Interesting Topic for Proofs Class

Question

I am teaching one of those university level classes where students learn enough proof theory to do their higher undergraduate courses. In such a class, you need things for the students to prove. What seems to be the norm in textbooks is a little bit of number theory (divisibility, odd/even numbers, etc.), some combinatory reasoning, and modular integers. My question concerns finding a subject of a more geometric flavor. It could be a subject from algebra, topology, or analysis, just as long as it is less discrete as the other subjects. Graph theory is taught in some books, but that boils down to combinatorics.

I understand if this question is removed for not meeting the standards of this site. I'm just hoping that it is interesting enough to the community to remain. Thanks.

Answer 1

Bezier curves (quadratic, and then cubic ; no need to go further) and splines curves provide a wealth of interesting subjects. The associated proofs are often short and rarely difficult and mix geometry (barycenters, 2D or 3D), analysis (parameterized curves), linear algebra (systems solving, matrix exponential, description of "Bezier patches" in 3D). The success is guaranteed under the condition that the students can program some of the techniques they have seen (I have tested it several years with Matlab programming).

There are many more subjects. I will hopefully come back later.

Answer 2

Perhaps the Kissing Circles problem?

I will try to update the answer if I think of more.

Answer 3

You can disguise a lot of problems coming from linear algebra or analysis by suppressing terminology. For example, rather than asking students to prove some $U $ is a subspace of a vector space $V $, have them prove closure of $U $ under addition and scalar multiplication. This is something which practices element chasing type arguments, for example. In a similar way, one can investigate whether some space $X $ is a Banach/Hilbert space. Some interesting examples could be $\mathbb{R}^n$ with the $\ell^p $ norms.