Leslie Saper, Professor and Director of Undergraduate Studies
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 L²growth 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.
 Contact Info:
Teaching (Spring 2018):
 MATH 605.01, ALGEBRAIC NUMBER THEORY
Synopsis
 Physics 205, WF 01:25 PM02:40 PM
 Teaching in Previous Semesters
 Office Hours:
 Tuesdays, 1:30  12:00, Fridays, 11:00  12:00, and by appointment
 Education:
Ph.D.  Princeton University  1984 
M.S.  Yale University  1979 
B.S.  Yale University  1979 
 Specialties:

Algebra
Topology Geometry
 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.
 Curriculum Vitae
 Current Ph.D. Students
(Former Students)
 Postdocs Mentored
 Recent Conferences Organized
 Workshop on Locally Symmetric Spaces, coorganizer with S. Kudla, J. Rohlfs, and B. Speh, Banff International Research Station, May 18, 2008  May 23, 2008
 Recent Publications
(More 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
 Selected Invited Lectures
 L^{2}cohomology of projective algebraic varieties, 2015, International Conference on Singularity Theory—in Honor of Henry Laufer's 70th Birthday, Tsinghua Sanya International Mathematics Forum, Sanya, China
 Perverse sheaves on compactifications of locally symmetric spaces, July 28, 2015, Isaac Newton Institute, Cambridge, England [html]
 Perverse sheaves and the reductive BorelSerre compactification, November 2123, 2014, Johns Hopkins University, Baltimore
 Raghunathan's Vanishing Theorem and Applications, December 28, 2011, Mumbai, India
 Cohomology of Locally Symmetric Spaces and the Moduli Space of Curves, June 09, 2011, Germany
 The congruence subgroup kernel and the reductive BorelSerre compactification, March 1, 2011, Algebraic Geometry and Number Theory seminar at John Hopkins University, Baltimore
 Selfdual sheaves and L²cohomology of locally symmetric spaces, June 24, 2010, Hausdorff Center for Mathematics, Bonn, Germany
 modules and the cohomology of locally symmetric spaces, December 15, 2008, International conference on Représentations des groupes de Lie et applications, Institut Henri Poincaré, Paris, France
 Quadratic Reciprocity from Euler to Langlands, September 28, 2007, Graduate/faculty Seminar, Duke University, [abstract]
 Geometry and Topology of Locally Symmetric Spaces, January 8, 2007, Instituto de Matemáticas Unidad Morelia, Morelia, Mexico
 Cohomology of Locally Symmetric Spaces, July 17  August 4, 2006, International conference and instructional workshop on discrete groups, Morningside Center of Mathematics, Chinese Academy of Science, Beijing, China [slides]
 L²Harmonic Forms on Locally Symmetric Sapces, January 19, 2006, Tata Institute for Fundamental Research, Mumbai, India
 L^{2}cohomology of locally symmetric spaces, International Conference In Memory of Armand Borel: Algebraic groups, arithmetic groups, automorphic forms and representation theory, 26  30 July, 2004, Center of Mathematical Sciences at Zhejiang University, China
 Cohomology of compactifications of locally symmetric spaces, 16  30 December 2003, The Graduate School of Mathematical Sciences, University of Tokyo, Japan
 On the Cohomology of Locally Symmetric Spaces and their Compactifications (two lectures), 1517 November 2002, Harvard University
 Rapoport's conjecture on the intersection cohomology of the reductive BorelSerre compactification, April 25, 2000, Institut Henri Poincaré, Paris, France
 L₂cohomology of Algebraic Varieties, August 23, 1990, International Congress of Mathematicians, Kyoto, Japan
