Vortices, Quilts, and Quasimaps

涡流、面组和拟图

• 批准号: 1207194
• 负责人: Christopher Woodward
• 金额: $ 43.67万
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-06-01 至 2017-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1207194&HistoricalAwards=false
• 关键词: Vortices; Quilts; Quasimaps

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.

PI将利用规范Floer理论研究可能开辛流形中Lagrange环面的非可积性，并与McDuff关于可积性的结果进行比较。 与F. Ziltener，他的前博士后，见E。Gonzalez和他现在的学生Sushmita Venugopalan PI将研究量子上同调和与规范Gromov-Witten理论的关系。 特别是，他将（a）证明量子版本的Kalkman的跨壁公式whichgoverning行为的Gromov-Witten不变量下的变化辛商和证明双有理等价下crepantflops在许多情况下，如阮（B）发现介绍量子上同调环的辛商，如toricorbifolds和gravity品种。 与K. Wehrheim和他的学生S. PI的Ma 'u将继续研究全纯被子，弗洛尔-卡塔尼亚理论中拉格朗日对应的功能性，以及与低维拓扑的关系。 特别是，PI将证明A-无穷版本和纤维德恩扭曲的精确三角形。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，尤其是在时间上，这些项目中的许多都与辛不变量在辛约化操作下的行为有关，在辛约化操作中，系统的自由度是通过使用对称性来减少。 这些不变量也出现在物理学家对量子力学“sigma模型”的研究中，他们对它们在辛约化下的行为做出了一些预测，其中一些将作为项目的一部分进行验证和推广。 全纯绗缝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建的不变量，这些规范理论被证明（在某些情况下已知）具有辛解释。 PI还将继续参与中学数学教师的工作。