FRG: Collaborative Research: Topological Invariants and Matrix Models

摘要奖：DMS 0244412首席研究员：谢尔登·卡茨弦理论对现代数学的许多领域产生了惊人的影响，包括代数几何、微分几何、拓扑学、表示论、分析和组合几何。特别是，弦对偶揭示了这些不同区域之间意想不到的关系，其中许多已经在数学上得到了证明。矩阵建模技术的最新进展表明了数学上的进一步联系。现在可以预期，数论也将变得相关。建议研究对偶对称中的统一主题，以寻求对这些对称的更深层次的理解。它还被建议致力于一系列的数学问题，这些二元性激发了。特殊的对偶包括镜像对称性，它最初是在Gromov-Witten理论的背景下阐明的，以及S对偶，它是超对称规范理论和弦理论的对偶对称。最近，两者都与矩阵积分有关。在超对称规范理论的情况下，这给出了微扰Feynman图与瞬子模空间上的某些计算之间的关系，PI建议进一步阐明这一点。在格罗莫夫-威腾理论的特殊情况下，发现了与纽结理论的联系。这导致了非紧Toric Calabi-Yauthreefolds的相应不变量的完整计算。他们将利用物理和数学之间的现有联系和建立新的联系来解决这两个领域的一系列前沿开放问题。这些想法有望在不同的数学领域之间建立新的联系，因为物理学中的某些想法等同于数学中重要的未解决问题。被引入数学的技术预计将对数学的核心领域产生革命性的影响，就像过去的相关技术一样。这个项目与弦理论社区正在进行的努力同时进行，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；这是爱因斯坦所回避的问题。预计许多不同的数学领域将以意想不到的方式相互联系，这将对数学和物理产生深远的影响。由于该项目的范围很广，建议采取多学科、多机构的方法。PI将把他们的合作者、博士后和研究生网络聚集在一起，并在这些领域领导一项紧张的合作努力。将创建一个跨学科的数学/物理课程，以培养未来在数学和物理相结合的领域的领导者。私人投资机构将扩大利用现有的互联网视频会议技术进行协作，并将组织关于该项目主题的讲习班。这是由几何分析和代数、数论和组合数学的数学科学系和数学物理系的物理系联合颁发的奖项。

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.