Tuesday, January 25, 2005

Growing up with C-C Spaces.

At some point I meant to write a blog post about this, but the damned preparations constantly get in the way. It might as well be now.

Anyways, I'm finally giving the talk that I've wanted to give for a while:
Tomorrow: an Introduction to C-C Spaces
[click here for an abstract at Student Analysis Seminar]

It's a lot of fun and a friendly mix of many a topic. Sometimes I believe that this area is what made me decide to be a mathematician, so many years ago.

Of course, it wouldn't be a very good conference talk or even a faculty-run seminar talk. There's the faux pas that I won't be talking about any of my own research, for one thing. Moreover at every conference I've attended, everyone seems to know this stuff already .. so like a little kid trying to run with the big kids, I learned about C-C spaces, too.

With that in mind, it feels like I'll be talking only about trivial things .. yet thinking about it further, it's taken me 4-5 years to sort out the scope of this topic. Each time I learn something new, something clicks in my head from C-C Spaces.
  • some differential and Riemannian geometry,
  • a little abstract algebra for Lie groups,
  • metric topology,
  • functional analysis and the calculus of variations,
  • a little dynamics,
  • measure theory and geometric analysis,
I wonder what that means. The perfectionist side of me is convinced that I could have learned these things in far shorter a time, but just this once I'll tell him (or me) to shut up. That's not the point, though.

Now I have to explain this stuff to someone else: a small audience, in fact, and composed of my own peers ..
[scratch that: they're all quite bright and far cleverer than me (;]

I feel like that little kid again, rushing back to the playground and telling all my friends about what I saw .. hoping to tell the story right and hoping they won't laugh at me if I tell it wrong.

How do I explain a theory to my friends, when it took me years to understand it? Will they understand? Will they think it's trivial or boring stuff?

I suppose everyone likes to be well-liked, even mathematically. Perhaps I've seen too many math grad students who grumble at the thought of analysis, or undergrads who groan at the sight of calculus. After a while it gets to you and leaves you uneasy and introspective.

Anyways, back to preparing notes for tomorrow's talk. Wish me luck!


Anonymous said...

So, how was it?

Kevin said...

It was great! Really, a very fine presentation. I particularly enjoyed your "calculus" example. I'm looking forward to next weeks continuation. Thanks!

janus said...

Thanks, K. Much appreciated! Before, I wasn't sure if the topic would go well.

As for the 'calculus' example, I figured that it would be something that everyone could appreciate: geometers, algebraists, analysts, or applied people, too .. well, as long as they remembered their calculus, that is.

Once upon a time I was participating in a summer research project, and that 'calculus' example was the first exercise that my undergrad advisor assigned to me. I ended up stuck at the last step, and that was when I first heard about the notion of an Isoperimetric Theorem.

Oddly enough, that was the original goal of that summer: to prove a sharp isoperimetric theorem for the Heisenberg group, and determine the extremal set. As far as I know, the problem remains unsolved.

Oh, as for the first comment on this post: I can say for certain that there exists one member of the audience who liked my talk. Let's hope that he is not unique!

Anonymous said...

An interesting idea, to assign the Heisenberg isoperimetric problem to undergraduates. Reportedly, Nash did the same with FLT, and justified it by saying that "people have a mental picture that this is a difficult problem. Maybe that's the stumbling block. Maybe, if people didn't realize that the problem was hard, they could solve it". Wasn't Nash running that REU by any chance?

janus said...

Perhaps Nash was right. Undergraduates are amazingly capable, because they may not know the technical difficulties of a specific problem, and are willing to try anything.

When someone first showed me this problem, I thought that it could be solved in a matter of weeks. After a few years of looking at the problem, I'd say it's somewhat harder than that.