Arithmetic and Topological Structures in Physics

这个由美国国家科学基金会奖资助的项目处于理论物理学和几何学的交汇点。它的主要目标是研究各种理论物理模型中的数学结构，从弦理论到量子场论和量子统计力学。使用数学领域的新方法，如算术几何、拓扑学和同伦理论，来处理物理问题，将有可能揭示物理模型背后隐藏的、更深层次的数学结构。该项目将有一个强大的教育部分，涉及几个研究生和本科生的研究培训。该项目的主要研究方向包括：使用非阿基米德方法和几何研究弦理论的ADS/CFT全息对应；发展非对易扭转空间理论和瞬子的相关结构；与算术对象如Dessins d‘enfant、Kuga和Shimura簇、模符号及其推广有关的量子统计力学系统的构建，以及它们在数论性质研究中的应用；量子统计力学系统的分类；量子场论中的基态结构，如SYK模型；经典和非对易动机的更高分类推广；使用非对易动机来研究重量场理论的Feynman积分；关于场上变量的Fermi-Pasta-Ulam动力学和Kam定理；非对易和范畴环境中的高度函数和Arakelov几何；非对易几何和分布式计算的立方集模型上的随机游动和热核计算。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。

This project funded by the NSF award is at the intersection of theoretical physics and geometry. Its main goal is the investigation of mathematical structures in various models of theoretical physics, ranging from string theory to quantum field theory and quantum statistical mechanics. The use of novel methods, arising from areas of mathematics such as arithmetic geometry, topology, and homotopy theory, to approach questions in physics, will make it possible to uncover hidden and deeper mathematical structures behind physical models. The project will have a strong educational component, involving the research training of several graduate and undergraduate students.The main research directions in this project include: the use of non-archimedean methods and geometries in the investigation of the AdS/CFT holographic correspondence of string theory; the development a theory of noncommutative twistor spaces and associated constructions of instantons; the construction of quantum statistical mechanical systems associated to arithmetic objects such as dessins d'enfant, Kuga and Shimura varieties, modular symbols and their generalizations, and their use in the investigation of number theoretic properties; categorifications of quantum statistical mechanical systems; motivic structures in quantum field theories such as SYK models; higher categories generalizations of classical and noncommutative motives; the use of noncommutative motives for the investigation of Feynman integrals of massive field theories; Fermi-Pasta-Ulam dynamics and KAM theorems on varieties over fields; height functions and Arakelov geometry in noncommutative and categorical settings; random walks and heat kernel computations on noncommutative geometries and on cubical sets models of distributed computating.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.