FRG: Collaborative Research: Gromov-Witten Theory

自牛顿时代以来，数学一直是帮助我们理解宇宙本质的关键工具。著名的例子是牛顿力学中的微积分和爱因斯坦广义相对论中的微分几何。在过去的二十年里，有大量的活动致力于建立一个所谓的宇宙弦论模型，它包含了一些最复杂的数学。格罗莫夫-威滕理论的主题诞生于20年前数学与物理密切互动的时期。从那时起，格罗莫夫-威滕理论在几何学和物理学中都确立了自己的中心地位。与此同时，它的范围也大大扩展到许多不同的数学领域，从经典的赫尔维茨理论到现代的唐纳森-托马斯不变量（格罗莫夫-威滕理论的束理论对应）。尽管取得了成功，但许多核心问题仍未得到解决。两个值得注意的例子是紧化Calabi-Yau流形的高格Gromov-Witten不变量的计算以及Gromov-Witten不变量与Donaldson-Thomas不变量之间的精确关系。这些问题的解决对几何学和物理学都具有重要意义。在这个建议中，一个由世界上最好的专家组成的团队被召集起来解决这些核心问题。此外，pi建议开发技术来研究与格罗莫夫-威滕理论有关的各种问题，包括枚举代数几何、辛几何和数学物理。pi希望对这些数学领域及其相互关系做出重要而实质性的贡献。这个项目本质上是跨学科的，因为物理和数学思想都起着核心作用。从这个意义上说，它增加了当前数学和物理之间相互作用的趋势。这个项目强调团队合作。通过研究研讨会，组织和参加国内和国际会议，本提案还将加强本科生和研究生以及博士后的培养。将会有一些研究出版物，将有助于向学生介绍这个令人兴奋的数学领域。该奖项由DMS的代数和数论以及拓扑程序共同资助。

Since the era of Newton, mathematics has been a key tool in helping us comprehend the nature of the universe. Famous examples are calculus via Newtonian mechanics and differential geometry via Einstein's theory of general relativity. During the last twenty years, there has been a great deal of activity devoted to building a so-called string-theoretic model of the universe, which incorporates some of the most sophisticated mathematics. The subject of Gromov-Witten theory was born twenty years ago during a period of intensive interaction between mathematics and physics. Since then Gromov-Witten theory has established itself as a central area in both geometry and physics. At the same time, it has expanded greatly in its scope to many diverse areas of mathematics, ranging from the classical topic of Hurwitz theory to the modern area of Donaldson-Thomas invariants (the sheaf-theoretic counterpart of Gromov-Witten theory). Despite its success, many central problems remain unsolved. Two notable examples are the computation of higer genus Gromov-Witten invariants of compact Calabi-Yau manifolds and the precise relation between Gromov-Witten and Donaldson-Thomas invariants. The resolution of these problems is of great importance for geometry and physics. In this proposal, a team of the best experts in the world is assembled to attack these central problems. In addition, the PIs propose to develop technology to study a variety of questions relating Gromov-Witten theory to enumerative algebraic geometry, symplectic geometry and mathematical physics. The PIs hope to make important and substantial contributions to these areas of mathematics, and their interrelations.This project is interdisciplinary in nature, in that both physical and mathematical ideas play central roles. In this sense it adds to the current trend of interaction between mathematics and physics. This project emphasizes teamwork and collaboration. Through research seminars, organizing and participating in national and international conferences, this proposal will also enhance the training of undergraduate and graduate students, as well as postdoctoral fellows. There will be a number of research publications that will help in introducing students to this exciting area of mathematics.This award is cofunded by the Algebra and Number theory and the Topology programs of DMS.