Research in Mathematical Physics and Algebraic Geometry

这个项目的目标是探索和推进代数几何与弦理论和量子场论之间的一些主要问题。这项研究将运用和结合代数几何、拓扑学、可积系统、弦论和量子场论的各种技术。该项目还包括许多更广泛的影响活动，如研究生和本科生的课程开发，以及为博士后、研究生和本科生提供的研究培训机会。几十年来，探索弦理论和量子场论与代数几何的相互作用一直是非常有成效的，这种工具和方法的组合似乎只会随着时间的推移而增强。在这个项目中，PI计划执行：(1)受量子场论的启发，通过非阿贝尔霍奇理论对几何朗兰兹猜想的攻击；(2)根据希钦系统的变化，对最近发现的S类物理理论进行数学研究；(3)将超几何的思想应用于弦理论中的更高回路计算；(4)探索代数几何中的模问题，其中一些是由量子场论猜想引起的，另一些纯粹是在代数几何范围内的；(5)将他构造的实现希钦系统的Calabi-Yau可积系统推广到亚纯和抛物线形式；(6)进一步探讨F-理论的各个方面，建立其数学基础。这些具体研究领域中的每一个都代表着数学和/或物理学中的一个重大悬而未决的问题，其解决方案将对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优势和更广泛的影响审查标准进行评估，被认为值得支持。

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 curricular development at the graduate and undergraduate level, and research training opportunities for postdocs, graduate and undergraduate students. Exploration of the interactions of string theory and quantum field theory with algebraic geometry has been extremely productive for decades, and the power of this combination of tools and approaches only seems to strengthen with time. In this project, the PI plans to carry out: (1) a quantum field theory-inspired attack on the geometric Langlands conjecture via non-abelian Hodge theory; (2) a mathematical investigation of the recently discovered physical theories of class S in terms of variations of Hitchin systems; (3) applications of ideas from supergeometry to higher loop calculations in string theory; (4) exploration of moduli questions in algebraic geometry, some of them motivated by a quantum field theory conjecture, others purely within algebraic geometry; (5) extension of his construction of Calabi-Yau integrable systems realizing Hitchin's system to meromorphic and parabolic versions; and (6) further exploration of aspects of F-theory and establishment of its mathematical foundations. 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.