Question

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

Answer 1

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.

Answer 2

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).

Answer 3

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.

Answer 4

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.

Answer 5

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.

Answer 6

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.

Answer 7

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.

Answer 8

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

Answer 9

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.

Answer 10

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.

Answer 11

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.

Answer 12

I see my previous answer got a negative vote. Can someone explain why? This is a very serious issue. The mathematical community should be ashamed of the way math is presented to the public in the only place most people ever see it: Required courses taken by students who are there only because it's required.

Mathematicians say of "college algebra" courses and the like that the students should have learned the material in high school and irresponsibly dropped tbe ball and neglected to do so. Therefore mathematicians don't want to devote effort to it. They assign the most inexperienced people in math departments to teach it, or they assign it to people with Ph.D.s in education who are brilliant at teaching technical skills, and do brilliant psychological research, but don't know that math doesn't consist of technical skills any more than music does (music requires long hours of honing technical skills, but everybody knows that's not what it's all about). Or when the latter do know it, it's just something they've heard, not something in which they have professional expertise. There is work here that ONLY mathematicians can do, that ONLY mathematicians have reason to want done, and mathematicians have been unwilling to do it.

Answer 13

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

Answer 14

Math and design: let's solve the world we have trashed.