Sunday, 3 March 2013

Mirror Symmetry!

Wednesday last week, my friends and I decided that we're going to take a stab at learning mirror symmetry this semester. We decided this based on a few talks that we gave to each other about geometric Langlands, cobordismy stuff, mirror symmetry and enumerative geometry. In the course of preparing for my talk about mirror symmetry, I stumbled across this in a paper:

"In this section I will attempt to outline the mirror symmetry principle in mathematical terms, and describe some of the mathematical evidence for it. I apologize to physicists for my misrepresentations of their ideas, and I apologize to mathematicians for the vagueness of my explanations."

-David R. Morrison.

Frickin' awesome.

I suppose that I should really write up something here that gives a flavour of (homological) mirror symmetry but I really don't understand this nearly well enough yet. I mean, okay, sure - (a small part of the mathematical side of) the story revolves around two (families of) spaces $$X$$ and $$X^{\vee}$$, one of which is symplectic and one of which is complex, that demonstrate dual Hodge structures dimensions.

Much of the current mathematical perspective on homological mirror symmetry is about the equivalence of the derived Fukaya category on $$X$$ and the derived category of coherent sheaves on $$X^{\vee}$$. Although sitting through Motohiko Mulase's (*) talk on Friday, you can see that he really thinks of mirror symmetry as...the Laplace transform.

Again, this is something that I'd love to be able to explain here, but I just don't get it. I mean, an example of this phenomenon is applying the Laplace transform to the cut and join relation governing certain characteristic classes on the (DM-compactified) moduli space of curves to obtain certain recursion relations (which, awesomely enough, prove Witten's conjecture and the $$\lambda_g$$-conjecture). And it's infuriating, because I spoke to him for a fair while after his wonderfully engaging and understandable talk, and yet here I am...still not quite catching how he thinks of mirror symmetry.

I guess I'mma just have to find him on Tuesday, admit that I'm a total idiot and ask him to explain it in greater detail (assuming that I can find him =P).

It's good that there's no room for pride in maths. =)

By the way #1: Cyberdyne systems from the Terminator vs Cyberdyne Inc. of robot legs fame, seriously Skynet? You couldn't have been a teeny bit subtler?

By the way #2: And here are roughly 450 million reasons to learn your maths right.

*: the guy is just...so...effervescent, especially for a uni professor! Totes likeable, and an excellent speaker. It's going to take a great deal of commitment to hate him. =)