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From mirror symmetry of the Catalan numbers to algebraic geometry of quantization
Faculty Research Seminar| Speaker: | Motohico Mulase, UC Davis |
| Location: | Zoom Zoom (see below) |
| Start time: | Tue, Jan 11 2022, 12:10PM |
One upon a time (just about a decade ago), Boris Dubrovin and I had the following conversation at MSRI:
MM: Hi Boris, I finally understood what mirror symmetry is!
BD: Oh, yeah? Then tell me, what is it?
MM: It's a Laplace transform!
We spent some time discussing what I meant by the Laplace transform (not Fourier-Mukai!) and he was intensely listening. Finally he said,
BD: Yes, I agree!
MM: You do? Great! Then let's check if our understanding is the same. What is the mirror symmetric dual to a point?
BD: That's easy! It is Kontsevich's Lax operator of the KdV equation!
MM: A differential operator is the mirror dual of a point? Hmm... Oh, I see! You mean, it is x = y^2, right?
BD: A curve x = y^2? Er... Oh, yes, it is indeed! Now it's my turn. What do you think is the mirror of the sine curve?
MM: Sine curve? A, ha! That's Mirzakhani's volume function of the moduli space of hyperbolic surfaces!
BD with a big smile: Yes, it is! Then you say the mirror of Hurwitz numbers is...
MM: The Lambert curve!
BD: Exactly!
MM: Now I understand that you and I have the same understanding!
BD: Indeed! But Motohico, I have already been saying that mirror symmetry is the Laplace transform for 15 years!
MM: Oh, have you been? But you never said what we have been talking about today!
With big smiles, we shook hands. A young researcher at MSRI who was listening to our conversation then exclaimed:
"What? You guys say you understood each other after such a cryptic conversation?" Boris and I bursted into laughter.
In this seminar talk, I present what I would have told him about the mirror of Hitchin systems, if Dubrovin were still with us.
This meeting will be on Zoom: https://ucdavis.zoom.us/j/96864394419?pwd=REppQWJK... Meeting ID: 968 6439 4419 Passcode: 631425