The investigator intends to work on three projects. The first, joint with Avner Ash (Ohio State), deals with arithmetic subgroups G of SL(3,Z). A cuspidal cohomology class for G which is a Hecke eigenclass generates an automorphic representation which should be attached to a family of Galois representations. They will study these Galois representations in two special cases. The second project involves a general approach to understanding the cohomology of G(N) in SL(3,Z) for prime N. The third project is joint with Robert MacPherson (M.I.T.). For any torsion-free finite-index G in Sp(4,Z), they have built a G-equivariant deformation retract (a "spine") inside the symmetric space for Sp(4,R); it has properties like those of the well-rounded retract for SL(n). They intend to attempt to extend their construction to non-torsion-free G, and to build a similar object for Sp(6) and higher-rank groups. The intricate computations envisioned here are intended to shed light on structures which arise in important ways in several different mathematical areas. The so-called arithmetic locally symmetric spaces interest number theorists as well as geometers, while some of the questions about them have been transformed by the investigator into pure topology. No transformation removes their difficulty, but when looked at in the right way, some difficulties can be seen to have been addressed by earlier generations of geometers. Seventeenth and nineteenth century geometers studied a beautiful class of objects known as projective configurations, and bearing out the mathematical dictum that beautiful objects repay study because they recur again and again in different guises, properties of these configurations have turned out to bear on the cohomology of certain arithmetic groups, normally considered a purely twentieth century invention.
