Algebraic Geometry and Strings

几十年来，对物理理论（弦理论和量子场论）与数学（尤其是代数几何）相互作用的探索一直非常富有成效，而且这种工具和方法组合的力量似乎只会随着时间的推移而增强。该项目的目标是探索和推动代数几何与弦理论和量子场论接口的一些主要问题。该研究将采用并结合代数几何、拓扑、可积系统、弦理论和量子场论等多种技术。该项目还包括许多更广泛的影响活动，例如会议和学校的指导和组织、国际委员会和奖项委员会的成员资格、修订宾夕法尼亚大学的研究生课程、研究生和本科生课程开发、为博士后、研究生和本科生提供咨询、编辑几本公共服务卷以及编辑期刊和会议记录卷。更具体地说，该项目包括以下主题：通过非阿贝尔霍奇理论对几何朗兰兹猜想进行 QFT 启发的攻击；根据希钦系统的变体对 S 类物理理论进行数学研究；将超几何思想应用于弦理论中的高级循环计算；探索代数几何中的模问题，其中一些是由 QFT 猜想引发的，另一些则纯粹在代数几何中；进一步探索 F 理论的各个方面并建立其数学基础；以及分类对称性和缺陷对称性 TFT 的探索。这些具体研究领域中的每一个都代表了数学和/或物理学中的一个重大开放问题，其解决方案将为该领域做出重大贡献。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

Exploration of the interactions of physical theories (string theory and quantum field theory) with mathematics (especially algebraic geometry) has been extremely productive for decades, and the power of this combination of tools and approaches only seems to strengthen with time. The goal of this project is to explore and push forward some of the major issues at the interface of algebraic geometry with string theory and quantum field theory. The research will employ and combine a variety of techniques from algebraic geometry, topology, integrable systems, String theory, and Quantum Field theory. The project also includes many broader impact activities such as steering and organization of conferences and schools, membership of international boards and prize committees, revising Penn’s graduate program, curricular development at the graduate and undergraduate level, advising postdocs, graduate and undergraduate students, editing several public service volumes and editing of journals and proceedings volumes.More specifically, the project includes, among other topics: a QFT-inspired attack on the geometric Langlands conjecture via non-abelian Hodge theory; a mathematical investigation of physical Theories of class S in terms of variations of Hitchin systems; applications of ideas from supergeometry to higher loop calculations in string theory; exploration of moduli questions in algebraic geometry, some of them motivated by a QFT conjecture, others purely within algebraic geometry; further exploration of aspects of F theory and establishment of its mathematical foundations; and exploration of categorical symmetries and defect symmetry TFTs. Each of these specific research areas represents a major open problem in math and/or in physics, whose solution will make a major contribution to the field.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.