DMS 0554254, PI: John Millson, Co-PI: Thomas Haines DMS 0554349, PI: Michael Kapovich DMS 0554247, PI: Shrawan Kumar, Co-PI: Prakash Belkale
Research in Lie theory has undergone striking advances in a number of directions spurred by and giving rise to discoveries in topology, symplectic and algebraic geometry and combinatorics. The recent solutions by Klyachko of the eigenvalues of a sum of Hermitian matrices problem and by Knutson and Tao of the saturation and Horn conjectures are of particular relevance to this proposal. Both these problems are associated to the group of invertible n by n matrices. The discovery of quantum cohomology and the quantum Schubert calculus led to the solution of analogous problems for the group of unitary n by n matrices. The PIs P.Belkale, T. Haines, M. Kapovich, S. Kumar and J. Millson propose to attack for a general reductive group G the problems previously solved for the groups of invertible unitary n by n matrices. The history of Lie theory has shown that it is of critical importance to understand in the context of general Lie groups results proved initially for the group of invertible n by n matrices.
A large part of mathematics and physics has been involved with the study of eigenvalues, for example determining the modes of vibration of a violin string or the energy levels of an atom amounts to finding eigenvalues of a Hermitian linear operator. A fundamental problem is to determine the possibilities for the eigenvalues of the sum of two operators given the eigenvalues of each one. Another fundamental problem with its roots in physics is the problem of studying the representations of an abstract group as a group of matrices. This problem in turn has been organized into subproblems. One of the most important of these is the problem of decomposing (tensor) products of representations as sums of representations. The point of this proposal is that these two basic problems, the eigenvalue of the sum problem and the decomposing (tensor) products problem are very closely related. The authors propose to pin down this relationship (already well-understood for special cases) for the general case. This FRG grant will play a fundamental role in the further development of a national group of scientists working on the area common to Lie theory, topology, algebraic and symplectic geometry, combinatorics and the theory of buildings. A class of young mathematicians especially graduate students and postdocs in the mid-Atlantic area (including Washington and Chapel Hill) and the greater (San Francisco) bay area (including Davis) will have the opportunity to learn about and work on exciting and fundamental problems through the Meetings and Workshops envisaged by the PIs. This class already includes the nine graduate students presently advised by the PIs. We expect to include other graduate students and postdocs associated to the very strong programs in representation theory and geometry at the University of Maryland, the University of North Carolina at Chapel Hill and the University of California at Davis. We also expect that graduate students, postdocs and faculty from neighboring universities such as Johns Hopkins, Duke, North Carolina State University, UC-Berkeley and Stanford will participate in and profit from the FRG grant. This award is jointly funded by the programs in Analysis, and Algebra, Number Theory, & Combinatorics, and Geometric Analysis.
