Arithmetic and Topological Structures in Physics

物理学中的算术和拓扑结构

基本信息

批准号：
2104330
负责人：
Matilde Marcolli
金额：
$ 44.13万
依托单位：
California Institute of Technology
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2021
资助国家：
美国
起止时间：
2021-07-01 至 2024-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2104330&HistoricalAwards=false
关键词：
Arithmetic Topological Structures Physics

项目摘要

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.

由NSF奖资助的该项目是在理论物理和几何学的交集中。它的主要目标是对各种理论物理学模型中数学结构的研究，从字符串理论到量子场理论和量子统计力学。源于算术几何，拓扑和同型理论等数学领域而引起的新方法将使物理学的问题探讨问题，这将使您可以揭示物理模型背后的隐藏和更深的数学结构。该项目将具有强大的教育成分，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：在研究ADS/CFT全息弦理论的调查中，使用非架构的方法和几何形状；开发一种非交流性扭曲空间和相关结构的理论；与算术对象相关的量子统计机械系统的构建，例如Dessins d'Enfant，Kuga和Shimura品种，模块化符号及其概括，以及它们在数量理论特性研究中的使用；量子统计机械系统的分类；量子场理论（例如SYK模型）中的动机结构；经典和非交通动机的较高类别的概括；使用非共同动机来研究大规模田野理论的Feynman积分； Fermi-Pasta-Ulam动力学和kam定理在田野上的品种；在非共同和分类设置中的高度功能和Arakelov几何形状；对非交通性几何形状和分布式计算的立方组模型的随机步行和热内核计算。该奖项反映了NSF的法定任务，并且使用基金会的智力优点和更广泛的影响评估标准，被认为值得通过评估。