9401611 Harer This proposal concerns research on three related topics. The first is a continuation of the principal investigator's work on the computation of the homology of the moduli space of curves. New techniques are proposed which involve the interplay between the known triangulations of moduli spaces and the Arbarello flag. The second topic is the generalization of the results on moduli space to its level-n covers and to Torelli space. The third is joint work with David Epstein and Steve Kerckhoff in which they are looking for automatic structures for the outer automorphism group of a free group. This research is part of an ongoing attempt by a number of mathematicians and physicists to understand the mathematics which underlies certain recent developments in theoretical physics. In particular, gravity is the only one of the four fundamental forces of nature for which there is no reasonable quantum theory (quantum theories describe the behavior of matter in the small, while ordinary theories describe behavior in the large). An approach to quantum gravity was suggested by the physicists (primarily Witten), but the mathematics necessary to understand it involves the moduli space of curves and is not yet understood sufficiently. ***
