Mathematical Sciences: Theta Functions and Moduli Spaces

这个项目将集中在黎曼曲面的解析扭转和经典解析扭转领域中出现的问题。主要研究Riemann曲面上的theta-函数、解析挠率和向量丛中的经典模问题。将研究由解析挠率构造的tau-theta函数，从使用Szego核的循环群方法构造的tau-theta函数，以及使用阿贝尔柯西核构造的矩阵奇异积分算子。Szego-核和theta-函数将被发展用于穿孔黎曼曲面上的单调群，这将有助于进一步理解高阶亏格黎曼曲面上的等单调形变形和秩二丛的模空间的谱曲线(双覆盖)方法。还将研究经典的Bergman核函数和Szego核函数到具有连通的平坦丛的模空间的自旋零挠因子和亚纯扩张。这项工作既涉及经典的数学函数理论，也涉及数学物理中与保持单调变形有关的深层次问题。

This project will focus on problems arising in the field of analytic torsion and the classical analytic torsion of Riemann surfaces. The work centers on classical moduli problems in theta-functions, analytic torsion and vector bundles on Riemann surfaces. Studies of tau-theta functions constructed from analytic torsion, from the loop-group approach using the Szego-kernel and from matrix singular integral operators using abelian Cauchy kernels will be carried out. The Szego-kernel and theta-functions will be developed for monodromy groups on punctured Riemann surfaces; this will help in the further understanding of isomonodromy deformations on higher-genus Riemann surfaces and the spectral curve (double-cover) approach to the moduli space of rank-two bundles. There will also be research on the spin-zero torsion divisor and the meromorphic extension of the classical Bergman and Szego kernel-functions to the moduli space of flat bundles with connections. This work is related with both classical mathematical function theory as well as deep questions arising in mathematical physics concerned with monodromy preserving deformations.