Dan M. Barbasch - US grants

9401176 Speh/Barbasch Speh will continue her joint work with Rohlfs on cuspidal cohomology. She also will investigate the analytic torsion of locally symmetric spaces. In joint work the PI's will study the residual spectrum of the split orthogonal and symplectic group. Barbasch will continue his research on unipotent representations for real groups, in particular their character theory and unitarity. He also will study spherical unitary dual for split p- adic groups. The theory of Lie groups, named in honor of the Norwegian mathematician Sophus Lie, has been one of the major themes in twentieth century mathematics. As the mathematical vehicle for exploiting the symmetries inherent in a system, the representation theory of Lie groups has had a profound impact upon mathematics itself, particularly in analysis and number theory, and upon theoretical physics, especially quantum mechanics and elementary particle physics. ***

ABSTRACT Speh/Barbasch In joint work the PI's will study the residual spectrum of the split classical groups. Barbasch will continue his research on unipotent representations for both real and p-adic groups, in particular their character theory and unitarity. Speh will continue her work with J.Rohlfs on the stabilisation of geometric invariants of locally symmetric spaces. The theory of Lie groups, named in honor of the Norwegian mathematician Sophus Lie, has been one of the major themes in twentieth century mathematics. As the mathematical vehicle for exploiting the symmetries inherent in a system, the representation theory of Lie groups has had a profound impact upon mathematics itself, particularly in analysis and number theory, and upon theoretical physics, especially quantum mechanics and elementary particle physics.

We propose to continue two successful local outreach activities at the high school level, improving and diversifying our collection of activities and materials. The Math Explorer's Club (MEC), meets once a week at Ithaca High School in the 2:45-3:30 time slot set aside for after school clubs. Activities are organized into six to eight week modules, each of which is devoted to a specific topic and led by a Cornell graduate student. The aim of MEC is to show high school students that there is more to mathematics than the centuries old progression from algebra to calculus, and to introduce them a wide spectrum of mathematical ideas, through games, puzzles, model building, and other explorations. The Senior Seminar is a class on advanced mathematics designed for students who have taken most of the available math classes. It meets for one period during school hours three days week at the high school and introduces students to topics such as cryptography, number theory, graph theory, game theory, combinatorics, probability, and topology that they would typically not see until their junior or senior years in college.

The intellectual merit of this proposal lies in the development of high quality materials that introduce advanced mathematical concepts to high schools in an engaging and entertaining manner. The broader impact of this proposal is to export these successful activities to a number of other communities across the country. To facilitate the development of similar activities in other locations, the faculty coordinator and some of the graduate student members will make presentations each year at the Joint Membership meetings of the MAA and AMS, and work with individuals from other universities that want to use our materials. We will create a web-based archive of activities for the MEC and of lecture notes for the Senior Seminar that can be used by students and teachers throughout the country. The availability of tested materials will make it much easier to start similar activities in other locations. The Math Explorer's Club and the Senior Seminar aim to interest more high school students in majoring in math in college, and to increase the number of young women and men who choose mathematics as a career, by increasing the flow into the pipeline.

Abstract
Barbasch

There are two major themes in this proposal, the determination of the unitary dual, and the structure of unipotent representations. Barbasch will continue and extend his previous work to find necessary and sufficient conditions for unitarity for classes of Langlands parameters such as principal series in real and p-adic groups. The aim is to find proofs that are as uniform and conceptual as possible, and express the answers in a closed form. Unipotent representations are (conjectured to be) the building blocks of the unitary dual. Their character theory plays an important role in the theory of automorphic forms. Barbasch will study the character theory of unipotent representations in the real case, with an emphasis on relations to endoscopic groups, particularly twisted ones. Realizations of unipotent representations as local factors of automorphic forms, and in the context of the dual pairs correspondence are also envisioned. This research will provide a deeper understanding of representation theory of reductive Lie groups, in particular unitary representations, which play an essential role in the applications of representation theory to automorphic forms, mathematical physics analysis and geometry.

This research will generate problems suitable for Ph.D. theses as well as combinatorial problems for undergraduate research. Together with several faculty members at Cornell, Barbasch will run the ``Sophus Lie Days at Cornell'' an instructional conference designed to expose undergraduates and graduate students in mathematics and other sciences to the theory of Lie groups and their representations, one of the aims being to get them interested and involved in research at an early stage. The results of this research will be disseminated through research papers, talks at seminars and conferences, and various web sites.

This project has two primary goals. The first is to solve the problem of the unitary dual: to describe the irreducible unitary representations of real reductive Lie groups. The primary tool is an algorithm to compute the unitary dual of any given group, which we are implementing inside the "atlas" software. We plan to use this information to prove results about the unitary dual, beginning with the unitarity of Arthur's unipotent representations. The second primary goal is to make information about representation theory of real groups accessible to non-specialists, via the software, a web site, public workshops, and other means. The atlas software is freely available on the atlas web site, and will continue to be maintained there indefinitely.

The idea of using symmetry to study problems in mathematics and science dates back to Fourier's work on heat nearly two hundred years ago. In the hands of Hermann Weyl, Eugene Wigner, and Andre Weil, symmetry has come to play a central role in quantum mechanics and in number theory. Lie groups, named after the Norwegian mathematician Sophus Lie, are the mathematical objects underlying symmetry. Representation theory studies all of the ways a given symmetry, or Lie group, can manifest itself. The problem of understanding all "unitary" representations (in which the symmetry operations preserve lengths) is one of the most important unsolved problems in the subject, and has potential applications in many areas; for example, it is an abstract version of the question, "what quantum mechanical systems can admit a certain kind of symmetry?"

Laws of nature, in particularly those of physics, satisfy basic symmetries. In mathematical language, these symmetries are most often expressed in terms of Lie groups and their representations (named after Sophus Lie, a 19th century mathematician who pioneered these concepts in his study of solutions of differential equations). This project is concerned with properties of unitary representations. The modern study of unitary representations stems from quantum physics. In addition, they play a crucial role in many other applications such as tomography, crystallography, and signal processing.

In more technical terms, this project is focused on properties of unipotent representations, which are the building blocks of the unitary duals of real and p-adic reductive groups. The determination of the unitary dual is a major problem in the representation theory of such groups. The PI will formulate and sharpen conjectures on the signatures of hermitian forms for modules of the affine Hecke algebra. These are essential for the determination of the unitary duals of p-adic groups. Another focus of this project is to make explicit the role of unipotent representations in the description of the unitary dual. In addition to linear groups, the PI will study these notions for nonlinear and disconnected groups that arise in mathematical physics and in the study of automorphic forms.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.