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What's one class that mathematics that should be offered to undergraduates that isn't usually? One answer per post.

Ex: Just to throw some ideas out there Mathematical Physics (for math students, not for physics students) Complexity Theory

Michael Hoffman
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  • Can someone here explain why my first answer below got a negative vote? I raised a more serious and important question than whether calculus on manifolds or asymptotic enumeration, etc. etc. etc., should be taught.

    Is a truthful statement that there is a crucially important part of the curriculum that mathematicians should be ashamed of unwelcome within math overflow?

    Or if someone disagrees with it or thinks it's not important, can they say so in words in a dignified manner?

     – Michael Hardy Jun 15 '10 at 02:38
  • Your answer is polemical and not too the point, it does not answer the question. – Michael Hoffman Jun 15 '10 at 23:46
  • I liked the question originally, but in view of the latest answer I fear it has reached the point where it is not productive (signal/noise ratio), and so I am casting the final vote to close. – Yemon Choi Jan 08 '12 at 20:11
  • My undergraduate mathematics classes involved zero critical thinking and zero play. I don't think which subject matters, but typical areas could be approached so that students ask & answer questions like "Why do we want it to be this way?", "What would happen if we did it another way?", and "What would be the pro's and con's of doing it that way instead?". Time for this could come at the expense of covering more material. – isomorphismes Sep 14 '15 at 19:58
  • Generations of people try to find antiderivatives, inverses and equation solutions in terms of elementary functions and/or Lambert W although these don't exist. The theorems of Liouville, Ritt 1925, Lin 1983 and Chow 1999 about solutions in finite terms or elementary numbers should be taught to undergraduates of all disciplines. – IV_ Jan 01 '22 at 14:54

42 Answers42

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Programming. I think it varies a lot from department to department but some places seem to do a bad job of teaching programming and it can be a really important skill.

Kim Greene
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    I think programming is not only useful to be able to compute something with the aid of a computer, but programming implies a special way of thinking about problems. Debugging experience and software engineering has helped me much in mathematics. (So, my +1) – Konrad Voelkel Dec 07 '09 at 16:08
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    I would also add some bit of "Experimental mathematics" to programming---excellent combination. – Suvrit Aug 25 '11 at 23:55
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    And especially, programming well. There are a lot of techniques and tricks that make programming much less painful: source version control, how to write tests, good comments vs bad comments, how to refactor code so that it's readable and "makes sense", use the right language for the right problem and don't worry about premature optimizations... – Federico Poloni Aug 26 '11 at 07:43
56

I would have loved a class on how to write mathematical papers and what goes into mathematics research. Everything from neat Latex tricks to how to organize and structure ideas, theorems, etc, going over bad vs good papers, even perhaps discussing what makes good math books. As well, an overview of what goes into a PhD thesis would have been extremely useful. It's a shame that one usually has to pick up these various bits of info on their own. The class could coincide with a current undergraduate senior project for example and act as a supplement. I took a similar class like this in the physics department and it helped me immensely with my work.

Alex R.
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  • I hope more mathematics bloggers write on such subjects for us who haven't had such a class. – Yoo Nov 05 '09 at 10:16
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    There's not a whole lot of magic when it comes to writing good papers or books. Usually the magic is an immense amount of hidden effort. Allen Hatcher's "Algebraic Topology" is considered a pretty good book. I have dot-matrix printouts of some of the first versions, dating back to about when Hatcher arrived at Cornell. So the book represents about 30 years of teaching algebraic topology, and many iterates of revising the lecture notes. – Ryan Budney Nov 06 '09 at 22:43
  • Allen Hatcher's "Algebraic Topology" is considered a pretty good book. <- Experience with its Leray-Hirsch section tells me otherwise. But it's in one of the appendices; the main sections seem to be much better. – darij grinberg Mar 15 '10 at 11:11
  • Such classes do exist in at least some places: at MIT, we have 18.100C, which is an introductory analysis class (typically using Baby Rudin) with an additional hour of class time each week devoted to communication tutorials; they learn LaTeX and have writing assignments in paper format in addition to their weekly problem sets. In addition to the math part of the course staff, we get a lot of help from an instructor from the "Writing Across the Curriculum" office who specializes in math and related writing (but is not herself a mathematician). – JBL Mar 18 '10 at 15:45
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    I now wish I had such a class as an undergrad, incidentally, although I think some of the undergrads actually in the class view it (incorrectly!) as "fluff." (Many students take 18.100B, which gives the analysis without the writing.)

    I disagree with Ryan Budney: just like giving presentations, or teaching, or even conducting research, there are lots of specific, teachable skills that can improve one's writing. Before helping to teach this class, I could diagnose a paragraph as unclear, and with enough work I could fix it. ...

     – JBL Mar 18 '10 at 15:50
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    The content of this class includes both a language to describe certain types of bad writing, and tools for quickly identifying and correcting what about it loses the reader. In other words, there are techniques to make good writing easier, and people who are good writers typically either know these techniques or have a strong intuitive feel for them. Just as is the case with any other endeavor, teaching the techniques makes it easier to do it well, and having really good heuristics lets one write better without having to work any harder at it. – JBL Mar 18 '10 at 15:58
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Motivation.

I have seen many students dropping out of math because they didn't get an answer to the question "why should I learn this?". Of course, one could say, a good student should have intrinsic motivation and/or figure out the motivations by himself, but this seems to me like wasting potential.

I don't want to say that mathematics courses don't provide any motivation, but in undergraduate courses (and even textbooks), especially in linear algebra and calculus (when there isn't so much time), I haven't seen enough motivation.

This motivation should go beyond "we want to model the physical world" and/or "with this theorem you can calculate the Eigenvalues". Students need the story between these two extremal answers, they need to know how calculation of Eigenvalues is really applied in modelling the physical world. (This is just an example, I would appreciate to see more motivation for abstract, non-applied mathematics, too)

Konrad Voelkel
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    When teaching eigenvalues the textbook I used gave a simple predator prey example and then looked at the diagrams of the phase space . (I think it was woodrats and spotted owls (oops that is a prey-predator problem!)) The oscillations in population levels could be seen to be interesting as a simple explanation of things students have heard about. That lead o to how on earth can you calculate these eigenvalues and eigen vectors and to some numerical methods. The point was that the example chosen was simple enough to comprehend without knowing a whole lot of some other subject area. – Tim Porter Mar 14 '10 at 19:38
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    Two points. 1) To make this happen, you need to get motivation into the textbooks. 2) I don't think motivation always needs to be real-world application. Just two sentences foreshadowing how a new theorem will be used or a motivational example problem would do it. Stephen Abbott's book "Understanding Analysis" does this wonderfully... and it spoiled me for other textbooks, I'm sure. – Neil Toronto May 11 '11 at 17:21
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Inequalities!

