Mathematical Sciences: Topological Trace Formula

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9303550 Goresky The Lefschetz trace formula of Arthur is an expression for the alternating sum of traces of the map induced on the L2 cohomology of a locally symmetric space Y by a Hecke Correspondence. This expression is given in terms of orbital integrals, volumes of stabilizer groups, and characters of discrete series representations. Arthur's formula will be rederived (in this joint project with R. MacPherson of M.I.T.) by interpreting it as the Lefschetz fixed point formula for the action of the Hecke correspondence on the weighted cohomology of the reductive Borel-Serre compactification of Y. This interpretation will be used (a) to extend Arthur's formula to include the case of the ordinary cohomology, (b) to derive a similar expression for the alternating sum of traces on the Hodge components of the cohomology, and (c) to conjecture a related expression which should be valid for Shimura varieties defined over fields of characteristic p 0. During the 1970's, R. Langlands (of the Institute for Advanced Study in Princeton) outlined a series of conjectures and ideas of enormous scope and depth which, when fully explored and verified, will result in a "grand unification" of several branches of mathematics, including number theory, representation theory of Lie groups, and harmonic analysis. During the last twenty years enormous progress has been made on this program and very difficult obstacles have been overcome. Nevertheless, it is commonly believed that it may take another fifty years (or more) before Langlands' ideas are fully explored. The present project concerns a newly discovered geometric and topological interpretation and proof of the "trace formula," one of the basic tools in the subject. Using this interpretation, the trace formula has been generalized and refined to the point where it may be applied to the central objects of interest (namely, Hecke correspondences on Shimura varieties). Although these geometric techniques have b een developed primarily to carry out this step in Langlands' program, they have already been applied to problems in other areas of mathematics. ***