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首席研究员：Eleny-Nicoleta Ionel该提案旨在通过结合几何，拓扑和分析方法来增加对辛流形的Gromov-Witten不变量结构的理解。第一个项目的目的是定义一个辛流形相对于奇异子空间（具有轻度奇异性，例如正常交叉）的Gromov-Witteninvariants，并使用这些不变量来证明一个广义辛求和公式。该项目使用几何和分析方法来研究在某些类型的自然退化过程中全纯映射的模空间会发生什么，揭示了令人惊讶的新特征，但也带来了新的挑战和复杂性。这项工作有几个有趣的应用，其中之一是作为一个单独的项目提出的，它涉及到使用唐纳森因子给出虚基本周期的简单几何定义的想法。 第三个项目是由两个弦理论家提出的一个猜想所激发的，R。Gopakumar和C.瓦法PI一直在与PennasParker合作研究6维空间中Gromov不变量的结构定理;这将有许多与Gopakumar-Vafa猜想相同的结果，并且它与C很好地结合在一起。Taubes深入研究了4维空间中Seiberg-Witten变量和Gromovin变量之间的关系，这项工作处于弦理论和辛拓扑学的交叉点。 弦理论是作为统一广义相对论和粒子物理学的潜在候选者而发展起来的。这个理论的细节已经被证明是异常丰富的，并激发了数学界许多非凡的成果。但数学的结果也引导和启发了弦理论的许多新发现。 人们希望这个项目将产生更广泛的影响，为数学家和理论物理学家之间日益增长的互动增加动力。 特别是，贯穿所有项目的主题之一是辛拓扑学可以对弦理论有所贡献。PI的研究和其他活动也对下一代数学家的教育产生了影响。 其中一个提议的项目将涉及研究生，让他们在研究生生涯的早期从事前沿研究。 PI还积极鼓励和指导女研究生，并将继续大力支持年轻女性进入数学领域。