# Leslie Saper

- Professor of Mathematics

**External address:**120 Science Dr, Room 110 Physics Building, Durham, NC 27708-0320

**Internal office address:**Box 90320, Dept. Of Mathematics, Durham, NC 27708-0320

**Phone:**(919) 660-2843

**Office Hours:**

Tuesdays, 1:30 - 12:00, Fridays, 11:00 - 12:00, and by appointment

### Research Areas and Keywords

##### Algebra & Combinatorics

##### Geometry: Differential & Algebraic

##### Number Theory

##### Topology

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.

### Selected Grants

Cohomology of Locally Symmetric awarded by National Science Foundation (Principal Investigator). 2005 to 2009

(98-0380) Cohomology of Locally Symmetric Spaces awarded by National Science Foundation (Principal Investigator). 1998 to 2001

(94-0093) Presidential Young Investigator Award: Mathematical Sciences awarded by National Science Foundation (Principal Investigator). 1989 to 1994

(90-0295) 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.* Ed. L Ji. Somerville, MA: International Press, 2017. 555-581. (Chapter)

Borel, A. "Lie Groups and Linear Algebraic Groups I. Complex and Real Groups." *Lie Groups and Automorphic Forms, AMS/IP Studies in Advanced Mathematics.* Ed. L Saper. American Mathematical Society/International Press, 2006. 1-50. (Chapter)

Borel, A. "Introduction to the Cohomology of Arithmetic Groups." *Lie Groups and Automorphic Forms, AMS/IP Studies in Advanced Mathematics.* Ed. L Saper. American Mathematical Society/International Press, 2006. 51-86. (Chapter)

Saper, L. "ℒ-modules and the conjecture of Rapoport and Goresky-Macpherson." *Formes Automorphes (I) — Actes du Semestre du Centre Émile Borel, printemps 2000.* Ed. J Tilouine, H Carayol, M Harris, and M-F Vignéras. 2005. 319-334. (Chapter)

Saper, L. "On the Cohomology of Locally Symmetric Spaces and of their Compactifications." *Current developments in mathematics, 2002.* Ed. D Jerison, G Lusztig, B Mazur, T Mrowka, W Schmid, R Stanley, and S-T Yau. Somerville, MA: International Press, 2003. 219-289. (Chapter)

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.* Ed. C-F Bödigheimer and R Hain. Providence: Amer. Math. Soc., 1993. 345-360. (Chapter)

Saper, L, and Stern, M. "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.* Ed. R Langlands and D Ramakrishnan. Montréal: Centre De Recherches Mathématiques, 1992. 489-491. (Chapter)

Saper, L, and Zucker, S. "An introduction to L²-cohomology." *Several Complex Variables and Complex Geometry.* Providence: Amer. Math. Soc., 1991. 519-534. (Chapter)

Saper, L. "L₂-cohomology of algebraic varieties." *Proceedings of the International Congress of Mathematicians, August 21-29, 1990, Kyoto.* Ed. I Satake. New York: Springer-Verlag, 1991. 735-746. (Chapter)

Ji, L, Murty, VK, Saper, L, and Scherk, J. "The fundamental group of reductive Borel–Serre and Satake compactifications." *Asian Journal of Mathematics* 19.3 (2015): 465-486.
Full Text

Saper, L. "L²-cohomology of locally symmetric spaces. I." *Pure and Applied Mathematics Quarterly* 1.4 (2005): 889-937.
Full Text

Saper, L. "Geometric rationality of equal-rank Satake compactifications." *Mathematical Research Letters* 11.5 (2004): 653-671.

Saper, L. "Tilings and finite energy retractions of locally symmetric spaces." *Commentarii Mathematici Helvetici* 72.2 (1997): 167-201.

Saper, L. "L²-cohomology of Kähler varieties with isolated singularities." *Journal of Differential Geometry* 36.1 (1992): 89-161.

Habegger, N, and Saper, L. "Intersection cohomology of cs-spaces and Zeeman's filtration." *Inventiones Mathematicae* 105.1 (1991): 247-272.
Full Text

SAPER, L, and STERN, M. "L2-COHOMOLOGY OF ARITHMETIC VARIETIES." *ANNALS OF MATHEMATICS* 132.1 (July 1990): 1-69.
Full Text

Saper, L, and Stern, M. "L(2)-cohomology of arithmetic varieties." *Proceedings of the National Academy of Sciences of the United States of America* 84.16 (August 1987): 5516-5519.
Full Text

Saper, L. "L₂-cohomology and intersection homology of certain algebraic varieties with isolated singularities." *Inventiones Mathematicae* 82.2 (1985): 207-255.
Full Text

Saper, L. "ℒ-modules and micro-support (Accepted)." *to appear in Annals of Mathematics*.

## Pages

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