讲座摘要: Generating functions of intersection numbers of certain tautological classes on moduli spaces of stable curves provide geometric solutions to integrable systems. Notable examples are the Kontsevich-Witten tau function and Brezin-Gross-Witten tau function. Both of them are tau-functions of the KdV hierarchy. Using matrix models, Mironov-Morozov gave a formula expressing Kontsevich-Witten tau function as an expansion of Schur's Q-polynomial with simple coefficients. This formula was called Mironov-Morozov conjecture by Alexandrov. A similar formula was also conjectured by Alexandrov for Brezin-Gross-Witten tau function. In this talk I will describe two proofs of these formulas using Virasoro constraints and cut-and-join operators. The talk is based on joint works with Chenglang Yang.

主讲人简介：北京大学讲席教授，北京国际数学中心副主任，北京数学会理事长，Peking Mathematical Journal 主编。曾任美国圣母大学教授，获得美国Sloan Research Fellowship。 2006年受邀在西班牙马德里国际数学家大会上作45分钟报告。研究方向为微分几何与数学物理，多篇文章发表在Annals of Mathematics, Duke Math. J.,  J. Diff. Geom 等国际著名数学期刊上。