Mathematical Sciences: Finite Groups and Elliptic Cohomology

本课题是将Thompson级数理论推广到椭圆模。这是一个将有限群的表示理论与椭圆模形式联系起来的装置。它在数学的各个领域都有重要意义，如零星单群理论、流形的拓扑学和椭圆上同调。特别地，我们将考察Hecke算子在椭圆上同调中作为运算的作用，以及Monster单群的猜想零亏格椭圆模的存在性。本课题属于有限群论的一般领域。它将考察群论和数论之间的联系。这项工作有望对理论物理和数学产生影响。

This project is concerned with an extension of the theory of Thompson series to elliptic modules. This is a device which relates representation theory of finite groups with elliptic modular forms. It has consequences in various areas of mathematics such as the theory of sporadic simple groups, topology of manifolds, and elliptic cohomology. In particular, the role of Hecke operators as operations in elliptic cohomology, and the existence of a conjectured elliptic module of genus zero for the Monster simple group will be examined. This project is in the general area of finite group theory. It will examine connections between group theory and number theory. This work promises to have consequences for theoretical physics, as well as, mathematics.