Gromov Witten Invariants of Singular Spaces

奇异空间的 Gromov Witten 不变量

• 批准号: 0605003
• 负责人: Eleny-Nicoleta Ionel
• 金额: $ 33.31万
• 依托单位: Stanford University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-07-01 至 2011-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0605003&HistoricalAwards=false
• 关键词: Gromov; Witten; Invariants; Singular; Spaces

AbstractAward: DMS-0605003Principal Investigator: Eleny-Nicoleta IonelThe proposal is aimed at increasing understanding of thestructure of the Gromov-Witten invariants of symplectic manifoldsby combining together geometric, topological and analyticalmethods. The first project aims to define the Gromov-Witteninvariants of a symplectic manifold relative a singular subspace(with mild singularities, e.g. normal crossings) and to usethose invariants to prove a generalized symplectic sumformula. The project uses both geometric and analytical methodsto investigate what happens to the moduli spaces of holomorphicmaps during certain kinds of natural degenerations, revealingsurprising new features but also new challenges andcomplications. There are several interesting applications of thiswork, one of them presented as a separate project, which involvesthe idea of using a Donaldson divisor to give a simple geometricdefinition of the virtual fundamental cycle. The third projectis motivated by a conjecture made by two string theorists,R. Gopakumar and C. Vafa. The PI has been working with ThomasParker on a structure theorem for Gromov invariants in 6dimensions; this would have many of the same consequences as theGopakumar-Vafa Conjecture, and it ties in nicely with C. Taubesdeep work on the relation between Seiberg-Witten and Gromovinvariants in 4 dimensions.The proposed work lies at the intersection of string theory andsymplectic topology. String theory developed as a potentialcandidate for unifying general relativity and particle physics.The details of this theory have turned out to be extraordinarilyrich, and have inspired many remarkable results inmathematics. But results in mathematics have also guided andinspired many new discoveries in string theory. It is hoped thatthis project will have the broader impact of adding momentum tothe growing interaction between mathematicians and theoreticalphysicists. In particular, one of the themes running through allthe projects in this proposal is that symplectic topology cancontribute insights into string theory. The research and otheractivities of the PI also have impact on the education of nextgeneration of mathematicians. One of the proposed projects willinvolve graduate students, engaging them in cutting-edge researchearly in their graduate career. The PI has also been active inencouraging and guiding women graduate students and will continueto strongly support young women entering mathematics.

Abstractaward：DMS-0605003原理研究者：Eleny-Nicoleta ionelth The The提案旨在增加对Symplectic formoltsby的Gromov-indentures的理解，将几何学，拓扑，拓扑，拓扑，拓扑，拓扑，拓扑，拓扑，拓扑，拓扑，拓扑，分析和分析方法结合在一起。第一个项目旨在定义一个相对奇异子空间（具有轻度奇异性，例如正常杂交）和用户不变的不变式的simblectic歧管的gromov-wittenInvariants，以证明具有广义的象征性sumformula。该项目同时使用几何方法和分析方法研究了在某些类型的自然变性，揭示新功能以及新的挑战和复杂性的情况下，研究全体形态图的模量空间会发生什么。这项工作有几个有趣的应用，其中一个是一个单独的项目，其中涉及使用唐纳森除数来给出简单的几何学定义的虚拟基本周期。 第三个项目是由两个字符串理论家r的猜想所激发的。 Gopakumar和C. Vafa。 PI一直与Thomasparker合作，在6Dimons中为Gromov不变的结构定理。这将产生与Gopakumar-Vafa猜想的许多相同后果，并且与C. Taubesdeep的工作很好地联系在一起，在4个维度中，Seiberg-Witten和Gromovinvariants之间的关系。拟议的工作在于弦乐理论和符号拓扑的相交。 弦理论是作为统一一般相对性和粒子物理学的潜在范围而开发的。该理论的细节已被异常富含，并激发了许多显着的结果。但是数学的结果也指导了弦理论中的许多新发现。 希望这项项目将在数学家和理论生理学家之间增加势头的更大影响。 特别是，通过本提案中所有项目运行的主题之一是，象征性的拓扑可以使弦理论的见解。 PI的研究和其他活性也会影响数学家的下一代教育。 拟议的项目之一Willinvolve研究生，使他们从事研究生生涯的尖端研究。 PI还积极地失去了妇女研究生，并将继续支持年轻女性进入数学。