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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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! 


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



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


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. 


What should one teach to liberalarts 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 secondyear calculus, although they'll never take firstyear 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. 


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. 


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 prerequisites are basic probability and logic and complex numbers and basic Hilbert space theory, and the content is philosophy and noncommutative 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. 


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:
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. 


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. Padics would be great (Gouvea even laments in his book that padics 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. 


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):



"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 liberalarts majors who will take only one math course in their postsecondary 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. 


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). 


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 CurryHoward 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. 


I am actually thinking about functional analysis and modern Fourier analysis. 


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. 


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. 


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) 


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? 


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. 


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. 


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. 


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. 


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. 


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


Igor Rivin has a whole diatribe on this topic! 


A calculus class that goes very slowly. 


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:



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. 


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 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. 

