Mathematical Sciences: Riemann Surfaces and Analytic Torsion

John Fay将研究与theta函数和Riemann曲面相关的最新发展。这是一个繁荣的领域，因为它与Kadomtsev-Petviashvili微分方程组和弦理论有联系。它对数学也有很大的内在兴趣，因为它将有助于黎曼曲面上的高阶向量丛的理论。本文主要研究Riemann曲面上向量丛的模空间上的Ray-Singer解析挠率。这将包括三个方面的研究。第一个是关于Jacobi簇上的挠率的全纯分解和消失性。第二部分研究了挠率对黎曼曲面模数的显式依赖关系。最后一个方面将讨论二阶丛的广义Szego核和由挠率定义的theta函数的性质。

John Fay will carry out research into recent developments relating theta functions and Riemann surfaces. This is an area which has flourished because of its connections with the Kadomtsev-Petviashvili hierarchy of differential equations and with string theory. It also has great intrinsic interest for mathematics since it will contribute towards a theory of higher rank vector bundles on Riemann surfaces. The research will center on Ray-Singer analytic torsion as defined on the moduli spaces of vector bundles on Riemann surfaces. This will include study of three aspects. The first concerns the holomorphic factorization and vanishing properties of torsion on the Jacobi variety. The second involves investigation of the explicit dependence of torsion on the moduli of the Riemann surface. The final aspect will be concerned with the properties of the generalized Szego kernel and theta- functions defined by torsion for rank-two bundles.