9304279 Griess This award supports research aimed at developing a theory which will unite all the sporadic simple groups by means of infinite dimensional groups. Specifically, the principal investigator will investigator the general structure of infinite dimensional groups associated to lattices. He will study finite structures associated to Lie groups and develop a general theory of automorphism groups of a class of nilpotence class 2 groups. He will further classify finite subgroups of finite dimensional simple Lie groups by data such as labeled Coxeter-Dynkin diagrams. The principal investigator will also work on a new version of a moonshine module. A group is an algebraic structure with a single operation. It appears in many areas of mathematics, as well as, physics and chemistry. The fundamental building blocks of finite groups are finite simple groups. One of the major results in mathematics of the past decade is the classification of the finite simple groups, the proof of which would require 10,000 to 15,000 journal pages. This research is aimed at using this classification in the study of arbitrary finite groups. ***
