- Professor of Mathematics
Mondays 3:30 pm – 4:30 pm, Thursdays 4:30 pm – 5:30 pm, Fridays 2:30 pm – 3:30 pm, and by appointment.
Research Areas and Keywords
Algebra & Combinatorics
intersection cohomology, combinatorics of Weyl groups, K-theory
Geometry: Differential & Algebraic
Locally symmetric spaces, L2-cohomology, geometric analysis of singularities
automorphic forms, arithmetic of algebraic varieties
locally symmetric spaces, intersection cohomology, topology of compactifications, K-theory
A central theme in mathematics has been the interplay between topology and analysis. One subject here is the representation of topological invariants (such as cohomology) by analytic means (such as harmonic forms). For compact manifolds this is the well-known Hodge-deRham theory. Professor Saper studies generalizations of these ideas to singular spaces, in particular complex algebraic varieties. In these cases, an appropriate replacement for ordinary cohomology is Goresky and MacPherson's intersection cohomology, while on the analytic side it is natural to impose L²-growth conditions.
When one deals with varieties defined by polynomials with coefficients in the rationals, or more generally some finite extension, this theory takes on number theoretic significance. Important examples of such varieties are the locally symmetric varieties. One may reduce the defining equations modulo a prime and count the number of resulting solutions; all this data is wrapped up into a complex analytic function, the Hasse-Weil zeta function. This should be viewed as an object on the topological side of the above picture. On the analytic side, Langlands has associated L-functions to certain automorphic representations. The issue of whether one may express the Hasse-Weil zeta function in terms of automorphic L-functions, and the relation of special values of these functions to number theory, are important fundamental problems which are motivating Professor Saper's research.
Geometry and Topology of Locally Symmetric Spaces. Conference on Geometry, Topology, and their Interactions in honor of Farrell-Jones. Instituto de Matemáticas Unidad Morelia, Morelia, Mexico. January 8, 2007
L²-cohomology of locally symmetric spaces. International Conference in Memory of Armand Borel: Algebraic groups, arithmetic groups, automorphic forms and representation theory. Center of Mathematical Sciences at Zhejiang University, China. July 27, ...
On the Cohomology of Locally Symmetric Spaces and their Compactifications (two lectures). Current Developments in Mathematics 2002 conference. Harvard University. November 15, 2002 - November 17, 2002
Rapoport's conjecture on the intersection cohomology of the reductive Borel-Serre compactification. Conference on Galois representations and automorphic representations. Institut Henri Poincaré, Paris, France. April 25, 2000