Grant: Discovery Grants Program - Individual

This proposal concerns the connection between finite simple groups and modular forms. These connections were conjectured by Conway and Norton who coined the term "monstrous moonshine" to describe the link between the monster sporadic simple group and modular functions and forms. Since their formulation in the late 1970s, the moonshine conjectures have spawned new branches of mathematics. The work of Frenkel, Lepowsky and Meurman and Borcherds showed the deep links with mathematical physics via Vertex Algebras. Borcherds also introduced generalized Kac-Moody algebras in his proof of the moonshine conjectures. The moonshine conjectures, and the new mathematics needed for their proof, continue to have an important impact on mathematics. It seems certain that the study of moonshine will continue to be a fruitful source of new mathematics. The proposed research program will contribute to this study through continued work on the classification and properties of congruence subgroups - particularly those of small genus. The project will use these result to study replication, Hecke operators and generalized moonshine. Recent work of Witten also suggests new connections between monstrous moonshine and physics. Part of this project will be to investigate the links between the sporadic simple groups, modular functions and mathematical physics.