FRG: Collaborative Research: Gromov-Witten Theory

FRG：合作研究：格罗莫夫-维滕理论

• 批准号: 1159964
• 负责人: Renzo Cavalieri
• 金额: $ 16.78万
• 依托单位: Colorado State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-04-01 至 2017-03-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1159964&HistoricalAwards=false
• 关键词: FRG; Collaborative; Research; Gromov; Witten

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.

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。著名的例子是通过牛顿力学和通过爱因斯坦的一般相对论理论进行微积分。在过去的二十年中，有很多活动致力于构建宇宙的所谓弦乐理论模型，该模型结合了一些最复杂的数学。 Gromov-Witten理论的主题诞生于二十年前的数学与物理学之间的密集互动期间。 从那时起，格罗莫夫（Gromov-Witten）理论就将自己确立为几何和物理学的中心领域。 同时，它的范围已大大扩展到许多不同的数学领域，从Hurwitz理论的古典主题到Donaldson-Thomas-Thomas不变性的现代领域（Gromov-Witten理论的荒凉理论对应物）。尽管取得了成功，但许多中心问题仍未解决。两个值得注意的例子是higer属Gromov-witten的Calabi-yau歧管的不变性以及Gromov-Witten和Donaldson-Thomas不变性的精确关系。这些问题的解决对于几何和物理学非常重要。在此提案中，一支由世界上最好的专家组成的团队被组成，以攻击这些核心问题。此外，PIS建议开发技术，以研究各种问题，将Gromov-Witten理论与枚举代数几何，符号几何和数学物理学有关。 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。该项目本质上是跨学科的，因为物理和数学思想都扮演着核心角色。 从这个意义上讲，它增加了数学与物理学之间当前相互作用的趋势。该项目强调团队合作和协作。通过研究研讨会，组织和参加国家和国际会议，该建议还将增强对本科生和研究生以及博士后研究员的培训。将有许多研究出版物有助于向学生介绍这一令人兴奋的数学领域。该奖项由代数理论和数字理论以及DMS的拓扑计划获得了奖项。