Research in Group Theory

California Institute Of Technology, Pasadena CA

Investigators

Abstract

The investigator will focus on two projects. First, a program to extend the current description of the subgroup structure of the finite simple groups, and to use that information to establish a negative answer to an important question in universal algebra, open for over thirty years: Is each finite lattice an interval in the lattice of subgroups of a finite group? Second, a program to classify the simple saturated 2-fusion systems of component type. One consequence of the second project should be a simplification of the proof of the theorem classifying the finite simple groups. The classification of the finite simple groups is one of the premiere achievements of modern mathematics, providing the algebraic foundation for the study of the symmetry of finite objects. Unfortunately the proof is very long and complicated, consisting of as many as ten thousand pages, spread amongst hundreds of articles. There are at least two books, aimed at a lay audience, describing the human aspects of the effort. An emerging area of mathematics, studying objects called fusion systems, offers hope of simplifying the proof of the classification using an innovative approach.