The Geometry, Topology and Combinatorics of Hamiltonian Lie Group Actions

摘要：项目编号：dms -0604807项目负责人：Tara S. holm项目负责人主要研究辛几何中的群和约简问题。辛几何的一个基本问题是将哈密顿系统的几何和拓扑与矩多面体的组合联系起来，反之亦然。几何方面的主要参与者包括辛流形本身及其辛约化。结合起来，顶点、边缘、k维面，甚至是矩多面体的腔室都起着重要的作用。一般来说，辛约化具有轨道奇异性。首席研究员概述了一个程序，以了解这些轨道的几何和拓扑结构。这些结果将为chen -阮轨道上同环的结构提供深入的见解，并得到关于轨道Gromov-Witten不变量的信息。这项工作也自然导致了关于在轨道上的群的基本问题。最后，PI将哈密顿系统的几何和拓扑提取成与矩多面体相关的组合数据。这将为最近关于任何coxeter群的Bruhat序间隔的组合结果提供新的线索。辛几何是描述数学物理现象的数学框架，从经典力学到弦理论。矩图是将物理系统的对称性转化为离散数据的重要工具。作为辛几何的专家，PI将对辛流形的几何和矩图数据的组合之间的关系有更深的理解。她的工作将阐明在这种背景下的字符串不变量。这些活动将促进我们在辛几何和组合学领域的知识，并将其应用于代数几何、代数拓扑和数学物理。该计划更广泛的目标包括提高女性在数学研究中的参与度和可见度，以及提高大学生在数学方面的经验。

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.