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

主讲教授

Prof. Tonghai Yang, University of Wisconsin, USA

日期时间

9:00-11:00AM - June 18/June 19/June 22/June 23/June 24

2:30-4:30PM - July 2

授课地点

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 see them as examples of Shimura varieties of orthogal type. These materials are also background for the summer program `Shimura Varieties' at the Morningside Center of Mathematics this coming July. In the last couple of lectures, I will explain some of my recent work on arithmetic intersection on Hilbert modular surfaces and/or my joint work with Jan Bruinier on Falting's height of CM cycles and a new proof of the Gross-Zagier formula.

Introduction to mathematical conformal field theory

主讲教授

Prof. Yizhi Huang (Rutgers University, USA)

日期时间

9:00-11:00AM - July 1/July 2

2:30-4:30PM - June 22/June 24/June 26/June 29

授课地点

Resource Building 1328

Abstract

I will start with the definition of conformal field theory by Kontsevich and Segal and its natural generalization to open-closed conformal field theory. There has been a long term program to construct conformal field theories satisfying these definitions from representations of vertex operator algebras. I will discuss the main results obtained in this program, including those on operator product expansions, modular invariance, the Verlinde conjecture, the Verlinde formula for fusion rules, modular tensor categories and open-closed conformal field theories. Open problems will also be discussed. The purpose of these lectures is to provide a quick introduction to people who are interested in doing research work in the area of mathematical conformal field theory.

Topics in String Geometry

主讲教授

Prof. Yanghui He (Oxford University, UK)

日期时间

9:00-11:00AM - July 9/July 13/July 20/July 22

2:30-4:30PM - July 3/July 7/July 15/July 17/July 24

授课地点

Resource Building 1328

Abstract

We walk the students, intended to be a mixture of beginning mathematicians and physicists, each interested in the other discipline, through some interactions between string (gauge) theory and geometry. Topics will include rudiments of mirror symmetry, enumerative geometry, and quiver representations.

Enumerative Combinatorics

主讲教授

Prof. Catherine H. Yan (Texas A&M University, USA)

日期时间

9:00-11:00AM - July 2/July 3/July 6/July 8/July 10/July 14/July 16/July 17

授课地点

Resource Building 1213

Abstract

This is an introduction at the graduate level to the fundamental ideas and results of combinatorics. The course moves quickly but does not assume prior study in combinatorics. It is intended for graduate students from mathematics or related areas wanting a good one-semester background in fundamental and applicable discrete mathematics. It also provides solid preparation for advanced combinatorics and for various courses in computer science. The course will cover structures and methods of combinatorics, including enumerative techniques, sieve methods, partially ordered sets, and generating functions. We will introduce algebraic tools and approaches to enumerative problems, and develop a combinatorial theory for many elementary combinatorial structures and data sets. In particular, we will emphasize the bijective proofs illustrating the frequent, and often surprising, interrelations between those structures.

Structures and Representations of extended affine Lie algebras

主讲教授

Prof. Yun Gao, York University, Canada

日期时间

9:00-11:00AM - July 10/July 13/July 24/July 27

2:30-4:30PM - July 6/July 7/July 8/July 9/July 23

授课地点

Resource Building 1213

Abstract

We will give a series of introductory talks on a class of the newly developed infinite dimensional Lie algebras---extended affine Lie algebras. They are a high dimensional generalization of the finite dimensional complex simple Lie algebras (being dimensiona zero) and affine Kac-Moody Lie algebras(being dimensiona one) which were introduced by some mathematical physicists. Their structures are rich and involve some mathematical aspects such as nonassociative algebras (Cayley algebras, Jordan algebras, Tits-Kantor-Koecher algebras), non-commutative geometry(quantum tori, cyclic homology). They are also closely related to the extended affine root systems introduced by Kyoji Saito in the study of elliptic singularities. These talks will include all basics, classifications, constructions, and module realizations for some extended affine Lie algebras.