Properties of Gromov-Witten Invariants

摘要奖：DMS-0306299首席研究员：Eleny-Nicoleta Ionel这个建议旨在通过结合来自不同研究领域的想法来理解辛流形的Gromov-Witten不变量的结构。第一个项目的动机是两位物理学家Gopakumar和Vafa的猜想。这个猜想暗示了对Calabi-Yau3-折叠的Gromov不变量的几个令人惊讶的限制。其目的是通过采用Taubes发展的一些分析技术来证明猜想公式，将Seiberg-Witten和Gromov不变量联系在4维空间中。第二个项目的目标是在有标记的黎曼曲面上复杂结构的模空间的上同调环上获得新的关系。使用前一篇论文中介绍的技术，PI发现了几个有趣的关系族，其中一个证明了Faber十年前的一个猜想。最后一个项目是将Gromov-Witten不变量的求和公式推广到比从辛和中出现的变形更一般的变形。在一般情况下，似乎已经出现了几个有趣的新现象。拟议的工作位于弦理论和辛拓扑的交叉点上。弦理论是作为宇宙统一理论的潜在候选者而发展起来的，它扩展了艾斯坦的相对论。它基于这样一种想法，即基本粒子(如电子、光子)应该被认为不是点，而是更小的振动环。计算出这一理论的细节被证明是相当微妙的，而哈辛反过来又启发了许多数学上的显著结果。但数学上的基本成果也启发了物理学上的许多新发现。希望这个项目将有助于增加数学和高能物理之间的互动。与此同时，其中一个项目将涉及研究生。

AbstractAward: DMS-0306299Principal Investigator: Eleny-Nicoleta IonelThis proposal aims to understand the structure of theGromov-Witten invariants of symplectic manifolds by combiningtogether ideas coming from different fields of research. Thefirst project is motivated by a conjecture made by twophysicists, Gopakumar and Vafa. The conjecture implies severalsurprising restrictions on the Gromov invariants of a Calabi-Yau3-fold. The goal is to prove the conjectured formula by adaptingsome analytical techiques developed by Taubes to relate theSeiberg-Witten and Gromov invariants in 4-dimensions. The secondproject's goal is to obtain new relations in the cohomology ringof the moduli space of complex structures on a marked Riemannsurface. Using the techniques introduced in a previous paper, thePI found several families of interesting relations, one of whichproves a ten year old conjecture of Faber. The last project seeksto extend the sum formula for Gromov-Witten invariants todeformations more general then those appearing from a symplecticsum. There already seems to be several interesting new phenomenaappearing in the general case.The proposed work lies at the intersection of string theory andsymplectic topology. String theory developed as a potentialcandidate for a unifying theory of the universe, which extendsEistein relativity theory. It is based on the idea thatelementary particles (like electrons, photons) should be thoughtnot as points, but rather small vibrating loops. Working out thedetails of this theory turned out to be quite delicate, and hasin turn inspired many remarkable results in mathematics. But alsofundamental results in mathematics have inspired many newdiscoveries in physics. It is hoped that this project willcontribute to the increased interaction between mathematics andhigh energy physics. In the same time, one of the projects willinvolve graduate students.