The Geometry, Topology and Combinatorics of Hamiltonian Lie Group Actions

University Of Connecticut, Storrs CT

Investigators

Abstract

Abstract Award: DMS-0604807 Principal Investigator: Tara S. Holm The principal investigator's principal focus in this proposal is group actions and reduction in symplectic geometry. A fundamental problem in symplectic geometry is to relate the geometry and topology of a Hamiltonian system to the combinatorics of the moment polytope, and vice versa. The principal players on the geometric side include the symplectic manifold itself and its symplectic reductions. Combinatorially, the vertices, edges, k-dimensional faces, and even the chambers of the moment polytope play a significant role. Generically, a symplectic reduction has orbifold singularities. The principal investigator outlines a program towards understanding the geometry and topology of such orbifolds. These results will provide insight into the structure of the Chen-Ruan orbifold cohomology ring, yielding information about orbifold Gromov-Witten invariants. This work has also naturally led to foundational questions regarding group actions on orbifolds. Finally, the PI will distill the geometry and topology of the Hamiltonian system into combinatorial data associated to the moment polytope. This will shed new light on recent combinatorial results concerning intervals in the Bruhat order on any Coxeter group. Symplectic geometry is the mathematical framework for describing phenomena in mathematical physics, from classical mechanics to string theory. The moment map is an important tool that translates the symmetries of a physical system into discrete data. An expert in symplectic geometry, the PI will achieve a deeper understanding of the relationship between the geometry of a symplectic manifold and the combinatorics of the moment map data. Her work will shed light on stringy invariants in this context. The proposed activities will advance our knowledge in the fields of symplectic geometry and combinatorics, with applications to algebraic geometry, algebraic topology and mathematical physics. The PI's broader objectives include increasing the participation and visibility of women in research mathematics, and enhancing the undergraduate experience in mathematics.