Research Interests for Leslie Saper
Research Interests: Locally symmetric varieties, Number theory and automorphic forms, L^{2}cohomology and intersection cohomology, Geometrical analysis of singularities
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 wellknown HodgedeRham 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 L^{2}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 HasseWeil zeta function. This should be
viewed as an object on the topological side of the above picture. On the analytic side, Langlands has associated
Lfunctions to certain automorphic representations. The issue of whether one may express the HasseWeil zeta
function in terms of automorphic Lfunctions, and the relation of special values of these functions to number theory,
are important fundamental problems which are motivating Professor Saper's research.  Recent Publications
 Saper, L, ℒmodules and microsupport,
to appear in Annals of Mathematics
(2018)
 Saper, L, Perverse sheaves and the reductive BorelSerre compactification,
in Hodge Theory and L²analysis, edited by Ji, L, vol. 39
(2017),
pp. 555581, International Press [abs]
 Ji, L; Murty, VK; Saper, L; Scherk, J, The fundamental group of reductive Borel–Serre and Satake compactifications,
Asian Journal of Mathematics, vol. 19 no. 3
(2015),
pp. 465486 [arXiv:1106.4810], [available here], [doi] [abs]
 Borel, A, Lie Groups and Linear Algebraic Groups I. Complex and Real Groups,
in Lie Groups and Automorphic Forms, AMS/IP Studies in Advanced Mathematics, edited by Saper, L, vol. 37
(2006),
pp. 150, American Mathematical Society/International Press
 Borel, A, Introduction to the Cohomology of Arithmetic Groups,
in Lie Groups and Automorphic Forms, AMS/IP Studies in Advanced Mathematics, edited by Saper, L, vol. 37
(2006),
pp. 5186, American Mathematical Society/International Press
