Abstract

An LG model (M, f) is given by a noncompact complex manifold M and the holomorphic function f defined on it, which is an important model in string theory. Recently LG models attract more attention due to their important applications in mirror symmetry and complex geometry. Typical examples include Laurent polynomials defined on log Calabi-Yau manifolds, which are mirror to different Fano manifolds.This talk gives a survey on the study of LG models via the methods of differential geometry. We start with some examples, gives the geometric and topological information contained by a LG model, and then consider the Schrodinger equations and their deformation theory of a family of LG models. The output is the Hodge varlation structures. Gauss-Manin connections, index theory and torsion invariants.

Biography

Huijun Fan got his PhD in Peking University in 1998, and then did postdoctoral works in Max-Planck Institute in Leipzig and in Mathematics Institute of AMSS. He joined Peking University in 2003 and became a full professor in 2010. He was the Chair of Math. department in 2017-2021. Currently he is the director of the Key Laboratory of Mathematics and Applied Mathematics of the Ministry of Education of Peking University and the deputy director of the Sino-Russian Math Center. He has won National Outstanding Youth Grant and the Second Prize of the National Natural Science Award. He is the plenary speaker of the 2021 annual meeting of the Chinese Mathematical Society. His research direction is geometry and mathematical physics, concentrating on problems arising from mirror symmetry.