So I signed up to give a talk at HMMT some time back, which will be this Saturday. As expected, I procrastinated preparation for it until now. The problem is, I’m not sure what to talk about. In high school, I wasn’t really into math contests such as HMMT — my mind was never able to find creative solutions with the necessary speed, and I’d consistently turn in abysmal performances. So as a result, I was never exposed to much of the culture of high school math contests (the existence of which I found out not that long ago). Anyway, as a result I’m not completely sure how to prep this talk, or even what to talk about. Some topics that I consider talk-worthy and interesting are:

1. Lecture one of algebraic geometry class. Define varieties and algebraic sets, and state (or even prove) the Nullstellensatz. But I suspect this will use too much commutative algebra than I should assume. I understand that plenty of extremely accomplished HMMTers may not know what a ring is.
2. The p-adic numbers.  This has the benefit of my being able to recycle an old talk.  But I might have to re-tool it.
3. Quadratic reciprocity. Perhaps the proof via Gauss sums, for instance. But this is something that people will tend to know, right?
4. A brief intro to computability theory (as in — Turing machines, unsolvability of the halting problem, complexity classes, maybe say something about Kolmogorov complexity)

The basic problem is that such topics essentially amount to picking your favorite textbook on subject X, choosing five or six pages, and reading them aloud to the students — in short, a normal class. Which is probably not what they’re looking for.

But some of you readers have better ideas than I.  So, any thoughts? Pretend, or not, that you were in high school. What would you wish to know that I could cover in an hour?

(If I end up using your topic, I’ll mention you in the talk!)