The Geometry, Topology and Combinatorics of Hamiltonian Lie Group Actions

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AbstractAward: DMS-0604807Principal Investigator: Tara S. HolmThe principal investigator's principal focus in this proposal is groupactions and reduction in symplectic geometry. A fundamental problemin symplectic geometry is to relate the geometry and topology of aHamiltonian system to the combinatorics of the moment polytope, andvice versa. The principal players on the geometric side include thesymplectic manifold itself and its symplectic reductions.Combinatorially, the vertices, edges, k-dimensional faces, and eventhe chambers of the moment polytope play a significant role.Generically, a symplectic reduction has orbifold singularities. Theprincipal investigator outlines a program towards understanding thegeometry and topology of such orbifolds. These results will provideinsight into the structure of the Chen-Ruan orbifold cohomology ring,yielding information about orbifold Gromov-Witten invariants. Thiswork has also naturally led to foundational questions regarding groupactions on orbifolds. Finally, the PI will distill the geometry andtopology of the Hamiltonian system into combinatorial data associatedto the moment polytope. This will shed new light on recentcombinatorial results concerning intervals in the Bruhat order on anyCoxeter group.Symplectic geometry is the mathematical framework for describingphenomena in mathematical physics, from classical mechanics to stringtheory. The moment map is an important tool that translates thesymmetries of a physical system into discrete data. An expert insymplectic geometry, the PI will achieve a deeper understanding of therelationship between the geometry of a symplectic manifold and thecombinatorics of the moment map data. Her work will shed light onstringy invariants in this context. The proposed activities willadvance our knowledge in the fields of symplectic geometry andcombinatorics, with applications to algebraic geometry, algebraictopology and mathematical physics. The PI's broader objectivesinclude increasing the participation and visibility of women inresearch mathematics, and enhancing the undergraduate experience inmathematics.