Properties of Gromov-Witten Invariants

[摘要]获奖：dms -0306299首席研究员：Eleny-Nicoleta ionel本提案旨在通过结合来自不同研究领域的思想来理解辛流形的gromov - witten不变量的结构。第一个项目是由两位物理学家Gopakumar和Vafa提出的猜想推动的。这个猜想暗示了Calabi-Yau3-fold的Gromov不变量的几个令人惊讶的限制。目标是通过采用Taubes开发的一些解析技术来证明猜想公式，以联系四维中的theseberg - witten和Gromov不变量。第二个项目的目标是在标记黎曼曲面上得到复结构模空间的上同环中的新关系。利用之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十年前的猜想。最后一个项目试图将Gromov-Witten不变量的和公式扩展到比辛和更一般的变形。在一般情况下，似乎已经出现了一些有趣的新现象。提出的工作是弦理论和辛拓扑的交叉。弦理论作为宇宙统一理论的潜在候选者而发展起来，它扩展了爱因斯坦的相对论。它基于这样一种观点，即基本粒子（如电子、光子）不应该被认为是点，而应该被认为是小的振动环。研究这一理论的细节被证明是相当微妙的，并反过来激发了许多数学上显著的成果。但数学的基本结果也激发了物理学的许多新发现。希望这个项目将有助于增加数学和高能物理之间的相互作用。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.