Arithmetic and Topological Structures in Physics

California Institute Of Technology, Pasadena CA

Investigators

Abstract

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.