I don't think I've ever seen a course on inequalities, and there's certainly enough elementary material to cover in a one-semester course. Very few undergrads know much about inequalities.

Jason Polak
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    I agree. And the book "Cauchy-Schwarz Master Class" by Michael Steele would be a great textbook for the class. – John D. Cook Nov 06 '09 at 22:41
  • @John: I'll take you one better,John-if such a course was required of all incoming freshman and it was made a prequisite to real analysis,I'll bet thier performance in that course would be VASTLY improved. The transition from exact solutions to finding bounds is usually what trips up most math majors in thier first rigorous analysis course.This would greatly allieviate this. – The Mathemagician May 07 '10 at 05:17
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Approximation/asymptotics. It amazes me how many otherwise good students don't have a sense for which parts of an expression are large and which are small.

David E Speyer
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  • At least at Wisconsin, we teach this (big O, little o, and all that) in the required math course for computer science majors. – JSE Nov 04 '09 at 05:22
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    My EE education in Romania was 10% about how to approximate. Of course, that's not an exact figure. – rgrig Mar 15 '10 at 10:45
  • Absolutely. I have no idea how to use any of these notations! – Harry Gindi May 20 '10 at 01:24
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Category theory!

Andrew Stacey
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    IMO category theory is best kept until the student has a context for it. So picking up bits and pieces as they learn algebraic topology is a natural thing to do, making it graduate level material for most students. – Ryan Budney Nov 06 '09 at 22:40
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    I disagree completely. I'm teaching a course this semester that introduces, in quick succession, metric spaces, normed vector spaces, hilbert spaces, and abstract vector spaces. So often I've found myself essentially teaching about categories without actually saying the words. And these are mainly non-mathematics students. – Andrew Stacey Nov 07 '09 at 21:32
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    I think there is a big danger in making undergraduate classes (especially classes with a strong algebraic flavor) consist entirely of definitions and trivial lemmas. A good class needs some deep theorems! Dwelling on things like category theory often ends up making this problem worse. – Andy Putman Nov 09 '09 at 08:04
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    Okay, so make it "category theory done right"! But I wouldn't teach a class that was called "Category Theory". That would be like offering a class in, oh I don't know, "Rings and modules". What's the point of that? But category theory could easily be introduced into a more general course well before they reached the "Algebraic Topology" stage where we can finally use the word "functor" without shame. And I disagree completely with your statement "A good class needs some deep theorems". No! A good class needs a story. – Andrew Stacey Nov 09 '09 at 08:54
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    Hi Andrew, I understand where you're coming from but what real advantage would you get by having categories around at that level? I tend to avoid terminology unless it's necessary for a major theorem. In that regard, most everything categorical at the undergraduate level can be de-categorified without any real loss. Without the language of category theory, students see the trend building up. My experience is that most students that are inclined that way teach themselves basics of categories without any formal instruction, nor do they even need to look at a book. – Ryan Budney Nov 09 '09 at 23:59
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    Gosh! Two words can really spark a debate! My serious point is that there is a tendency to view mathematics courses as separate entities. This is particularly the case in universities that run on modular systems. It can be hard for students to see mathematics as a single thing within the morass that we throw at them. If we allow the ideas of category theory to infect our courses, there's a chance that we give the students the opportunity to see how things relate to each other. Exactly how to do this is more complicated (and I'm running out of charac – Andrew Stacey Nov 10 '09 at 07:51
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    Using categories achieves this unification at only a superficial level. A better strategy is to have "capstone" courses on subjects like, say, semisimple Lie algebras that are pretty accessible but use real ideas and results from several different undergraduate areas. – Andy Putman Nov 11 '09 at 06:51
  • By the way, I spent a few minutes confused as to why Ryan was complaining that I liked category theory too much! You have a great name, Andrew... – Andy Putman Nov 11 '09 at 06:52
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    If I were feeling belligerent, then I would say that the sentence "Using categories achieves this unification at only a superficial level." shows that you've not grokked category theory. However, I'm not, so I shan't. As I said, I wouldn't teach a course called "Category Theory", but then I'm equally suspicious about "Rings and Modules". My feeling is more that we deliberately go out of our way to avoid talking about categories and so miss out on a wonderful opportunity to show students that mathematics is really one subject and not a load of disconnected, disjointed, random topics. – Andrew Stacey Nov 11 '09 at 07:50
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    And (how annoying is this character limit!) I think a course like "semi-simple Lie algebras" could, if done right, be a great way to bring topics together (if done badly it could be down right awful!). There's room for both strategies, indeed for lots of different strategies. Just as not everyone likes topology (hard to believe, I know), not everyone likes learning from case studies. One of our roles as educators is to make ourselves redundant (in any particular student's life, that is) in that we equip students with the tools to go out an learn for themselves. – Andrew Stacey Nov 11 '09 at 07:53
  • Oh, and same to you, Andy! – Andrew Stacey Nov 11 '09 at 07:54
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    The point is that category theory does enable you too see how much of what you are doing is trivial. It is a huge conceptual organizer: Free groups, tensor products, sheafification, ... These things used to be hard to understand (and compute with!) before people routinely used universal mapping properties. You don't have to teach a course on category theory, but emphasizing universal properties first, before giving the crazy constructions particular to each subject area, is a great unifying tool that helps to organize any subject. – Steven Gubkin Nov 12 '09 at 19:44
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    Hi Andy. There's nothing wrong with liking category theory. But I don't see it as something that'd be very useful as an undergraduate course at many institutions. Here at U.Vic at least, there would be no natural place to put such a course into the undergraduate curriculum because you'd have so few, and largely really boring examples. IMO categorical notions are better to be tossed into the mix in algebra, topology, analysis and geometry courses. Most kids that get that far can start connecting the dots on their own. – Ryan Budney Nov 18 '09 at 04:36
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    Ryan, I agree about not having a dedicated course on category theory. But then I'm not convinced about the value of a course on, say, groups or rings-and-modules. I think we compartmentalise our courses too much and leave it up to the students to see the connections. However, in the case of category theory (and also set theory) there's a tendency to not even mention it when it would be useful to do so because it's "too abstract" and so, by implication, too difficult for our poor students to cope with. – Andrew Stacey Nov 18 '09 at 07:48
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    IMO it's not so much an issue of being too abstract, it's an issue of wanting to have courses where there's a deep, sustained development. A major theme where you start from something basic and build up a theory with some depth. Groups, rings and modules give you an avenue for that kind of development -- start off talking about the notion of groups, you can build up a familiarity with mathematical formalism and eventually get to Galois theory, or the classification of modules over PIDs, Jordan Canonical Form, etc. – Ryan Budney Nov 18 '09 at 22:27
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    Courses where you touch lightly on many topics run the risk of students not learning anything. I want courses to be things where there are multiple layers of redundancy, so that if something doesn't stick, the students have several other passes at the same ideas from other points of view. I think that's much of the reason people choose a narrow, deep topic, rather than many "unifying" topics. – Ryan Budney Nov 18 '09 at 22:28
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    It is downright dangerous to do abstract nonsense before you have enough examples. Unless you have a really amazing taste you'll figure out that worthy mathematics is only what's true for (almost) every category, simply because these are the only things you know how to do, and you have no examples for anything else. – David Lehavi Dec 07 '09 at 12:09
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    I tried an interesting experiment some years ago. I introduced discrete event systems and Petri nets into an OR course. The version of `linear algebra' that that needed was over the natural numbers and positive reals since negative quantities of actual things to be produced (I will have -5 icecreams thank you!) don't make much everyday sense. Students did seem to get the point that we had to think of vectors, linear transformations etc in a new way. I actually was feeding categorical methods into things in a light way to get definitions that worked at the PRACTICAL level. – Tim Porter Mar 14 '10 at 17:50
  • (to continue) Although a related topic was only done in student projects, I did get students calculating in the (MAX,+) semiring i.e. tropical algebra! Again they had to grapple with rings and modules in this context and some categorical style definitions in order to understand how to construct quite a neat model for a simplified real life system. These are good examples of ALGEBRA at work and as Andrew seems to say you can treat the constructions categorically without making a fuss about it. We may tend too often to go for very safe options on traditional lines! – Tim Porter Mar 14 '10 at 17:56
  • Maybe a functional programming course using Haskell would suffice instead – galois Apr 25 '16 at 02:23
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I think undergraduates should take problem-solving classes. I don't think such classes are widely available, but bright students who didn't do a lot of problem-solving in high school would definitely get a lot out of them.

