Momentum maps in symplectic, algebraic and discrete geometry

Tara Holm在这个提案中的主要关注点是群体行为和商在辛几何和代数几何中的作用。虽然霍尔姆和她的合作者计划解决的问题起源于辛几何和代数几何，但他们的解决方案将涉及代数拓扑、数学物理和组合学的方法并具有应用。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将使用这个工具为Lam、Schilling和Shimozono对仿射格拉斯曼粒子的组合k理论计算创建一个一般框架。Holm和合作者S. Tolman将使用动量映射来增强我们对辛商的整上同环的理解。霍尔姆将与A. Bertiger和K. Taiplae一起研究辛商的量子不变量，并将其应用于组合学和代数几何。有许多被赋予群作用的空间的例子，它们不是哈密顿系统，但具有它们的许多拓扑性质。霍尔姆将与M. Harada， N. Ray和G. Williams一起研究环面拓扑环境下的轨道不变量。Holm和A. Pires将研究环折纸流形的拓扑结构。这些是通过折纸模板分类的，其组合学应该提供对原始流形的拓扑不变量的洞察力。最后，Holm和Y. Karshon正在撰写一本研究专著，将从等变辛几何中获得的强大方法传播给广泛的研究数学家，这些方法在整个数学领域都有应用。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。动量图是将物理系统的对称性转化为离散数据的重要工具。作为辛几何方面的专家，霍尔姆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。与合作者一起，她将研究的问题，其根源是辛几何和代数几何，其解决方案将有广泛的应用。这些活动将提高我们在辛几何、代数几何、代数拓扑、组合学和数学物理等领域的知识。霍尔姆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。

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.