Why so few posts of late?… a thesis synopsis

May 31, 2010

Perhaps you’ve noticed the dearth of blogging lately here at mathforlove. Here’s the story: I’m defending my thesis—“On the Number of FM Partners of a K3 Surface”—this Thursday afternoon, so at the moment, I’m ensconced in preparation. Or I would be, had I not also gotten sick this past week, and been pretty much knocked out.

But not to worry! I’m almost all better now, and getting to work again. Fortunately, most of what I need to do is done. I’ve written the thesis, my committee has more or less signed off on it. The final okay comes Thursday, hopefully, but no one has raised any red flags thus far. I’ve also written my talk. And I’ve even starting picking out the juiciest parts from my thesis for a paper, to be submitted to a journal later this summer. I’ve also been reading through the literature, and trying to anticipate what my thesis committee might ask.

So what is left to do? Strangely, not too much. The defense will be an hour talk, followed by an hour of questions. I just need to be ready to answer whatever they might ask. As my advisor noted, what I’ll be facing is some extremely smart people who don’t know much about what I’m doing, so they’ll ask me about whatever it reminds them of that they’re interested in.

Let me take one opportunity to describe what my thesis is about here. (When I defend, I’m going to consciously try to remember not to be overly, metaphorically simple—mathematicians in this context want to see me be as dense as necessary, and handle the heavy language and the power tools. Whereas I’ve been trying for so long to explain mathematically deep ideas in simple language and stress the accessibility that I sometimes forget that I’m talking to an audience of geniuses who’ve been thinking about mathematical ideas for a living for years.)

A K3 surface… well, it’s a little technical to define. It’s sort of doughnut-like. But, weirdly, it’s actually not too important. So don’t worry about it. (Okay, an example of one is: graph x^4+y^4+z^4+w^4=0 in projective 3 dimensional space.)

An FM partner… well, the cool thing about K3 surfaces is that there’s a bunch of different ways of looking at them. There’s a way to keep track of the types of geometric objects that can live on them. There’s also a rather abstract way to keep track of a type of curves that might be on them. There’s also a completely different way to keep track of the curves that are hard to track algebraically. The details aren’t important, but what’s amazing is that all of these processes involve another K3 surface, which does all of this at the same time: it keeps track of the geometric objects on the first, its curves have the same structure in the abstract algebraic way, and the curves that are hard to describe algebraically have the exact same structure too. The new K3 surface is called an “FM Partner” of the first.

The point is, we’ve got these K3 surfaces that are partners of each other. The question I was trying to explore is: can you figure out how many partners a K3 surface has?

Well, it’s a hard question. People knew a bunch of stuff before I started, and I’ve added a little bit to that. I looked into a specific example of K3 surfaces and gave a pretty decent count for those. I also showed that they could have a lot of FM partners—as many as any number you pick (not infinitely many though). I also showed how to study partners of partners. Turns out there’s a pretty nice structure to them.

There are a few things I would have liked to get that I haven’t yet. But they’ll have to wait till later.

Anyway, that’s my first and last attempt to explain what I’ve been up to for the past two years in a public place. I like problems that are accessible and understandable; it’s kind of a shame that the work I’ve been involved in is so opaque it’s difficult to share even with other mathematicians. But, alas, that’s how the field tends to be. It’ll be nice to get to devote more energy in the future to math that I can share with others.

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