003 Recent Developments in Mathematical Physics

2021-03-12 10:32:45





报告题目:Kontsevich-Witten tau function and Schur Q-polynomials

报告摘要:Kontsevich-Witten tau function is the generating function for intersection numbers of psi-classes on moduli spaces of stable curves. Witten’s conjecture, proved by Kontsevich, states that this function is a tau-function of the KdV hierarchy. Q-polynomials were introduced by Schur in 1911 for the study of projective representations of symmetric groups. Recently Mironov-Morozov gave a formula representing Kontsevich-Witten tau function as a linear expansion of Schur's Q-polynomial with very simple coefficients. Alexandrov called this formula the Mironov-Morozov conjecture. In this talk, I will describe a proof of this formula which does not rely on matrix model. This talk is based on a joint work with Chenglang Yang.



报告题目:The physics and mathematics of BCOV theory

报告摘要:In this talk, I will first introduce the mirror symmetry for Calabi-Yau threefolds, which describes the genus zero structures of the Gromov-Witten theory. Then I will talk about the higher genus structures of the Gromov-Witten theory of the Calabi-Yau threefolds, conjectured by Bershadsky-Cecotti-Ooguri-Vafa. I will explain how to use these higher genus structures determine the higher genus Gromov-Witten invariants.



报告题目:Q-polynomial Expansion for Brezin-Gross-Witten tau function

报告摘要:Brezin-Gross-Witten (BGW) tau function first appeared in a matrix model in the study of lattice gauge theory in 1980. An enumerative geometric interpretation of this function was found by Norbury in 2017. The generalized BGW models were introduced by Mironov-Morozov-Semenoff in 1996 as a one-parameter deformation of BGW model. The partition functions of all these models are tau functions of the KdV hierarchy. Recently Alexandrov conjectured that these functions are hypergeometric tau functions of BKP hierarchy and their linear expansions in terms of Schur's Q-polynomials have simple coefficients. In this talk I will describe a proof of these conjectures given in a joint work with Chenglang Yang.



报告题目:Virasoro conjecture for FJRW theory

报告摘要:Virasoro conjecture is one of the most fascinating conjecture in Gromov-Witten theory, which is introduced by Eguchi-Hori-Xiong. It states that the Gromov-Witten potential Z is a solution of a sequence of nonlinear differential equation: L_k(Z)=0, k>=-1.

And L_k satisfies the following Virasoro relation

[L_m, L_n]=(m-n)L_{m+n}

In this talk, I will give a survey on Virasoro conjecture. I will also talk about the explicit form of Virasoro constraints on FJRW theory and prove it in some simple case, base on the joint work with Yefeng Shen.