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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Sign up to join this communityWhat'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!
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.
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.
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.
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.
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.
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.
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.
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.
I had a course on "Asymptotic Enumeration" that was an advanced graduate level course that was fun and wish had an undergrad form.
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):
To have a hard time : not all the problems are easy to solve.