Qiaochu Yuan
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    I agree that problem solving is good. But the current phenomenon of having "problem solving" classes implies that the rest of mathematics is not about problem solving. Problem solving should be incorporated into the standard curriculum. – Michael Lugo Nov 03 '09 at 18:27
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    Fair point. What would be ideal is if classes more strongly emphasized problems and less strongly emphasized the stating and proving of theorems without motivation. – Qiaochu Yuan Nov 03 '09 at 18:35
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    To be contrarian, I've found that problem solving classes tend to be prep classes for exams like the IMO and the Putnam. Although you can teach students to do well on exams, I think there's better things you can do with their time. – Ryan Budney Nov 06 '09 at 22:24
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    I was thinking more along the lines of the third excerpt from Halmos here: http://qchu.wordpress.com/2009/08/05/halmos-on-writing-and-education/ – Qiaochu Yuan Nov 06 '09 at 22:47
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I'm going to change the question slightly. What topics do we all think are taught in undergraduate mathematics, but often leak out of the curriculum so that students see too little of them? I have in mind the standard situation at large research universities, where there is a mix of good and not-so-good students.

My pet peeves:

  1. Complex numbers as they should appear in standard calculus and linear algebra. They tend to be postponed to upper division courses. Complex numbers greatly simplify both trig identities and partial fractions, but calculus students aren't told.
  2. Complex analysis. It tends to float to the top of upper division and disappear.
  3. Full multivariate calculus: The Jacobian of a general change of coordinates, the derivative of a multivariate inverse function, maybe also the multivariate Newton's method. The calculus sequence often chickens out and just does special cases of the first of these.
  4. Higher-dimensional Euclidean geometry. Like, the definition of an n-cube and the fact that it has 2n vertices.
  5. Multivariate probability, especially with both discrete and continuous features.
Greg Kuperberg
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    Linear algebra. At least, it's not taught to math majors. They tend to see little bits of it in the calculus sequence (in order to solve systems of ODEs, for example) and then again in an abstract algebra class (matrices being a nice source of examples of groups) but never see a coherent treatment of it. At least this is true at the universities I'm familiar with. In both cases there is a linear algebra course mostly taken by nonmajors, but it's not possible for majors to get credit for both linear algebra and first-semester abstract algebra (basically the group theory course). – Michael Lugo Nov 04 '09 at 13:31
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    Adding emphasis to your comment (3), I find especially in the States there's a rather artificial distinction between "analysis" and "calculus". To the point that calculus is seen as mindless symbol manipulation, but in analysis thought is allowed. John Hubbard's textbook does a great job of blurring those boundaries, especially when it comes to things like the inverse function theorem and Newton's method. – Ryan Budney Nov 06 '09 at 22:32
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    My son took a really good AP calculus course at the local high school, using the most standard textbook and in some sense the most standard syllabus. So I've seen it from the other side. I don't think it's fair to call it "mindless symbol manipulation". But it is true that American universities sell calculus short in various ways. One reflection of that is that a good AP calculus course can be harder than the university product. – Greg Kuperberg Nov 06 '09 at 22:59
  • @ Micheal It sounds like you weren't trained in the states,Micheal.A lot of universities in the US DO offer both kinds of linear algebra courses.Sadly,what's missing and would be great is a FULL YEAR COURSE in linear algebra.There's so much students don't see because a semester just isn't enough to cover much of this vast topic. – The Mathemagician May 07 '10 at 05:12
  • @Michael: We have done the exact opposite. Our sequence covers linear algebra in the second semester, with a bit of the overflow into the third semester. However, I have never been assigned a system of ODEs problem in my entire career (of being a student). – Harry Gindi May 20 '10 at 01:33
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    In Germany, linear algebra is emphasized very much. Every university I know has a compulsory one-year linear algebra course along with the one-year analysis course (whose second semester is devoted to multivariate calculus) in the first year. Complex numbers are usually one of the first things which are introduced. Sometimes, they are even introduced in the first week of both courses since the professors talk less to each other than they should ;). But then, in most courses in analysis one is not really told what to do to solve an integral. – Lennart Meier Jun 16 '10 at 08:35
  • In my undergrad, after 2 classes on calculus, we had complex analysis (in the 2nd year of undergrad)---I still like that subject a lot, but unfortunately, did not get to meet it again, and have largely forgotten the best bits. – Suvrit Aug 25 '11 at 23:48
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Following Greg's lead, I wish that undergrads who want to become math majors didn't "skip out" of differential equations classes (in their eagerness to get to the good stuff like Drinfeld chtoukas, or whatever). I can't think of a more important foundational subject that tends to be systematically avoided by the "best" undergraduate mathematics students.

