FRG: Collaborative Research: Topological Invariants and Matrix Models

AbstractAward：DMS 0244412首席研究员：谢尔顿卡茨弦理论对现代数学的许多领域都产生了巨大的影响，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，弦的对偶性在这些不同的领域之间存在着出乎意料的关系，其中许多已经被数学证明。 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。 它提出了调查统一主题的对偶对称性，在搜索这些对称性的更深入的理解。 它还建议工作的一系列数学问题，这些对偶性的启发。 特殊的对偶包括镜像对称，最初在格罗莫夫-维滕理论的背景下阐明，和S-对偶，这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，这给出了一方面微扰费曼图与另一方面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关系。 在Gromov-Witten理论的特殊情况下，已经发现了与knot理论的联系。 这导致了一个完整的计算相应的不变量的非紧复曲面Calabi-Yauthreefolds。物理学研究者们建议将这些想法扩展到紧凑的情况。物理学研究者们将利用物理学和数学之间现有的联系，并建立新的联系，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。 被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，因为相关技术已经在过去。 这个项目发生在弦理论社区持续努力的同时，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，数学的许多不同领域将以意想不到的方式相互联系，这将对数学和物理产生深远的影响。 由于该项目的范围广泛，因此需要采用多学科和多机构的方法。 PI将把他们的合作者、博士后和研究生网络聚集在一起，并在这些领域进行密切的合作。 一个跨学科的数学/物理课程将被创建，以培养未来的领导者在数学和物理的接口领域。 PI将扩大现有的互联网视频会议技术的使用，以实现合作目的，并将组织关于该项目主题的研讨会。这是一个联合奖的数学科学部门计划在几何分析和代数，数论，组合数学，和物理部门计划在数学物理。

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.