Gromov Witten Invariants of Singular Spaces

摘要奖：DMS-0605003主要研究员：Eleny-Nicoleta Ionel该建议旨在通过结合几何、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辛流形相对于奇异子空间的Gromov-Wittenn不变量(具有温和的奇性，例如正常交叉点)，并利用这些不变量证明一个广义辛和公式。该项目同时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发生了什么，揭示了令人惊讶的新特征，但也揭示了新的挑战和复杂性。这项工作有几个有趣的应用，其中一个是作为一个单独的项目提出的，它涉及到使用Donaldson除数来给出虚拟基本循环的简单几何定义的想法。第三个投影是由两位弦理论家R.Gopakumar和C.Vafa提出的猜想。PI一直在与Thomas Parker合作研究6维Gromov不变量的结构定理；这将具有许多与Gopakumar-Vafa猜想相同的结果，并且它与C.TaubesDeep关于4维Seiberg-Witten与Gromov不变量之间关系的工作很好地结合在一起。拟议的工作位于弦理论和辛拓扑的交叉点上。弦理论作为统一广义相对论和粒子物理的潜在候选者而发展起来。这个理论的细节已经被证明是非常丰富的，并在数学上激发了许多显著的结果。但数学上的结果也指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，贯穿本提案中所有项目的主题之一是辛拓扑可以为弦理论提供见解。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。其中一个拟议的项目将涉及研究生，让他们在研究生生涯的早期参与尖端研究。国际数学联合会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。

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.