主 题: 鍖椾含澶у?鐗瑰埆鏁板?璁插骇绗?崄浜屾湡
报告人: Prof. Todor E Milanov (Stanford University, USA)
时 间: 2009-06-13 08:00 - 2009-07-27 18:00
地 点: Resource Building 1218
The Gromov-Witten invariants of a complex projective manifold $X$ enumerate holomorphic maps from Riemann surfaces to $X$ satisfying various constraints. Even when $X$ is a point the invariants are quite interesting. According to a conjecture of Witten proved by Kontsevich, the Gromov-Witten invariants of a point are completely determined from a certain solution of the KdV hierarchy.It is natural to ask whether Witten's conjecture has its analogue for other manifolds as well.This is a very difficult question and it is still quite open. My main goal is to give an introduction to this problem based on Givental's quantization formalism.
It turns out that if the manifold $X$ admits sufficiently many rational curves then the higher genus invariants can be expressed in terms of the genus-0 ones and the higher genus theory of the point. The answer, due to Givental, can be given in a very elegant way in terms of certain Fock space formalism. This will be my first goal. In particular, I am planning to describe the so called {em Frobenius structures} -- in the setting of Gromov--Witten theory they are also known as {em quantum cohomology} -- and their geometric interpretation in terms of a certain Lagrangian cone. Recently, C. Teleman announced a proof of Givental's reconstruction formula, which however is still not quite accepted in the mathematical community. I am planning to spent some time explaining Teleman's ideas.
Givental's formula can be described in terms of the mirror model of $X$. The later consists of a family of oscillating integrals satisfying a system of differential equations. In particular, the ideas of Gromov--Witten theory can be naturally applied to singularity theory -- the study of holomorphic functions with an isolated critical point. It is a deep theorem that the space of miniversal deformations of such functions admits a flat structure which is the analogue of quantum cohomology in Gromov--Witten theory. I won't have the time to go over the proof of existence of such flat structures but I will point out the major steps. Finally, I would like to explain how the quantization formalism from Gromov--Witten theory and the flat structures from singularity theory fit together and give us a quite promising approach to representations of infinite dimensional Lie algebras and integrable hierarchies. If time permits, I will give the applications of these ideas to the mirror model of the projective line $mathbb{C}P^1$, which in particular leads to the proof of the so called Toda conjecture -- the Gromov--Witten invariants of the projective line are governed by the Extended Toda hierarchy.
No previous knowledge of Gromov--Witten theory, or integrable hierarchies will be assumed. However some knowledge of symplectic geometry (e.g. Hamiltonian vector fields, Poisson brackets, moment maps), complex geometry (e.g. sheaves, vector bundles, characteristic classes), and representation theory (simple Lie algebras) will be assumed.
An informal introduction to Shimura varieties
涓昏?鏁欐巿
Prof. Tonghai Yang, University of Wisconsin, USA
鏃ユ湡鏃堕棿
9:00-11:00AM - June 18/June 19/June 22/June 23/June 24
2:30-4:30PM - July 2
鎺堣?鍦扮偣
Resource Building 1328
Abstract
In first few lectures, I will discuss low dimensional Shimura varieties and modular forms on them, including
modular curves and classical modular forms, and Heegner points on modular curves
Hilbert modular surfaces, Hilbert modular forms, Hirzebruch-Zagier divisors, and CM points. Hirzebruch-Zaiger divisors and CM points are two difference generalizations of Heegner points.
Shimura curves associated to quaternion algebras, and CM points on SHimura curves (if time permits)
We will explain how to se
