- 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.
Cohomology of Locally Symmetric Spaces and Applications to Number Theory awarded by National Science Foundation (Principal Investigator). 2005 to 2009
Cohomology of Locally Symmetric Spaces awarded by National Science Foundation (Principal Investigator). 1998 to 2001
Presidential Young Investigator Award: Mathematical Sciences awarded by National Science Foundation (Principal Investigator). 1989 to 1994
L2-Cohomology of Singular Spaces awarded by National Science Foundation (Principal Investigator). 1990 to 1992
Saper, L. “Perverse sheaves and the reductive Borel-Serre compactification.” Hodge Theory and L2-Analysis, edited by Lizhen Ji, vol. 39, International Press, 2017, pp. 555–81.
Saper, L. “ℒ-modules and the conjecture of Rapoport and Goresky-Macpherson.” Formes Automorphes (I) — Actes Du Semestre Du Centre Émile Borel, Printemps 2000, edited by J. Tilouine et al., vol. 298, 2005, pp. 319–34.
Saper, L. “On the Cohomology of Locally Symmetric Spaces and of their Compactifications.” Current Developments in Mathematics, 2002, edited by D. Jerison et al., International Press, 2003, pp. 219–89.
Saper, L. “L²-cohomology of the Weil-Peterson metric.” Mapping Class Groups and Moduli Spaces of Riemann Surfaces Proceedings of Workshops Held June 24-28, 1991, in Göttingen, Germany, and August 6-10, 1991, in Seattle, Washington, edited by C. -. F. Bödigheimer and R. Hain, vol. 150, Amer. Math. Soc., 1993, pp. 345–60.
Saper, L., and M. Stern. “Appendix to: On the shape of the contribution of a fixed point on the boundary. The case of Q-rank one, by M. Rapoport.” The Zeta Functions of Picard Modular Surfaces Based on Lectures Delivered at a CRM Workshop in the Spring of 1988, edited by R. Langlands and D. Ramakrishnan, Centre De Recherches Mathématiques, 1992, pp. 489–91.
Saper, L. “L₂-cohomology of algebraic varieties.” Proceedings of the International Congress of Mathematicians, August 21-29, 1990, Kyoto, edited by I. Satake, vol. 1, Springer-Verlag, 1991, pp. 735–46.
Saper, L., and S. Zucker. “An introduction to L²-cohomology.” Several Complex Variables and Complex Geometry, vol. 52, Part 2, Amer. Math. Soc., 1991, pp. 519–34.
Ji, L., et al. “The fundamental group of reductive Borel–Serre and Satake compactifications.” Asian Journal of Mathematics, vol. 19, no. 3, 2015, pp. 465–86. Manual, doi:10.4310/AJM.2015.v19.n3.a4. Full Text
Saper, L. “Geometric rationality of equal-rank Satake compactifications.” Mathematical Research Letters, vol. 11, no. 5, 2004, pp. 653–71.
Saper, L. “Tilings and finite energy retractions of locally symmetric spaces.” Commentarii Mathematici Helvetici, vol. 72, no. 2, Dec. 1997, pp. 167–201.
Saper, L. “L²-cohomology of Kähler varieties with isolated singularities.” Journal of Differential Geometry, vol. 36, no. 1, 1992, pp. 89–161.
Habegger, N., and L. Saper. “Intersection cohomology of cs-spaces and Zeeman's filtration.” Inventiones Mathematicae, vol. 105, no. 1, Dec. 1991, pp. 247–72. Scopus, doi:10.1007/BF01232267. Full Text
Saper, L., and M. Stern. “L²-cohomology of arithmetic varieties.” Proc Natl Acad Sci U.S.A., vol. 84, no. 16, Aug. 1987, pp. 5516–19.
Saper, L. “L₂-cohomology and intersection homology of certain algebraic varieties with isolated singularities.” Inventiones Mathematicae, vol. 82, no. 2, 1985, pp. 207–55. Manual, doi:10.1007/BF01388801. Full Text
Saper, L. “ℒ-modules and micro-support.” To Appear in Annals of Mathematics.
L²-cohomology of projective algebraic varieties. International Conference on Singularity Theory — in Honor of Henry Laufer's 70th Birthday. Tsinghia Sanya International Mathematics Forum, Sanya, China. December 20, 2015
Students and faculty enjoyed a break before exams to share food and conversation at the annual math party on April 24, 2017. Those who competed in the MCM/ICM competition received Duke Math shirts and certificates. Officers of DUMU and research... read more »
The Julia Dale winners for 2016 are Lindsey Brown, Bryan Liu and Paul Ziquan Yang. Each of these students excelled in upper level and graduate math courses and wrote an innovative senior thesis. Under the direction of Mike Reed and... read more »