Momentum Maps in Symplectic, Algebraic, and Discrete Geometry

辛几何是一门广泛而深入的学科，它以数学物理为基础，在过去二十年中得到了广泛的研究。辛流形是带有基本几何结构的空间，它为每个二维平面分配一个面积。因为辛流形在局部都“看起来一样”，区分它们中的两个涉及到全局属性和测量，比如一个球可以以保持面积的方式嵌入到空间中有多大（这是辛容量的一个例子）。幸运的是，许多辛流形都有内置的对称性，本项目的主要焦点是对称在辛几何中的作用（特别是那些被组织成哈密顿群作用的对称），以及当对称被用来“折叠”流形时产生的商空间。研究这些对称性的一个关键工具是动量图，在这个项目中，PI将对哈密顿系统的几何形状和动量图像的组合学之间的关系有更深的理解。更广泛地说，该奖项支持的活动将推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。PI还将继续在康奈尔大学培养研究生和指导博士后，并继续开展对更广泛的数学界和社会产生重大影响的活动，例如发表公开演讲，描述几何和拓扑的“大思想”，并在寻求改善全国本科数学教育的专业组织和团体中发挥重要的领导作用。如上所述，汉密尔顿的群体行为产生了动量图。这允许我们构造辛约化，辛约化也可以用几何不变理论进行代数描述。伪全纯曲线为辛不变量的研究提供了强有力的分析工具。使用这些工具的一个特别好的软件包是Hutchings的嵌入式接触同源性（ECH）。辛几何中的一个基本问题是将哈密顿系统的几何和拓扑与动量多面体的组合联系起来，反之亦然。在该奖项支持的项目中，PI将研究有关辛嵌入问题的问题，包括环4流形的辛容量和使用ECH容量的有理直纹曲面。PI的分析将增加我们对等变辛几何拓扑不变量的理解，包括为辛商建立一个满性和形式包，以及对复杂性空间的研究。她还在编写一本研究生教材，向学生和研究人员介绍该领域的关键方面。最后，PI将解决计算环拓扑中的一些问题。答案将依赖于各种领域的工具，包括代数几何、交换代数和代数拓扑。这组专题包括环面折叠辛流形的研究，辛流形的近亲，以及关于环面轨道的普通和弦不变量的问题。

Symplectic geometry is a broad and deep subject, with foundations in mathematical physics, that has seen much activity over the last two decades. Symplectic manifolds are spaces that carry a basic geometric structure which assigns an area to each two-dimensional plane. Because symplectic manifolds all locally `look the same', distinguishing two of them involves global properties and measurements, such as how large a ball can be embedded into the space in an area-preserving way (this is an example of a symplectic capacity). Fortunately, many symplectic manifolds have built-in symmetries, and the main focus in this project is the role of symmetries in symplectic geometry (especially, those symmetries that are organized into Hamiltonian group actions) and the quotient spaces that result when the symmetries are used to 'fold up' the manifold. A key tool in studying these symmetries is the momentum map, and in this project the PI will achieve a deeper understanding of the relationship between the geometry of a Hamiltonian system and the combinatorics of the momentum image. More broadly, the activities supported by this award will advance our knowledge in the fields of symplectic geometry and combinatorics, with applications to algebraic geometry, algebraic topology and mathematical physics.  The PI will also continue to train graduate students and mentor postdocs at Cornell, as well as continue activities which have a substantial impact on the broader mathematical community and society as large, such as giving public lectures describing `big ideas' from geometry and topology, and playing significant leadership roles in professional organizations and groups that seek to improve undergraduate mathematics education across the nation.As mentioned above, Hamiltonian group actions give rise to the momentum map. This allows us to construct the symplectic reduction, which can also be described algebraically using geometric invariant theory. Pseudoholomorphic curves provide strong analytic tools to study symplectic invariants. A particularly nice package for using these tools is Hutchings' embedded contact homology (ECH). A  fundamental problem in symplectic geometry is to relate the geometry and topology of a Hamiltonian system to the combinatorics of the momentum polytope, and vice versa.  In the projects supported by this award, the PI will study questions about symplectic embedding problems, including symplectic capacities of toric 4-manifolds and rational ruled surfaces using ECH capacities. The PI's analysis will add to our understanding of topological invariants in equivariant symplectic geometry, including building a surjectivity and formality package for symplectic quotients, and a study of complexity one spaces. She is also writing a graduate textbook  introducing students and researchers to the key aspects of the field. Finally, the PI will address a number of questions in computational toric topology.  The answers will rely on tools from  a variety of fields, including algebraic  geometry, commutative algebra, and algebraic topology. This set of projects includes the study of toric folded symplectic manifolds, a close cousin of symplectic toric manifolds, and  questions about ordinary and stringy invariants of toric orbifolds.