Danny Calegari
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    I completely agree!!! Lots of places have "rigorous linear algebra" type classes (I'm teaching one right now) -- I wonder why more places don't have a differential equations class geared for math majors. There are even nice books available (like Hurewitz's beautiful little book, or Arnold's). – Andy Putman Nov 12 '09 at 19:35
  • I wonder the exact same thing,Andy.And when they ARE offered,it's usually taught by people that haven't seen differential equations since thier qualifying exams. – The Mathemagician May 07 '10 at 05:02
  • Of course I agree. But it should also don't hide that their are also functional and difference equations. – ogerard May 18 '10 at 22:33
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    I think part of the problem here is the types of course offerings universities have. For example, when I was an undergrad at U.Alberta, there was an honours intro ODEs course, but it was taught like a service course and was largely non-rigorous. It's difficult to attract students to material when the curricula is presented in such an unflattering light. – Ryan Budney Aug 31 '11 at 15:21
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Computer Science. I know programming has been said already, but computer science isn't programming. (There's the famous Dijkstra quote: “Computer science is no more about computers than astronomy is about telescopes.”)

There is a vast and beautiful field of computer science out there that draws on algebra, category theory, topology, order theory, logic and other areas and that doesn't get much of a mention in mathematics courses (AFAIK). Example subjects are areas like F-(co)algebras for (co)recursive data structures, the Curry-Howard isomorphism and intuitionistic logic and computable analysis.

When I did programming as part of my mathematics course I gave it up. It was merely error analysis for a bunch of numerical methods. I had no idea that concepts I learnt in algebraic topology could help me reason about lists and trees (eg. functors), or that transfinite ordinals aren't just playthings for set theorists and can be immediately applied to termination proofs for programs, or that if my proof didn't use the law of the excluded middle then maybe I could automatically extract a program from it, or that there's a deep connection between computability and continuity, and so on.

Dan Piponi
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    +1 for "Computer science isn't programming" and the great Dijkstra quote. – Gabe Cunningham May 20 '10 at 01:17
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    Numerical methods should be considered a very advanced topic, even though it was historically 'early'. The beauty of computer science, as far as I am concerned, generally lies in those parts where everything is exact. The connections between a multitude of areas of mathematics and computer science seem to be exploding in the last few years - and generally with no numbers in sight! – Jacques Carette May 20 '10 at 02:23
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    I'm one of a very large and growing number of graduate students whose training didn't require any serious computer science-and who deeply regrets it now. In many ways,computer science is one vast interrelated set of applications of mathematics to engineering.It DEFINITELY should be required of mathematics students and as early as possible in thier training.There's very good reason to unite the CS and mathematics departments as many universities do. – The Mathemagician Jun 12 '10 at 01:57
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    I agree wholeheartedly. I feel that Pure Maths is more relevant to, and has more in common with, CS than Stats, and yet my department is in the middle of combining with the Stats school. – ADL Jun 14 '10 at 13:36
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    @Alan Stats will make the joint department a lot more money then theoretical computer science will. It's all about the bottom line. – The Mathemagician Jun 14 '10 at 18:11
  • I wish I could vote this up twice. – Neil Toronto May 11 '11 at 17:24
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Computational Algebraic Geometry. Something like the book "Ideals, Varieties and Algorithms" by Cox, Little and O'Shea serves as a good bridge from high school algebra with lots of computations and polynomials to modern algebra with rings and groups, without assuming knowledge of the latter.

Harrison Brown
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Charles Siegel
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  • Stillman occasionally offers a course like that at Cornell. – Ryan Budney Nov 06 '09 at 22:22
  • There was one at Penn last year, but it was a special course, not a regular one (though there is discussion going on...) – Charles Siegel Nov 06 '09 at 23:37
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    At Bangor years ago we had an undergrad course on Grobner bases etc. Ronnie Brown taught it and it seemed to go down well with the students. He says: It is claimed that mathematics is so advanced that it is impossible for students to read research papers in the subject. However, in a Bangor course on Groebner Bases, one quarter of the continuous assessment was the following: use a bibiliographic database to search for papers on Groebner bases, choose on, and write an account of its use of Groebner bases to the best of your ability and knowledge in the time available. – Tim Porter Mar 14 '10 at 18:12
  • My undergrad offered exactly this course, and I took it. – Andrew Stout May 11 '11 at 17:30
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How to write on a blackboard! At the very least, how to write so that the chalk doesn't squeak.

(Declaration of interests: this was inspired by Kim Greene's answer to Tyler Lawson's question about getting fonts right on a blackboard.)

Slightly more seriously, we should teach our students how to present their ideas well.

Andrew Stacey
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I was never offered a geometry course as an undergraduate, and there's so much lovely geometry, from Euclidean and non-Euclidean geometries, to algebraic and differential geometry, and the rest. So...geometry.

Alasdair McAndrew
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Basic logic. Coming into university we all start from different backgrounds and some of us have been taught poorly in the past and haven't had the opportunity to learn the fundamentals of conditional statements or if and only ifs, etc. For example, knowing that proving the contrapositive is the same as proving the original statement is worthy knowledge indeed! Try proving that if x^2 is even then x is even without knowing this trick.

user1447
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Courses in physics and complexity theory were certainly available in my undergrad days, and were mandatory for some undergrads. I guess it depends on the country, and possibly the college...?

One glaring omission that was prevalent in my time (and place) was number theory. It was typical for math graduates to never have seen even the statement of quadratic reciprocity, which I find crazy.

Alon Amit
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A course that just attempts to define the current research areas of maths. If the landscape is so complex, why can't undergraduates be provided with a map, so to speak, in order to begin to decipher this subject?

