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
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 

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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. 


Bayesian Statistics. I think it's more useful in many practical situations than traditional statistics. 


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


Basics of numerical methods: What computers can and can't do and how they operate in general. 


I think undergraduates should take problemsolving classes. I don't think such classes are widely available, but bright students who didn't do a lot of problemsolving in high school would definitely get a lot out of them. 


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. 


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. 


Category theory! 


Having been offered a notallthat typical undergraduate curriculum, and having then proceeded to miss a lot of it through oversleeping, 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 undergradlevel combinatorics? If not, that'd be my suggestion: about the only mathematics I've succeeded in explaining to nonscientists in the pub, from R(3,3) to the idea of lower bounds via random colourings. 


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


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. 


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. 


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. 


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 undergradlevel 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... 


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. 


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 notsogood students. My pet peeves:



A calculus class that goes very slowly. 


Igor Rivin has a whole diatribe on this topic! 


I had a course on "Asymptotic Enumeration" that was an advanced graduate level course that was fun and wish had an undergrad form. 


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, SchörderBernstein 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. 


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. 


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. 


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. 


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


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? 


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. 


DESIDERATA:
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). 


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, nonapplied mathematics, too) 


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 (nonEuclidean 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. 


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. 

