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.

Abstractaward：DMS 024412原理研究者：Sheldon Katzstring理论对许多现代数学领域都产生了壮观的影响，包括代数几何，差分代，拓扑，拓扑结构，表示理论，分析，分析和combinatorial几何形状。 尤其是，弦二元性暗示了这些不同领域之间的关系，其中许多是数学上证明的。 矩阵模型技术的最新进展表明数学上的进一步关系。 现在可以预期，数字理论也将变得相关。 ITI提议研究对二元性对称性的统一主题，以搜索对这些对称性的更深入的理解。 ISALSO提议解决这些二元性启发的一系列数学问题。 特定的二元性包括镜像对称性，最初是在格罗莫夫 - 卫星的背景下阐明的，而s偶性是二元性的对称性，是对称的，对称性量表的理论和弦理论。 最近与矩阵积分有关。 在苏格对称仪表理论的情况下，这一方面给出了扰动的feynman图与另一方面的istantons模量空间上的某些计算之间的关系，PIS提出了进一步的阐明。 在特定的gromov-witten理论中，已经发现了与基理论的联系。 这导致了对非紧密曲折calabi-yauthreefolds的反应不变的完整计算。 PI提议将这些想法扩展到紧凑型。 这些想法被指望以在各种数学领域之间建立新的联系，因为某些物理学中的某些想法在数学中同等效率不可解决的问题。 预计将数学引入数学的技术将对数学的核心领域具有革命性的影响，因为相关技术已经过去了。 该项目与弦理论社区的持续富裕同时发生，并将有助于在该领域设定未来方向。 弦理论旨在将重力与电磁和核力量统一。这是爱因斯坦的问题。 可以预料，数学的多种领域将以意想不到的方式与彼此相关，这将对数学和物理学产生深远的影响。 由于该项目的广泛范围，指出了多学科和多机构攻击。 PI将将合作者，博士后和研究生的网络汇总在一起，并指导在这些领域进行激烈的合作努力。 将在数学和物理界面的领域中创建一个学科数学/物理课程。 PI将扩大其用于协作目的的现有Internet Videococontrencing Technologies的使用，并将愿意组织有关该项目主题的研讨会。这是数学科学计划Ingemetric Analysis和代数，数字理论和组合学和数学物理学物理部计划的A县奖。