The group will work on eigenvalue and saturation problems for reductive groups.

A large part of mathematics 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 the 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 is that of studying the representations of a group in terms of matrices. This problem in turn has been organized into subproblems. One of the most important of these is that of decomposing tensor products of representations into sums of representations. A crucial point is that these two fundamental problems are very closely related. This relationship is understood in special cases, and a major effort of this FRG will be to pin down this relationship in general cases.

This FRG grant will play an important role in the further development of a national group of scientists working in the areas common to Lie theory, topology, algebraic and symplectic geometry, combinatorics, and the theory of buildings. Graduate students and postdocs at the three universities involved, and also at such other universities as Duke, NC State, Johns Hopkins, UC Berkeley, and Stanford, will have the opportunity to learn and work on these exciting and fundamental problems through meetings and workshops to be organized by the PIs.