FRG: Collaborative Research: Topological Invariants and Matrix Models

摘要：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、拓扑学、表示理论、分析和组合几何。特别是，弦的对偶性暗示了这些不同领域之间意想不到的关系，其中许多已经被数学证明了。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，数论也会与之相关。我们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。它还被提议用于研究这些对偶性所激发的一系列数学问题。特殊的对偶性包括镜像对称，最初在Gromov-Wittentheory的背景下阐明，以及s对偶性，这是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有关。在超对称规范理论的情况下，这一方面给出了摄动费曼图与在实例的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gromov - witten理论的特殊情况下，已经发现了与结理论的联系。这导致了非紧环calabi - yauthrefold相应不变量的完整计算。pi建议将这些想法扩展到紧凑情况。pi将利用物理和数学之间的现有联系并建立新的联系，以解决这两个领域的广泛前沿开放问题。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相当于数学中重要的未解决问题。引入数学的技术有望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。该项目与弦理论社区正在进行的努力同时发生，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；这是爱因斯坦没有解决的问题。可以预见，许多不同的数学领域将以意想不到的方式相互联系起来，这将对数学和物理学产生深远的影响。由于该项目的范围广泛，表明了一种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，并在这些领域指导激烈的合作努力。将开设一个跨学科的数学/物理课程，以培养数学和物理交叉领域的未来领导者。各专业机构将扩大现有互联网视讯会议技术的使用，以达致合作目的，并将就该项目的主题举办工作坊。这是数学科学部的几何分析与代数、数论与组合学项目和物理部的数学物理项目的联合奖项。

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.