Andrew
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  • One section of the Princeton Companion to Mathematics consists of perhaps twenty-five articles of length ten to fifteen pages, each of which attempts to do this for a specific research area. This could be a useful resource for such a course. – Michael Lugo Nov 30 '09 at 16:21
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    I'm not sure I would have wanted to actually go to a course on such a topic. I did however read the Princton Companion in my spare time and would certainly recommend that interested undergraduates did so in order to put things in context. After all, an interested undergraduate should want to put things in context. However, in my opinion this is definitely the sort of thing that should be done in one's spare time. I think it's somehow too imprecise and intangible to be given as a course. – Spencer Mar 14 '10 at 17:24
  • I think this would be a GREAT capstone course for math majors if it could be organized properly.A great challenge for any teacher. – The Mathemagician May 07 '10 at 04:57
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Bayesian Statistics. I think it's more useful in many practical situations than traditional statistics.

Kim Greene
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    Bayesian statistics is more coherent too. You can cover the basics in a week (and spend a career looking at the implications of those basics!) – John D. Cook Nov 06 '09 at 22:46
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Basics of numerical methods: What computers can and can't do and how they operate in general.

Tommi
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  • We do a bit of that in some of our courses -- numerical methods for differential equations, for example. But we can't expect students to know much about programming so this severely limits the kinds of things you can ask the students to do. I'd love it if they had a solid C++ background, which would allow us to dive into things like arbitrary-precision number systems, interval arithmetic and such. – Ryan Budney Nov 06 '09 at 22:36
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I think a great class for undergrads (in particular, for seniors planning on grad school) would be a capstone "Comparative Mathematics" course. In my imagining, this would be a mix of math history, the "greatest hits", contrasting the fundamental objects of study and proof techniques, and an introduction to the map of modern mathematics. Think the Princeton Companion to Mathematics distilled into a semester.

Sean Clark
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  • You might be interested in the paper I wrote (with R.Brown), Mathematics in Context, A New Course, For the Learning of Mathematics, 10 (1990) 10-15. The course was student led! Ronnie and I took on whatever the student wanted to hear discussed. (With suggestions from us as well!)

    You can find a version on ROnnie's webpage http://www.bangor.ac.uk/~mas010/publar.html

     – Tim Porter Mar 14 '10 at 18:19
  • I've always thought the Princeton Companion to Mathematics should be read by every undergraduate! It's great because it exposes one briefly to a whole bunch of areas of math, which helps one both see where math is going and see if there are any areas of math they might be interested in but have never tried. Your idea of a course would be great. – David Corwin Jun 16 '10 at 08:06
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Maybe this is an overbroad answer, but I'd like to see more specialized subjects that are just really fun. Computational geometry (in the classical/Euclidean sense, not the computational algebraic geometry sense) is the example that leaps to mind -- I'm not aware of anywhere that offers it as an explicitly undergrad-level course, despite the fact that it's amazingly fun, quite simple (I suspect that bright undergrads could get to Arora's PTAS for Euclidean TSP within a semester, and certainly Christofides' algorithm is within the reach of anyone who's taken basic algorithms), and practically useful, although I guess this is more (T)CS than straight math...

Harrison Brown
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  • Again at Bangor, we used to teach Christofides' algorithm. I did it in an OR course. We also taught computational geometry in the sense you mention. The students in these courses seemed to appreciate that they could understand what the problems were (not necessarily the solutions!) – Tim Porter Mar 14 '10 at 18:22
  • Pardon the self-advertisement, but I am hoping this new undergraduate textbook will make it easier to follow your advice (and that of Alisdair McAndrew): Discrete and Computational Geometry. Satyan Devadoss and Joseph O'Rourke. Princeton University Press, to appear, 2010. – Joseph O'Rourke Jun 12 '10 at 14:46
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I am actually thinking about functional analysis and modern Fourier analysis.

7-adic
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  • If your school isn't in the top 20 mathematics programs,expect single digit enrollment if at all,7. – The Mathemagician May 07 '10 at 04:43
  • I consider my undergraduate school in the top 20, and even then we had single digit enrolment in modern Fourier analysis and "advanced analysis" (basic functional analysis, various embedding theorems etc.) – Willie Wong May 19 '10 at 21:57
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    I was a graduate student at Yale, and all but a few courses had single digit enrollment. I distinctly remember a course where I was the other student, and a course where our exchange student from Germany got stuck, being the only one attending and not wishing to offend the professor by dropping. – Victor Protsak May 20 '10 at 07:39
  • I don't know whether to laugh or be offended that Yale is that financially solvent that it can afford to do that,Victor.But I guess that's one of the advantages of attending a REAL university. – The Mathemagician Jun 12 '10 at 01:59
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Quantum mechanics, as in understanding the mathematics behind its foundational issues, and not just as in computing the spectrum of the Hydrogen atom (though that's good too).

It's hard to think of a topic that shakes one's image of the physical world harder than quantum mechanics. General relativity is easy to digest once you are not scared of things like manifolds. Quantum mechanics remains a challenge to one's worldview no matter how hard one tries to get used to it. You cannot count yourself scientifically literate if you were not exposed to the foundational issues of quantum mechanics.

And it's a math course at least as much as a physics course. The pre-requisites are basic probability and logic and complex numbers and basic Hilbert space theory, and the content is philosophy and non-commutative probability theory and (may as well, at the end) some spectra of some differential operators.

Mermin article "Is the moon there when nobody looks? Reality and the quantum theory" was an eye opener for me, the year after I finished my undergraduate studies.

Dror Bar-Natan
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Traditional Statistics. Many biology majors end up knowing more statistics than many mathematics majors which I think is a weird state of affairs.

Kim Greene
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    It's easy to teach statistics if you're sloppy, but if you want to be careful you quickly end up in deep water -- philosophical controversies, not to mention measure theory etc. So I think one reason stat is not often taught to math undergrads is that it's hard to do well. – John D. Cook Nov 06 '09 at 22:44
  • At the undergraduate level being sloppy is fine. After all, statistics is at least primarily an applied science. – David Harris May 11 '11 at 16:37
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A calculus class that goes very slowly.

Autumn Kent
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    If you don't teach calculus well, you can't have a decent math major. – Charlie Frohman Mar 15 '10 at 12:44
  • AMEN.Too bad most research based programs don't seem to realize that-that's why all thier calculus courses are taught by immigrant graduate students who can't speak English and use category theory to teach integration.....LOL – The Mathemagician May 07 '10 at 05:08
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"What's one class that mathematics that should be offered to undergraduates that isn't usually?"

