Arithmetic and Topological Structures in Physics

这个由美国国家科学基金会资助的项目是理论物理和几何的交叉。它的主要目标是研究各种理论物理模型中的数学结构，范围从弦理论到量子场论和量子统计力学。使用来自数学领域的新方法，如算术几何、拓扑和同伦理论，来解决物理问题，将有可能揭示物理模型背后隐藏的和更深层次的数学结构。该项目将有很强的教育成分，涉及几个研究生和本科生的研究培训。本项目主要研究方向包括：利用非阿基米德方法和几何方法研究弦理论的AdS/CFT全息对应；非交换扭转空间理论的发展及实例的相关构造与算术对象相关的量子统计力学系统的构建，如dessins d'enfant、Kuga和Shimura变种、模符号及其推广，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；量子场论中的动力结构，如SYK模型；经典动机和非交换动机的高范畴概括用非交换动机研究大质量场论的费曼积分；田间品种的Fermi-Pasta-Ulam动力学及KAM定理非交换和分类条件下的高度函数和Arakelov几何；非交换几何和分布计算的立方集模型上的随机漫步和热核计算。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值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.