New Geometric Models for Theoretical Physics and for Computational Linguistics

本计划的目的是研究物理学中的几何结构，重点是引力和物质模型、量子场论、弦理论和量子统计力学中几何、拓扑和算术之间的相互作用。调查的主要主题将是：量子场论中的动机结构，p-adic全息AdS/CFT对应，应用于数论的量子统计力学，引力谱模型中的算术结构，任意子和量子霍尔系统的几何结构，以及通过因果动力学三角剖分物理学来解决分布式计算中的优化问题的方法。该项目的另一部分侧重于语言学几何模型的构建，特别是理解世界语言的句法参数空间结构及其几何和拓扑结构。这是一种新的句法研究方法，它符合乔姆斯基的“原则和参数”的框架，但基于使用的数学方法，如持久同源性，系统发育代数几何，编码理论。该项目也有一个重要的教育组成部分，涉及研究生和本科生研究项目的监督。

The purpose of this proposal is to investigate geometric structures in physics, with focus on the interaction between geometry, topology, and arithmetic in models of gravity and matter, in quantum field theory, in string theory, and in quantum statistical mechanics. The main themes of investigation will be: motivic structures in quantum field theory, p-adic holographic AdS/CFT correspondence, quantum statistical mechanics applied to number theory, arithmetic structures in spectral models of gravity, the geometry of anyons and quantum Hall systems, as well as an approach to problems of optimization in distributed computing via the physics of causal dynamical triangulations. Another part of this project focuses on the construction of geometric models for Linguistics, and in particular on understanding the structure of the space of syntactic parameters of world languages and its geometry and topology. This is a novel approach to the study of syntax, which fits in the framework of Chomsky's "principles and parameters" but is based on the use of mathematical methods such as persistent homology, phylogenetic algebraic geometry, coding theory. The project also has a significant educational component, involving the supervision of graduate and undergraduate research projects.