OK, I'll rephrase my earlier answer. A class that should be offered to undergraduates that usually isn't is a "what is mathematics" course for those liberal-arts majors who will take only one math course in their post-secondary schooling. It would be a truthful course that would avoid telling them that mathematics consists memorizing algorithms whose utility can be seen only by taking later courses that they won't take. It would acquaint them with the fact that mathematics, like physics, is a subject in which new discoveries are constantly being made. It would tell them that one doesn't generally do math by taking a problem and feeding it into an algorithm that was given to one by a prophet who came down from Mount Sinai. It would tell them that mathematics is a subject that, like music, relies heavily on technical skills but does not consist of those alone. Among the goals would be that a student who takes only that course and becomes a professor of some liberal arts subject would not be among the many such professors who don't suspect the existence of such a field as mathematics.

Michael Hardy
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    Courses like this are offered at many universities. My understanding is that they're actually fairly common. – Charles Siegel Jun 16 '10 at 08:38
  • I think this answer is much better: +1

    Also, I haven't seen a course like this offered for undergrads where I go... they have a similar physics course, but not one in (where I think they would be helpful) CS and Math

     – Michael Hoffman Jun 16 '10 at 13:36
9

Igor Rivin has a whole diatribe on this topic!

HJRW
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  • Thanks for the link! His diagnosis is very frank, but proposed solution leaves something to be desired (no, I don't have an answer, either). – Victor Protsak May 19 '10 at 22:01
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I would love to see more differential geometry offered in undergrad. The course I envision would start with a review of vector calculus, move to studying hypersurfaces in $\mathbb{R}^n$, and then move into a study of manifolds. You could tie all these subjects together via the Fundamental Theorem of Curves, the Fundamental Theorem of Hypersurfaces, and the Fundamental Theorem of Riemannian Geometry. I feel such a course would help bridge the gap between undergrad and grad school; simultaneously reviewing the key ideas of calculus at a high level while also giving a solid foundation from which to study manifolds in grad school.

Here are some topics that could be covered:

  • Curves in $\mathbb{R}^n$

  • $k$-frames and curvature, leading to the Fundamental Theorem of Curves

  • Hypersurfaces in $\mathbb{R}^n$

  • Tangent spaces and curvature
  • First/second fundamental forms
  • Moving frames, Christoffel Symbols, Gauss Equations, Codazzi-Mainardi Equations.

  • Fundamental Theorem of Hypersurfaces, a word on curvature tensors

  • (Real) Manifolds, charts, multiple definitions of tangent vectors

  • Mention Lie Bracket and Lie Algebras (after doing derivations for tangent vectors)
  • Affine Connection, leading to the Fundamental Theorem of Riemannian Geometry

P.S. I am not a differential geometer. I study homotopy theory. I just thought a course like this was missing from the curriculum. Any feedback on other topics that could be covered or tangents that could be mentioned leading off from this material would be welcome.

David White
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I like the idea of promoting mathematics to students, i.e. more explanation of the contribution of mathematics to civilization, so that students have some language and background to justify their subject. See other articles on my web page.

On specific subjects, I have enjoyed showing first year UK students the power of the symbolic algebra packages on dealing with Grobner bases, i.e. solving polynomial equations in more than one variable. To more advanced students one can give exercises like:

Find a polynomial in x,y which has more than 5 critical points, classify them as max, min, saddle, and use the computer algebra package to draw your function and display or indicate the critical points. Verify with the package that the points you find are critical points.

(The last part gives useful lessons in rounding accuracy.) The whole exercise gives students a nice sense of power, as the machine manipulates vast expressions!

Ronnie Brown
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Having been offered a not-all-that typical undergraduate curriculum, and having then proceeded to miss a lot of it through over-sleeping, I'm not sure what is or isn't usually offered up. Does Ramsey theory (or even just Ramsey's theorem) get a mention in undergrad-level combinatorics? If not, that'd be my suggestion: about the only mathematics I've succeeded in explaining to non-scientists in the pub, from R(3,3) to the idea of lower bounds via random colourings.

Yemon Choi
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  • At Oxford the Graph Theory course has the basic Ramsey Theory stuff, and the course Probabilistic Combinatorics has Erdos' lower bound. I also definitely recommend it for places where it isn't included. – Thomas Bloom Dec 07 '09 at 10:18
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Courses aimed towards applied, rather than pure, mathematics. Like a modeling course related to environmental sciences, perhaps. Most math majors prepare the students for graduate school in pure mathematics, but offer less support for applied tracks, and there be some good careers there.

Anna Varvak
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  • I'd like to add that students should try to take these courses as much as possible from their engineering departments, physics departments, computer science departments, economics departments, etc. We want our trained mathematicians to be able to communicate with the people that use mathematics. Similarly, if mathematicians want to learn quantum mechanics, learn it from a physicist, not a mathematician. Whenever you do something, seek the beating heart of the material. – Ryan Budney Nov 18 '09 at 07:50
  • There are enough applied math courses. I'm glad that I will never have to take one. They seem dreadfully boring. – Harry Gindi Nov 18 '09 at 13:57
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    I did a course with Atomic and Nuclear Physics making up a third of the time. Apart from beautiful functional analysis and topology courses, one of my favorite courses was in Quantum Mechanics taught by an excellent Physics prof. I there understood about orthogonal functions, etc. He talked sense! The earlier courses on these subjects went straight past me. I could do the problems but did not grok them. – Tim Porter Mar 14 '10 at 18:27
7

DESIDERATA:

  • asymptotical analysis and its applications
  • analytic combinatorics
  • analytic number theory
  • complex analysis with focus on transforms (i.e laplace, inverse laplace, saddle-point method, stationary phase...)

WE NEED MORE OF THOSE BEFORE THE UGRADS THINK THAT CATEGORY THEORY IS THE PINNACLE OF MATH!!! (the last sentence is a joke, please do not get offended anybody out there).

danseetea
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mrm
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    Combinatorics seems like one of those subjects that you really have to have a natural affinity for. – Harry Gindi Dec 07 '09 at 10:15
  • analytic combinatorics is very different... – mrm Dec 08 '09 at 04:02
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    IMHO there is already too much unmotivated analysis in undergrad education. I rarely meet undergrads who think categories are the pinnacle of math, but I do meet more than enough undergrads who think that two-lines bounds involving epsilons, deltas, absolute values (as commonly seen in stochastics, diff. equations and asymptotics) are the pinnacle of maths. – darij grinberg Mar 15 '10 at 11:06
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    @Harry: I used to think so too. But then I saw analytic combinatorics, 'done right', and I learned to really enjoy it, while I hated my 2nd year combinatorics class. – Jacques Carette May 20 '10 at 02:31
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    @Jacques, which book did you use? or did you go from lecture notes? It always seems that combinatorics is almost never taught in an understandable way... – Michael Hoffman May 20 '10 at 21:14
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Caveat: My undergraduate & graduate studies were both not in Math but in Engineering. But I would love to have taken a course on the history of mathematics and I think this isn't a commonly offered course. There are lots of compelling stories here and it also gives a great perspective on how the different areas of math came to be born (non-Euclidean geometry from work on the parallel postulate to name one). Knowing how a subject evolved historically can give a nice perspective on the subject especially when a formal course on the subject might not necessarily follow the same order of ideas.

