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.
