Lie Group Actions on Symplectic Manifolds

Cornell University, Ithaca NY

Investigators

Abstract

Abstract Award: DMS-0071625 Principal Investigator: Reyer Sjamaar The principal investigator plans to study invariants of Hamiltonian Lie group actions using methods from differential topology, Lie theory and geometric invariant theory. Expected applications include: extensions and stronger versions of classical inequalities in matrix analysis, and new results on moduli spaces of parabolic vector bundles. The tools to be used include methods for handling the singularities that inevitably arise from Hamiltonian actions (desingularization, quantization), which were developed in a previous NSF-funded project. This project involves graduate student participation and collaboration with researchers at other US and Canadian institutions. The type of question I seek to answer is: given the singular values of two matrices (of the same size), what are the possible singular values of their sum? In this form the question goes back at least to H. Weyl, who early in the last century found a partial answer and from this obtained his famous estimates for the eigenvalues of a Laplacian. The question is not only related to such classical spectral problems, but also to mechanics (classical and quantum) and to the representation theory of Lie groups. Two ramifications of the problem which I particularly wish to pursue are versions for noncompact groups and for the infinite-dimensional groups known as Kac-Moody groups.