Dinesh
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    With the proviso that much of the older "folk history" is, I'm told, not accurate, or is misleading. I went to several entertaining history of mathematics lectures where this was pointed out vehemently. Also, the history of ideas is really tricky, because we have to try and understand how e.g. the Greeks thought, not how we would think about what they appear to describe. – Yemon Choi Mar 15 '10 at 06:40
6

When I was in my first year, I always missed order theory. I teached it to myself then and thought many times "Why didn't they teach us this - we would understand everything so much better!"

And I still think so today. Order theory starts off easy, when you lern about relations, preorders, lattices and so on, and then you get into Zorns Lemma, Schörder-Bernstein Theorem and stuff like that.

But I'm also on the category theory track. I think the notion of category will become, just like the notion of a group, more common sense, not just in mathematics but also in computer science, physics, chemistry and maybe even more.

Konrad Voelkel
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    That should cover that in an axiomatic set theory course-which ALL math majors should have as a required course. – The Mathemagician May 07 '10 at 05:05
  • Why should one study axiomatic set theory in a separate course? I disagree that all math majors need an axiomatic set theory course, but rather need a brief intro that you can get in an introduction to proofs class. – Michael Hoffman May 20 '10 at 21:12
  • Could be because I'm in my last year of undergraduate and haven't taken an axiomatic set theory course – Michael Hoffman May 20 '10 at 21:13
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Computing with Feynman diagrams in zero dimension, i.e., a graphical calculus for tensor contractions. It is very elementary yet can lead quickly into rather deep mathematics. It would make later studies in say mathematical physics or low dimensional topology much more congenial. Possible applications could be

  • redoing a good portion of linear algebra see e.g.: http://arxiv.org/abs/0910.1362

  • doing some basic representation theory following the formalism in the book by Cvitanovic: http://birdtracks.eu/

  • projective geometry on the line and on the plane and some elimination theory, Bezout's theorem is very easy to understand in this language.

  • computer graphics in the spirit of J. F. Blinn see, e.g., the account given in: http://www-m10.ma.tum.de/foswiki/pub/Lehrstuhl/PublikationenJRG/52_TensorDiagrams.pdf

  • since many answers in this post are about asymptotics, another application is to compute the higher order terms of the Laplace method.

  • combinatorial enumeration, e.g., a proof and examples of application of Lagrange inversion, explicit forms of the implicit function theorem, etc. etc.

Abdelmalek Abdesselam
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Once students have been exposed to linear algebra and vector calculus, build calculus on manifolds using many examples; i.e. go from real $\mathbb{R}$-abstract multilinear to the de Rham complex, illustrating in $\mathbb{R}^3$. All that easen differential geometry, differential topology Riemannian geometry, etc.

J W
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janmarqz
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    But, please, with motivation. Understanding manifolds is THE hardest part of "elementary" advanced mathematics. Just introducing them by an abstract definition and then doing proofs by "local-global" handwaving doesn't do the job; the students will neither have an idea what the definition signifies, nor why the handwaving is allowed. Maybe some non-Euclidean geometry as a nontrivial motivating example would be useful... – darij grinberg Mar 15 '10 at 11:18
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    Read John and Barbara Hubbard's VECTOR CALCULUS,LINEAR ALGEBRA AND DIFFERENTIAL FORMS:A UNIFIED APPROACH,2nd edition,to see how it's done,Jacques. – The Mathemagician May 07 '10 at 04:56
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    I actually would love to see non-euclidean geometry (from a more elementary perspective) and differential geometry unified into one course. You always see elementary books which give you bits and pieces of the idea that there are other geometries, and that the sophisticated way of doing this is differential geometry - then much later in life you get thrown into a course where you pick up from multivariable calculus and define manifolds, tangent spaces, tensor fields, etc... There should be an attempt to show the connection between the two. – David Corwin Jun 16 '10 at 08:12
  • on the beginning missing duality of finite dimensional vector spaces (preferable over the real field) garanties from minor to almost no understanding :P – janmarqz Aug 30 '12 at 18:24
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    I agree, David. Even after reading tomes like Spivak's first couple volumes, it took me decades to realize that the classical euclidean and non euclidean geometries were just those Riemannian surfaces which were simply connected and of constant curvature. Thus the natural progression would have been to learn Euclidean geometry as flat geometry, then spherical/projective and hyperbolic geometry as universal constant non zero curvature geometry, then quotients of these as other constant curvature geometries, and finally more generally curved surfaces. Nikulin/Shafarevich is a good source. – roy smith Jun 23 '13 at 19:57
4

What about "Mathematics with Computers"? Having modern computer algebra and symbolic computation tools available, one can use them to present and explore nontrivial examples in various fields of mathematics. Part of the course could also present basic algorithms and other techniques used.

Tomaž Pisanski
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  • We do offer a course like that using MATHEMATICA at Queens College,Tomaz. I've been after the department for awhile to offer it in the summertime to increase enrollment and get a cash cow. – The Mathemagician May 07 '10 at 04:44
  • Do use symbolic computation to present and explore nontrivial examples from mathematics. I don't think you'll get as much traction from the basic algorithms, and the algorithms that CASes actually use really need a lot more background [but then, they are rather interesting]. – Jacques Carette May 20 '10 at 02:29
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    I think you could fudge a bit and get a good idea of how the algorithms work, assuming they have a basic background in abstract algebra – Michael Hoffman May 20 '10 at 21:10
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What should one teach to liberal-arts students who will take only one math course, and that because it's required of them?

The conventional answer: Partial fractions. And various useless clerical skills that they'll need if they take second-year calculus, although they'll never take first-year calculus. Et cetera.

My answer: the truth.

E.g. in third grade you were told that $$ 3+3+3+3+3 = 5+5+5 $$ and so on. Why should that be true? Assign that as a homework problem. At this point they may think that means there's some formula to plug this information into to get the answer. They've been taught that memorizing algorithms and applying them is what math is. That's a lie. We should stop lying and level with them.

