(1) The trace formula of J. Arthur and A. Selberg (for the trace of the action of a Hecke correspondence on the L2 cohomology of a Hermitian locally symmetric space) will be reproved and interpreted in terms of the Lefschetz fixed point theorem. This will involve a new evaluation of the local contribution to the Lefschetz number at a fixed point and the construction of a new cohomology theory ("weighted cohomology") on the reductive Borel-Serre compactification of X. Applications will be made to the computation of characters of discrete series representations, the cohomology of discrete groups, and the Hodge (p,q) decomposition of the L2 cohomology. (2) Chern numbers of singular complex algebraic varieties will be defined by lifting Chern classes to intersection cohomology. The invariance properties of these new Chern numbers will be investigated.
