@Balarka, a child prodigy for sure, I find that interesting. When I was 14 years old, I was proud of the fact that I understood the theory of quadratic equations and the binomial theorem.
The problem was to find a polynomial, f(x) = g(x)h(x) with g and h over Z[x] such that coefficients of f(x) are all nonnegative and smaller or equals to 10.
@Ted I just mentioned Lie groups when discussing areas of mathematics with @Balarka. I think they would be good for him to know -- that was my context. Then, when he has mastered Lie groups, he can teach me about them.
Sure, Lie groups are important to know, but knowledge of a large variety of fields makes you more competent at applying their tools to your problems in general
I've always been proud of having a broad background in math, even if I wasn't a superstar in a narrow way as some have been. And then we have people like Griffiths and Tao who are super-broad and super-stars.
@Balarka, when you get to your 20s you should try to be more generally well-rounded :)
@Balarka ... When you get older, you'll discover that, in fact, differential topology and differential geometry have lots of interaction with number theory.
My thesis was in complex integral geometry, @Paul, and the warm-up for that field is Crofton's formula, which tells you that you can figure out the length of a curve by finding (in some sense) the average number of intersection points with lines.
A grad student some number of years ago countermanded his adviser and took my year-long differential geometry class because he knew he wanted to understand the connections with number theory.
yeah, @Mike, @Balarka wins on that one.
@Mike, many people think anything with the word "manifold" or "derivative" means differential geometry. Not so.
Hodge theory for Riemannian manifolds and Hodge theory in algebraic geometry are really very different.
Granted, the origins are the Laplacian and Hodge. But there is a big difference in what one studies. So don't lump anything with a differential form into differential geometry.
@Balarka, I wish half the undergraduates in my senior-level differential geometry course knew half as much math as you do. You might not know what you need for my course, but neither do they, even though it's all prerequisites.
But the second he was like "so looking at $1 - 1 + 1 - 1 + 1 - 1 + \cdots$, the partial sums are all either $0$ or $1$ so clearly we just take the average"
@Pedro: To start taking courses that will challenge you instead of the stuff you basically know. I would argue you should not shun computational skills, but otherwise you belong in courses 2 years ahead of what you're taking.
yes, @robjohn, of course, but even with our inconsiderable mass, we're outweighted by the young'uns ... It's just the 14-year olds that know more than most math majors who are really the factor :P
I'm doing Spivak with a high school senior who meets with me for 1 1/2 hours a week and is learning more and doing better than almost all students who've taken the course from me (13 times teaching it) with 4 hours a week. Exceptional talents are much fun.
Our regular college class is actually dying. So many kids take the BC course, there aren't enough students to take our Calculus with Theory class anymore.
Yes, @nsanger ... I still lecture at him, and we've done some applications that aren't in the book. But I've pushed him hard on the theory, too.
I'm starting to get annoyed at Spivak though, since I bought it to do truly rigorous Calculus, but it seems I would need measure theory to do it completely well.
I know, it just bothers me when Spivak makes claims where I'm like "yes that's probably true... but we have none of the tools to prove that so we can't really assert it yet."
