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:
- 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.
- 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.
- Quadratic reciprocity. Perhaps the proof via Gauss sums, for instance. But this is something that people will tend to know, right?
- 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!)
Climbing Mount Bourbaki 
November 1, 2010 at 10:56 pm
I think the p-adic numbers would be good if you could avoid algebraic terminology, which most of the participants won’t be familiar with.
November 2, 2010 at 6:05 am
I’ve had a quick look at past problems and solutions, in particular algebra ones, and clearly the focus is on neat proofs and useful tricks.
So perhaps you might have a talk organized around a few basic results of similar flavour, which turn out to have wide-ranging implications in more research-level math.
One example I have in mind is how the easy to prove inequality
is ultimately the basis of a proof of the Prime number theorem, see chapter 7 of Stein-Sharkashi where it’s lemma 1.4 and then used in corollary 1.5 with the zeta function. (In fact you could simply talk about that PNT proof alone, the next few propositions 2.1 and 2.2 are still ok for high schoolers and show how 
appears.)
Depends on the amount of time you have left, better talk about something you’ve rehearsed well, and something that most can follow till the end (except perhaps for a few very short technical asides for the more advanced ones)… Good luck!
November 2, 2010 at 6:11 am
This will take quite a bit of preparation, but I think the classification of surfaces is really interesting material and, with the appropriate amount of pictures and half-waving, could make a pretty engaging talk.
November 2, 2010 at 6:11 am
Er, hand-waving.
November 2, 2010 at 6:39 am
Akhil, I think you should talk about the failure of unique factorization in Z[sqrt(-5)], Z[sqrt(-3)] etc. and explain how one can use unique factorization in other rings like Z[i] to prove interesting number-theoretical stuff. Talk about lattices and how the failure or truth of unique factorization is intimately related to the geometry of lattices. That’s my suggestion.
November 2, 2010 at 7:34 pm
Being a high schooler, I can say that your presentation on p-adic numbers that you posted would go way over the head of most of my mathematically-inclined peers.
The algebraic geometry talk would seem interesting, except it would require about 10 minutes to explain the concept of a ring (I know that last year I just couldn’t grasp the concept of an algebraic structure).
Computability theory would be appropriate and interesting, although it might be too computer-science-y for a math contest.
Really, I think that just introducing the concepts of groups, rings, fields, and vector spaces could be fascinating for a high schoolers. If you wanted, you could even discuss maps between structures and possibly introduce representation theory.
It all depends on how much information you can relay in a short period of time.
November 2, 2010 at 8:34 pm
I wouldn’t recommend 1 as being too technical. p-adic numbers might be interesting if you make it non-technical (talk about them as “power series in p”) and show some applications to problems in number theory. Your ideas 3 and 4 sound fine to me.
You can also try talking about something like Hall’s Marriage theorem (which at least one nice proof and lot’s of neat applications), or Cayley’s formula for the number of labelled trees on n vertices (or more generally the determinant formula for the number of spanning trees), or Conway’s theory of games.
You can also pick a chapter from “Proofs from the book” almost at random. (Let me know if you want to borrow my copy, we could go get it tomorrow after Lurie’s lecture.)
November 4, 2010 at 10:24 am
Hey Akhil,
Dont you think that giving lecture on Algebraic students at High school, would really help them? Isn’t it too advanced.