Andrés E. Caicedo
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Michael Hardy
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    So why the negative vote? – Michael Hardy Jun 14 '10 at 20:16
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    I posed the question and asked for a course, not a rant on the substance of mathematics education for non-mathematicians: -1 – Michael Hoffman Jun 15 '10 at 23:48
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    There is no "the truth". It's much easier to blindly criticize than to offer constructive answers: -1. – André Henriques May 11 '11 at 18:02
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    +1, if only for displaying the eternal problem about why is $3+3+3+3+3=5+5+5$. I've asked that of many bright undergrad before, and rarely got and answer better than "that's the law" (the commutative law, they mean, which we all abide by except when we do powers). – Dror Bar-Natan May 11 '11 at 18:56
  • @André: I was not blindly criticizing and I was offering a constructive answer. Maybe you should try that yourself. @Michael Hoffman: At least I was addressing a more important question than what you seem to be saying you intended. – Michael Hardy May 11 '11 at 19:04
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    Michael: answering the questions you want to pose is surely something better done on a blog. Redefining the original question to one you can then be righteous about does not strike me as very courteous – Yemon Choi May 12 '11 at 23:37
  • So you're suggesting that I redefined the question?? – Michael Hardy May 14 '11 at 04:46
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Personally, I think the answer to this question is largely going to depend on one's particularly interests (whether they lie in algebra, analysis, topology, or whatever). This can be seen from many of the previous posts.

That being said, I do think that more number theory would be a great addition to the undergraduate curriculum. Many students take an introductory number theory course (or skip it because they learned it all in high school) and then don't do any more. There are lots of great areas of number theory which don't require too much background. P-adics would be great (Gouvea even laments in his book that p-adics aren't taught earlier - so maybe such a course should use his book). One could teach a basic semester of algebraic number theory, or a course in elliptic curves (following Silverman and Tate, for example). Both of these require no more than a basic course in undergraduate algebra. You can probably find these courses at many top universities, but they usually aren't emphasized as much to undergraduates. The reason why I think that these would be good is because number theory is a particularly beautiful area of math, and by getting glimpses of modern number theory early on, students get to see how beautiful is the math that's ahead of them. (Another possibility is to have a course on Ireland and Rosen's book A Classical Introduction to Modern Number Theory. Princeton had a junior seminar on this book, for example.)

I also think Riemann surfaces are a very beautiful topic which should be taught early on and aren't too complicated in their most basic form. For, you get to see the deep geometrical theory lying behind the $e^{2 i \pi}=1$ and the ambiguity of complex square roots which you learned about when you were younger. It shows the student that there can be very deep ideas lying behind a simple observation, and it shows the beauty and deep understanding that modern mathematics can lead you to.

David Corwin
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  • Good suggestions,Davidac-but again,the students need to be very well prepared (such as at Princeton) for these kind of courses.They need to be pretty comfortable with basic algebra and rigorous calculus.Topology would help,but isn't really necessary. – The Mathemagician Jun 16 '10 at 17:20
  • Ok, but this could work as long as the student takes a basic abstract algebra course in junior year, which is common at a lot of places. – David Corwin Jun 17 '10 at 13:31
  • That is, all you need is an introductory abstract algebra course for p-adics, elliptic curves, and algebraic number theory. The mention of class field theory was just to appease the person who said algebraic number theory at its most basic level doesn't get anywhere interesting. You do need basic rigorous calculus for p-adics, but a typical undergraduate senior has done that. – David Corwin Jun 17 '10 at 13:33
3

First, statistics, indeed, is not taught enough. I studied statistics in a good school, but when it came to actually using it, found that I don't understand it. Second, motivation: they have to show the student how much and how urgently (s)he will need these concepts while on the workplace, with good real examples.

Alex
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  • I agree that there could be much improvement in how statistics is used as well as presented. However, most undergraduate curricula that (in my opinion) deserve the title have an offering in statistics, however elementary it may be. Was there a particular theorem or application you feel is not offered? Gerhard "Ask Me About System Design" Paseman, 2011.08.25 – Gerhard Paseman Aug 26 '11 at 03:44
  • For example, how to compute confidence intervals when estimating many parameters? What if regularization is used? How to pick/estimate a prior? Some basic intuition about stochastic processes; an introduction to statistical machine learning; graphical models; how to model temporal data; how to estimate if you don't have enough data for asymptotic results to be valid; robust statistics; modeling interaction between multiple features, etc. Above all, some intuition that would make me feel how to proceed, what may work and how it may fail. – Alex Aug 26 '11 at 04:28
  • Alex: shouldn't this be in a statistics degree rather than a math one? – Thierry Zell Aug 26 '11 at 11:32
  • Where I studied there was just math degree, no special statistics degree. And I didn't realize I'd do much statistics later. One thing about students is that they misplace their priorities getting fascinated by weird, fashionable or super-general stuff, and don't pay enough attention to the basics. That's where some guidance by the elders would be most appreciated. BTW, I'm not sure why one should separate math from statistics. To learn one but not the other is very dangerous for the future career. And in pure math randomness gives a different kind of intuition, just like geometry. – Alex Aug 26 '11 at 21:20
  • I see. It is certainly the case in the US that students in non-math programs (certain business, social sciences or information technology) end up learning a lot more statistics (at least in advanced degrees) than is offered to math students. The funny thing is that many such programs don't advertise it too openly, giving courses neutral names like "research methods". But of course, the statistics make students very employable in other contexts, see e.g. this New York Times article: http://www.nytimes.com/2009/08/06/technology/06stats.html – Thierry Zell Aug 26 '11 at 23:47
2

As a college student myself, I wish to study these classes when I was in my college, but they are not offered(I took most of these in Moscow instead):

  1. Algebraic topology.
  2. Real analysis (graduate level)
  3. Complex analysis (graduate level)
  4. Measure theory, geometric measure theory.
  5. Commutative algebra and homological algebra (at least Ext, Tor, etc)
  6. Riemann Surfaces
  7. An intro course in algebraic geometry
  8. Algebraic number theory.
  9. Classical Mathematical Physics
  10. Some intro course in ODE, dynamical systems (like Smale's horseshoe), and PDE.
  11. Combinatorical game theory.
  12. Elliptic curves.
Kerry
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I had a course on "Asymptotic Enumeration" that was an advanced graduate level course that was fun and wish had an undergrad form.

JB King
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To have a hard time : not all the problems are easy to solve.

guillaume
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