This research project involves application of the methods of algebraic and differential topology to the study of certain algebraic varieties. Weintraub will use a number of techniques and, in particular, exploit the fact that the spaces studied often admit, or arise from, certain natural group actions. The varieties to be studied include the Igusa compactifications of degree two Siegel modular spaces corresponding to various congruence subgroups of the symplectic group of degree four (and related groups), moduli spaces of Riemann surfaces (sometimes with special or additional structure), ball quotients, and other varieties related to configuration spaces. The information to be obtained includes homology and cohomology (both the groups and geometric representatives of the classes), Hodge decompositions, representations of the groups which act on the (co)homology, intersection homology, zeta functions, and Lefschetz numbers. In addition there are applications to group cohomology and Siegel modular forms. One of Weintraub's techniques of interest in its own right, is an elaboration of the Tits building (called "scaffolding") which he has developed. Of particular interest is the application of this work to theta-functions. It clarifies some points about how theta-functions transform, and most interestingly of all, reveals hitherto unsuspected connections between geometry/topology (in the guise of holonomy/Rochlin invariants) and theta-functions, and raises interesting new questions about the latter. (The project represents a continuation of a long-running research program, much of it joint work with other mathematicians.)
