FRG: Collaborative Research: Topological Invariants and Matrix Models

FRG：协作研究：拓扑不变量和矩阵模型

• 批准号: 0244412
• 负责人: Sheldon Katz
• 金额: --
• 依托单位: University of Illinois at Urbana-Champaign
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-07-01 至 2010-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0244412&HistoricalAwards=false
• 关键词: FRG; Collaborative; Research; Topological; Invariants

AbstractAward: DMS 0244412Principal Investigator: Sheldon KatzString theory has had a spectacular impact on many areas ofmodern mathematics, including algebraic geometry, differentialgeometry, topology, representation theory, analysis andcombinatorial geometry. In particular, string dualities suggestunexpected relations between these diverse areas, many of whichhave been proven mathematically. Recent advances in matrix modeltechniques suggest further relations in mathematics. It can nowbe expected that number theory will become related as well. Itis proposed to investigate unifying themes in duality symmetriesin search of a deeper understanding of these symmetries. It isalso proposed to work on a range of mathematical problems whichthese dualities inspire. Particular dualities include mirrorsymmetry, originally elucidated in the context of Gromov-Wittentheory, and S-dualities, which are duality symmetries ofsupersymmetric gauge theories and string theories. More recentlyboth have been related to Matrix integrals. In the case ofsupersymmetric gauge theories this gives a relation betweenperturbative Feynman diagrams on the one hand and certaincomputations on moduli spaces of instantons on the other, whichthe PIs propose to elucidate further. In the particular case ofGromov-Witten theory, connections have been found with knottheory. This leads to a complete computation of thecorresponding invariants for non-compact toric Calabi-Yauthreefolds. The PIs propose to extend these ideas to the compactcase.The PIs will be exploiting existing connections and forging newones between physics and mathematics to address a wide range ofcutting-edge open problems in both fields. These ideas areexpected to create new links between diverse areas ofmathematics, as certain ideas in physics are equivalent toimportant unsolved problems in mathematics. The techniquesintroduced into mathematics are expected to have a revolutionaryinfluence on core areas of mathematics as related techniques havein the past. This project occurs at the same time as an ongoingeffort by the string theory community and will help set futuredirections in that field. String theory seeks to unify the forceof gravity with the electromagnetic and nuclear forces; this isthe problem that eluded Einstein. It is anticipated thatnumerous diverse areas of mathematics will become related to eachother in unexpected ways and that this will have profoundconsequences for mathematics and physics. Due to the broad scopeof the project, a multi-disciplinary and multi-institutionalapproach is indicated. The PIs will be bringing together theirnetworks of collaborators, postdocs, and graduate students anddirecting an intense collaborative effort in these areas. Aninterdisciplinary math/physics curriculum will be created totrain future leaders in areas at the interface of mathematics andphysics. The PIs will expand their use of existing internetvideoconferencing technologies for collaborative purposes, andwill organize workshops on the topics of this project. This is ajoint award of the Division of Mathematical Sciences programs inGeometric Analysis and Algebra, Number Theory, & Combinatorics, andthe Physics Division program in Mathematical Physics.

摘要奖项：DMS 0244412 首席研究员：Sheldon Katz 弦理论对现代数学的许多领域产生了巨大的影响，包括代数几何、微分几何、拓扑、表示论、分析和组合几何。 特别是，弦对偶性表明这些不同领域之间存在意想不到的关系，其中许多已经得到数学证明。 矩阵模型技术的最新进展表明了数学中的进一步关系。 现在可以预期数论也将变得相关。 建议研究对偶对称性的统一主题，以寻求对这些对称性的更深入的理解。 还建议研究这些二元性激发的一系列数学问题。 特定的对偶性包括镜像对称性（最初在格罗莫夫-维滕理论的背景下阐明）和 S 对偶性（超对称规范理论和弦理论的对偶对称性）。 最近，两者都与矩阵积分相关。 在超对称规范理论的情况下，这一方面给出了微扰费曼图与另一方面对瞬子模空间的某些计算之间的关系，PI 建议进一步阐明这一关系。 在格罗莫夫-维滕理论的特殊情况下，已经发现了与纽结理论的联系。 这导致了非紧复曲面 Calabi-Yau 三重的相应不变量的完整计算。 PI 提议将这些想法扩展到紧凑的情况。PI 将利用现有的联系并在物理和数学之间建立新的联系，以解决这两个领域中广泛的前沿开放问题。 这些想法有望在数学的不同领域之间建立新的联系，因为物理学中的某些想法相当于数学中未解决的重要问题。 引入数学的技术预计将像过去的相关技术一样对数学的核心领域产生革命性的影响。 该项目与弦理论社区的持续努力同时发生，并将有助于确定该领域的未来方向。 弦理论试图将引力与电磁力和核力统一起来。这是爱因斯坦未能解决的问题。 预计数学的许多不同领域将以意想不到的方式相互联系，这将对数学和物理学产生深远的影响。 由于该项目范围广泛，因此需要采取多学科和多机构的方法。 PI 将汇集他们的合作者、博士后和研究生网络，并指导这些领域的密切合作。 将创建跨学科数学/物理课程，以培训数学和物理交叉领域的未来领导者。 PI 将扩大对现有互联网视频会议技术的使用以实现协作目的，并将组织有关该项目主题的研讨会。这是数学科学部几何分析和代数、数论和组合学项目以及物理部数学物理项目的联合奖项。