Arithmetic and Topological Structures in Physics

这个由NSF奖资助的项目是理论物理和几何的交叉点。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，来解决物理学问题，将有可能揭示隐藏在物理模型背后的更深层次的数学结构。该项目将具有很强的教育性，涉及对几名研究生和本科生的研究培训。该项目的主要研究方向包括：在弦理论的AdS/CFT全息对应研究中使用非阿基米德方法和几何;发展非对易扭量空间理论和相关的瞬子结构;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简化;量子场论中的动机结构，如SYK模型;经典和非对易动机的更高类别概括;使用非对易动机研究大规模场论的费曼积分;费米-帕斯塔-乌拉姆动力学和KAM定理在各种领域;高度函数和阿拉克洛夫几何在非对易和分类设置;该奖项反映了NSF的法定使命，并被认为是值得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的支持。

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.