Vortices, Quilts, and Quasimaps

PI将使用量规Floer理论研究可能开放的辛流形（如辛环面轨道）中拉格朗日环面的不可置换性，并与McDuff关于可置换性的结果进行比较。PI将与F. Ziltener，他的前博士后导师E. Gonzalez，以及他现在的学生SushmitaVenugopalan一起研究商的量子上同调以及与测量Gromov-Witten理论的关系。特别是，他将(a)证明Kalkman的过壁公式的量子版本，该公式支配着辛商变化下的Gromov-Witten不变量的行为，并证明在许多情况下，如阮氏猜想的那样，在crepantflops下的双相等性(b)找到辛商的量子上同环的表示，如环曲和颤变。与K. Wehrheim和他以前的学生S. Ma'u一起，PI将继续研究全纯被子，flower - fukaya理论中拉格朗日对应的泛函性，以及与低维拓扑的关系。特别是，pii将证明a -∞版本和纤维dehn扭曲的确切三角形。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，特别是与时间相关的能量函数。这些项目中的许多都与辛约简操作下辛不变量的行为有关，辛约简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，他们已经对它们在辛还原下的行为做出了一些预测，其中一些将作为项目的一部分得到验证和推广。全纯被子的项目将应用于我们对三维和四维空间的理解，特别是使用推测（在某些情况下已知）具有辛解释的规范理论构造的不变量。PI还将继续与中学数学教师合作。

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.