduates Robert Jantzen Villanova University Spring 2014 version  (PG)A very long, chatty set of lecture notes that Jantzen has been developing as a textbook for many years to give an accessible introduction to the subject for mathematics and physics majors at Villanova- which the author himself admits, somewhat tongue-in-cheek, is not an "elite" university. The purpose of the notes is to provide a mathematically sound but very informal introduction to differential geometry that emphasizes the geometric i.e. spatial or visual aspects of the subject to lay a foundation for more precise,advanced treatments. The subtitle of this online book is very illuminating as far as intended level and prerequisites:  a slightly different approach based on elementary undergraduate linear algebra, multivariable calculus and differential equations. The "approach" Janzten refers to can be somewhat misleadingly be called a "physical" approach. By this, I mean he emphasizes the aspects of the  local theory of differential geometry which are of greatest importance to both mathematics and physics students studying it.  One of the  ways he does this is to  use linear algebra to make a substantial and careful introduction to tensor analysis the basic language of the course. Tensor analysis is one of the most annoyingly confusing tools in modern science and just about everyone agrees it's largely made that way by the crazy way physicists teach it. Teaching tensors as multilinear functions that are elements of the dual space of Rn is the cure for this madness and Jenzten does a very good job.  His other major method is to emphasize 2 tools of linear algebra in particular: the inner product and the determinant. The inner product gives the basic Euclidean geometry of lengths and angles while the determinant really allows the generalization of these properties to multilinear and non-Euclidean aspects of geometry such as hyperbolic space and volumes. He also incorporates many, many diagrams, good exercises with complete solutions, humorous sidebars and best of all, actual physical applications! He bemoans the separation of mathematics and physics over the last century and takes great pains to reunite them. The result is one of the best written, lucid and just flat out entertaining books on differential geometry you'll ever read and it's a fantastic introduction to the subject for both mathematics and physics majors. No, it's not as rigorous as the purists in mathematics would like, but this is easily fixed by supplementing it with a concise but rigorous treatment such as the Oxford or Cambridge notes linked above. Indeed, the 2 would complement each other very well. Jentzen is to be highly commended and here's hoping the book remains free for many years to come for all students of differential geometry to savor. Very highly recommended!
  24. Differential Geometry George Sparling University of Pittsburgh Fall 2012 Course Materials  (PG)   These notes are concise and very confusingly disorganized-they don't really form a full course text on classical differential geometry. They are very careful and well written-but there's nothing here you won't find in some of the other sources here. And the formatting is ridiculous-I got dizzy chasing the links. I'd pass.  
  25. Calculus and Differential Geometry: An Introduction to Curvature Donna Dietz Howard Iseri Mansfield University 2011(PG)This is another inventive set of notes for a course in classical differential geometry that focuses on the idea of curvature and how it generalizes the idea of angles in plane geometry. The author begins with the angles of a polygon and by defining the impulse angle or total curvature as the area swept out by the unit normal vectors at the vertex  under the Gauss map, he relates the curvature to critical points in calculus. He proceeds in this manner to define the Riemann and Gauss curvatures in 2 and 3 dimensions.These notes by the author are quite rough and not really detailed enough yet to use as a text in differential geometry-but they certainly are worth reading as a supplement to such a course as a new perspective. Recommended.
  26. A Modern Course on Curves and Surfaces Richard S. Palais Brandies University Fall 2003 (R) This is one of the early computer programming driven courses in classical differential geometry by one of the world's eminent mathematicians and it is distinguished from the others by the sheer rigor, depth and clarity of it's mathematical content. In fact, it may be too difficult for the average undergraduate-it's really for serious students with strong backgrounds in real analysis and linear algebra.The course develops basic curve and surface theory using the Kleinian group theoretic approach and the deep geometry of metric and inner product spaces. Differential calculus on Banach spaces rather then Euclidean spaces is used to develop surface theory, ordinary differential equations and their solution spaces and the curvature  tensor.If it sounds rough-well, it is.The course is uncompromisingly abstract with no pictures-but intentionally, it has no pictures. The visual component of the course is supplied by the second half, which is composed of both problem sets and substantial MathLab projects. The problem sets are meaty and difficult, involving both major pieces of theory such as the Banach contraction theorem and Runge-Kutta methods of approximation along curves as well as programming code. The MathLab projects involve constructing algorithms for computer generated graphics and animations of curves and surfaces in R2 and R. I'm sure these exercises could easily be adapted to modern computer algebra systems today like Mathematica and Maple. It would be a very strong undergraduate indeed who could solve every problem and construct appropriate code for every project. That being said- for an honors course on the subject, I can't think of a better source right now. An excellent and challenging course for serious undergraduates and first-year  graduate students.
  27. Geometry II Knots and Surfaces Shahn Majid Imperial College 2014  (PG-13) A very nice handwritten set of notes for a course combining elementary courses in topology, knot theory and classical differential geometry.( I'm always impressed when someone's handwritten lecture notes are legible enough to be scanned in and posted online as a text-probably has to do with the fact my own handwriting resembles that of a baboon with multiple sclerosis. ) The course presumes a good background in linear algebra and calculus The notes cover a ridiculous amount of material and do it visually and very well- with strong exercises to boot. The author is well known for his textbooks on quantum groups which he penned while at Cambridge. These notes have the same beauty, concision and clarity of those texts as well as command of the various interrelations of the subject matter. For example, he develops the whole of basic knot theory using knots as the canonical example of a simple closed parametrized curve in the plane or space and then uses those definitions and examples when moving on to a detailed study of more general curves. He finishes with a very good introduction to surfaces with geodesic curves. An excellent course and well worth checking out. Highly recommended.