Selected Presentations & Appearances
Invited Lectures
Invited Lectures ; Let G be a reductive algebraic group defined over a number field. The congruence subgroup kernel quantifies to what extent is every arithmetic subgroup of G a congruence subgroup. It has been studied extensively. On the other hand, the reductive Borel-Serre compactification of an arithmetic quotient of the symmetric space associated to G reflects the geometry of this quotient at infinity. Its cohomology has been extensively studied in view of applications to automorphic forms. After describing these two seemingly disparate objects I will show how the congruence subgroup kernel can be related to the fundamental group of the reductive Borel-Serre compactification. There is a generalization to S-arithmetic subgroups. This is joint work with Lizhen Ji, V. Kumar Murty, and John Scherk.
Invited Lectures ; Goresky and MacPherson were motivated to introduce intersection cohomology in part to recover a generalized form of Poincare duality for singular spaces: for each pair of dual perversities, such as the lower middle and the upper middle, there is a nondegenerate pairing between the corresponding intersection cohomology groups. However for singular spaces with even codimension strata, or more generally for Witt spaces, the upper middle and the lower middle theories coincide, yielding a nondegenerate pairing on what is simply called middle perversity intersection cohomology. This enabled Goresky and MacPherson to define an L-class for Witt spaces. For non-Witt spaces, Banagl has shown there exist a well-defined L-class provided there exists a self-dual sheaf that interpolates the lower middle and the upper middle intersection cohomology sheaves. In the case of the reductive Borel-Serre compactification of a Hilbert modular surface, Banagl and Kulkarni show that such a self-dual sheaf exists. In this talk I will address the existence of such self-dual sheaves on the reductive Borel-Serre compactifications of general locally symmetric spaces, a question raised by Banagl and Kulkarni. Note that a completely independent analytic approach to restoring Poincare duality and producing characteristic classes was developed by Cheeger using L²-cohomology. I will also relate the existence of these self-dual sheaves to L²-cohomology.
Invited Lectures ; The homology of a compact closed n-manifold X satisfies Poincare duality: the intersection pairing between degree i and degree n-i homology is a perfect pairing over a field. When X has singularities, Poincare duality may fail to hold. Nonetheless, in the 1980's Goresky and MacPherson defined a topological invariant, the intersection homology, of a space X which satisfies Poincare duality even if X is singular; for a smooth space X, intersection homology agrees with ordinary homology. Intersection homology crops up in many places, from analysis to representation theory. In this talk I will give an informal introduction to intersection homology and some of its applications.
Invited Lectures: The theory of ℒ-modules was developed to solve the conjecture of Rapoport and Goresky-MacPherson: the intersection cohomology of the Baily-Borel-Satake compactification of a Hermitian locally symmetric space is isomorphic to the intersection cohomology of the reductive Borel-Serre compactification. However it applies more generally and is an powerful combinatorial tool to study constructible sheaves on the reductive Borel-Serre compactification of a general locally symmetric space. We will survey the theory and give applications to several areas, including cohomology of arithmetic groups, L²-cohomology, L²-harmonic forms, and weighted cohomology.