Aschbacher will investigate finite projective planes linearly coordinatized by right distributive rings. He will continue his project to produce a unified, accessible, and complete treatment of the existence, uniqueness, and structure of the sporadic groups. He will complete joint work on coverings of Riemann surfaces with a classical monodromy group, and on a project to produce a self-contained and accessible treatment of the Schur multipliers of the finite simple groups, and related cohomological questions.
The Classification of the finite simple groups is one of the major achievements of twentieth century mathematics, and has made possible many applications of finite group theory to other branches of mathematics. The projects in the proposal either improve and make more accessible the proof of the Classification, or apply the Classification and the theory of simple groups to problems in geometry, most particularly to the study of projective planes. Projective planes are one of two classes of geometric objects which demonstrate the necessity of the ``parallel axiom" in ordinary plane geometry; they play a central role in several modern approaches to geometry. The study of projective planes has a long history, but the most fundamental question about finite projective planes remains open: Does each plane possess at least one well behaved coordinatization, or are finite planes chaotic? Aschbacher hopes to shed some light on this question.
