Vortices, Quilts, and Quasimaps

PI将使用规范的Floer理论来研究拉格朗日环面在可能的开辛流形上的不可移动性，并与McDuff的结果进行比较。PI将与F.Ziltener、他的前博士后导师E.Gonzalez和他现在的学生Sushmita Venugopalan一起研究商的量子上同调以及与规范的Gromov-Witten理论的关系。具体地说，他将(A)证明Kalkman穿越墙公式的一个量子版本，该公式控制了Gromov-Witten不变量在辛商变化下的行为，并证明了在许多情况下在Crepantflop下的二元性等价。(B)找到辛商的量子上同调环的表示，例如Toricorbilold和箭形簇。与K.Wehheim和他以前的学生S.Ma‘u一起，PI将继续研究全纯被子，Floer-Fukaya理论中拉格朗日对应的函数性，以及与低维拓扑的关系。这些研究项目将提高我们对辛几何的理解，辛几何是理解经典力学的数学框架，特别是对于依赖时间的能量函数。这些项目中的许多都与辛约化操作下的辛不变量的行为有关，辛约化是通过使用对称性来减少系统的自由度。这些不变量也出现在物理学家对量子力学“西格玛模型”的研究中，他们对它们在辛约化下的行为做出了许多或预测，其中一些将作为该项目的一部分得到验证和推广。关于全纯被子的项目将应用于我们对三维和四维空间的理解，特别是使用规范理论构造的不变量，这些规范理论被猜测(在某些情况下是已知的)具有辛解释。PI还将继续与中学数学教师打交道。

The PI will use gauged Floer theory to study non-displaceability ofLagrangian tori in possibly open symplectic manifolds viacompactifications such as symplectic toric orbifolds, and compare withresults of McDuff on displaceability. With F. Ziltener, his formerpostdoctoral advisee E. Gonzalez, and his current student SushmitaVenugopalan the PI will study the quantum cohomology of quotients andrelationships with gauged Gromov-Witten theory. In particular he will(a) prove a quantum version of Kalkman's wall-crossing formula whichgoverns the behavior of Gromov-Witten invariants under variation ofsymplectic quotient and proves birational equivalence under crepantflops in many cases, as conjectured by Ruan (b) find presentations ofquantum cohomology rings of symplectic quotients such as toricorbifolds and quiver varieties. With K. Wehrheim and his formerstudent S. Ma'u the PI will continue to study holomorphic quilts,functoriality of Lagrangian correspondences in Floer-Fukaya theory,and relationship with low-dimensional topology. In particular, the PIwill prove the A-infinity version and the exact triangle for fiberedDehn twists.These research projects will improve our understanding of symplecticgeometry, which is a mathematical framework for understandingclassical mechanics, particularly for time-dependent energy functions.Many of these projects are related to the behavior ofsymplectic invariants under the operation of symplectic reduction inwhich the number of degrees of freedom of a system is reduced by usingsymmetry. These invariants also appear in the study ofquantum-mechanical ``sigma models'' by physicists, whohave made a number or predictions about their behavior undersymplectic reduction, some of which will be verified and generalizedas part of the project. The project on holomorphic quilts will haveapplications to our understanding of three- and four-dimensionalspaces, especially invariants constructed using gauge theories whichare conjectured (and in some cases known) to have symplecticinterpretations. The PI will also continue his involvement with middleschool mathematics teachers.