Moduli Spaces Relative Singular Divisors and Lagrangians

AbstractAward：DMS-0905738首席研究员：Eleny Ionel该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。该提案旨在通过结合来自几个不同研究领域的想法，增加对稳定全纯映射的模空间结构的理解。该项目的核心是试图使用几何和分析方法来调查在某些自然退化过程中这些模量会发生什么。特别地，它导致定义相对于某些类型的奇异子空间的Gromov-Witten不变量的厌恶，包括相对于一个正常的交叉辛因子和一个Lagrangian以规定的方式相交。 它还研究了这些不变量的行为下适当的光滑的环境空间或thedivisor和Lagragian.The拟议的工作在于弦理论和辛拓扑的交叉。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证明是非常丰富的，并激发了许多显着的数学结果。但数学的结果也引导和启发了弦理论和镜像对称的许多新发现。希望这个项目将有助于数学和其他科学的各个领域之间日益增长的相互作用。特别是，贯穿这一提议的主题之一是辛拓扑可以为弦论提供见解。

AbstractAward: DMS-0905738Principal Investigator: Eleny IonelThis award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). The proposal is aimed at increasing understanding of thestructure of the moduli spaces of stable holomorphic maps bycombining ideas from several different fields of research. Thecore of the project seeks to use geometric and analytical methodsto investigate what happens to these moduli during certainnatural degenerations. In particular it leads to defining aversion of Gromov-Witten invariants relative certain type ofsingular subspaces, including relative both a normal crossingsymplectic divisor and a Lagrangian intersecting in a prescribedway. It also investigates how these invariants behave underappropriate smoothings of either the ambient space or of thedivisor and the Lagragian.The proposed work lies at the intersection of string theory andsymplectic topology. String theory developed as a potentialcandidate for unifying general relativity and particlephysics. The details of this theory have turned out to beextraordinarily rich, and have inspired many remarkable resultsin mathematics. But results in mathematics have also guided andinspired many new discoveries in string theory and mirrorsymmetry. It is hoped that this project will contribute to thegrowing interaction between various fields of mathematics andother sciences. In particular, one of the themes running throughthis proposal is that symplectic topology can contribute insightsinto string theory.