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.

**Contact Info:**

Office Location: | 110 Physics |

Office Phone: | (919) 660-2843 |

Email Address: | |

Web Page: | https://www.math.duke.edu/faculty/saper |

**Teaching (Spring 2018):**

- MATH 605.01,
*ALGEBRAIC NUMBER THEORY*Synopsis- Physics 205, WF 01:25 PM-02:40 PM

**Teaching in Previous Semesters**

**Office Hours:**- Mondays, 11:00 - 12:00, Wednesdays 2:00 - 3:00, Fridays, 1:00 - 2:00, 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 singularitiesA 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 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 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.

**Curriculum Vitae**

**Current Ph.D. Students**(Former Students)- Joshua Cruz

**Postdocs Mentored**- Mike Lipnowski (2013 - 2016)
- Jonathan Hanke (2003 - 2007)

**Recent Conferences Organized***Workshop on Locally Symmetric Spaces,*co-organizer with S. Kudla, J. Rohlfs, and B. Speh, Banff International Research Station, May 18, 2008 - May 23, 2008

**Recent Publications**(More Publications)- Saper, L,
*ℒ-modules and micro-support*, to appear in Annals of Mathematics (2018) - Saper, L,
*Perverse sheaves and the reductive Borel-Serre compactification*, in Hodge Theory and L²-analysis, edited by Ji, L, vol. 39 (2017), pp. 555-581, 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. 465-486 [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. 1-50, 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. 51-86, American Mathematical Society/International Press

- Saper, L,

**Selected Invited Lectures***L*, 2015, International Conference on Singularity Theory—in Honor of Henry Laufer's 70th Birthday, Tsinghua Sanya International Mathematics Forum, Sanya, China^{2}-cohomology of projective algebraic varieties*Perverse sheaves on compactifications of locally symmetric spaces*, July 28, 2015, Isaac Newton Institute, Cambridge, England [html]*Perverse sheaves and the reductive Borel-Serre compactification*, November 21-23, 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 Borel-Serre compactification*, March 1, 2011, Algebraic Geometry and Number Theory seminar at John Hopkins University, Baltimore*Self-dual sheaves and L²-cohomology 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, July 17 - August 4, 2006, International conference and instructional workshop on discrete groups, Morningside Center of Mathematics, Chinese Academy of Science, Beijing, China [slides]*Cohomology of Locally Symmetric Spaces**L²-Harmonic Forms on Locally Symmetric Sapces*, January 19, 2006, Tata Institute for Fundamental Research, Mumbai, India, 26 - 30 July, 2004, Center of Mathematical Sciences at Zhejiang University, China*L*, International Conference In Memory of Armand Borel: Algebraic groups, arithmetic groups, automorphic forms and representation theory^{2}-cohomology of locally symmetric spaces*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)*, 15-17 November 2002, Harvard University*Rapoport's conjecture on the intersection cohomology of the reductive Borel-Serre compactification*, April 25, 2000, Institut Henri Poincaré, Paris, France*L₂-cohomology of Algebraic Varieties*, August 23, 1990, International Congress of Mathematicians, Kyoto, Japan