Momentum maps in symplectic, algebraic and discrete geometry

塔拉霍尔姆的主要重点是在这一建议的作用，群行动和conquients辛和代数几何。 虽然问题，霍尔姆和她的合作者计划解决起源于辛和代数几何，他们的解决方案将涉及方法，并有应用代数拓扑，数学物理和组合。 在所有这些问题中的一个基本工具是动量映射，它将哈密顿系统的几何和拓扑与动量多面体的离散几何联系起来。 PI将使用此工具为Lam、Schilling和Shimozono的仿射格拉斯曼组合K理论计算创建一个通用框架。 霍尔姆和合作者S.托尔曼将使用动量映射来增强我们对辛同调的积分上同调环的理解。 与A. Bertiger和K. Taiplae，霍尔姆将调查量子不变量的辛同分，与应用组合数学和代数几何。 有许多例子表明，具有群作用的空间不是哈密顿系统，但具有它们的许多拓扑性质。 与M。Harada，N. Ray和G.威廉姆斯，霍尔姆将研究不变量的orbifolds的背景下，环面拓扑。 霍尔姆和A.皮雷斯将研究复曲面折纸流形的拓扑。 这些分类的折纸模板，其组合应该提供洞察到原来的流形的拓扑不变量。 最后，霍尔姆和Y. Karshon正在写一本研究专著，将传播强大的方法，从等变辛几何，有应用程序在整个数学研究的数学家广泛的受众。辛几何是数学物理现象描述的数学框架，从经典力学到弦理论。 动量映射是将物理系统的对称性转化为离散数据的重要工具。 专家在辛几何，霍尔姆将实现一个更深入的理解之间的关系的几何辛流形和组合的时刻地图数据。 与合作者一起，她将调查的问题，其起源是在辛和代数几何，其解决方案将有广泛的应用。拟议的活动将推进我们在辛几何，代数几何，代数拓扑，组合学和数学物理领域的知识。 霍尔姆的更广泛的目标包括增加参与和知名度的妇女在研究数学，并提高本科生的经验，数学。

Tara Holm's principal focus in this proposal is the role of group actions and quotients in symplectic and algebraic geometry. While the questions that Holm and her collaborators plan to address have origins in symplectic and algebraic geometry, their solutions will involve methods from and have applications to algebraic topology, mathematical physics, and combinatorics. A fundamental tool in all of these problems is the momentum map, which relates the geometry and topology of a Hamiltonian system to the discrete geometry of the momentum polytope. The PI will use this tool to create a general framework for Lam, Schilling and Shimozono's combinatorial K-theory calculations of the affine Grassmannian. Holm and collaborator S. Tolman will use the momentum map to enhance our understanding of the integral cohomology rings of symplectic quotients. Together with A. Bertiger and K. Taiplae, Holm will investigate quantum invariants of symplectic quotients, with applications to combinatorics and algebraic geometry. There are many examples of spaces endowed with group actions that are not Hamiltonian systems, but enjoy many of their topological properties. Together with M. Harada, N. Ray and G. Williams, Holm will study invariants of orbifolds in the context of toric topology. Holm and A. Pires will investigate the topology of toric origami manifolds. These are classified by origami templates, whose combinatorics should provide insight into the topological invariants of the original manifolds. Finally, Holm and Y. Karshon are writing a research monograph that will disseminate the powerful methods from equivariant symplectic geometry that have applications across mathematics to a wide audience of research mathematicians.Symplectic geometry is the mathematical framework for describing phenomena in mathematical physics, from classical mechanics to string theory. The momentum map is an important tool that translates the symmetries of a physical system into discrete data. An expert in symplectic geometry, Holm will achieve a deeper understanding of the relationship between the geometry of a symplectic manifold and the combinatorics of the moment map data. Together with collaborators, she will investigate questions whose origins are in symplectic and algebraic geometry, and whose solutions will have wide applications. The proposed activities will advance our knowledge in the fields of symplectic geometry, algebraic geometry, algebraic topology, combinatorics and mathematical physics. Holm's broader objectives include increasing the participation and visibility of women in research mathematics, and enhancing the undergraduate experience in mathematics.