Topology, Geometry, and Physics

拓扑、几何和物理

• 批准号: 1611957
• 负责人: Daniel Freed
• 金额: $ 31.5万
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-08-01 至 2020-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1611957&HistoricalAwards=false
• 关键词: Topology; Geometry; Physics

This research project forms a part of an ongoing vigorous interaction between geometry and high energy theoretical physics. The engagement of mathematics with other sciences, and of science with mathematics, is a continual source of fruitful ideas with long-term benefits -- our current technology and economy rely on the foundation of basic research from past decades and centuries. For many years, the interaction of physics with geometry has concerned quantum field theory and string theory, and now condensed matter theory is added to the mix. A major issue in the latter is the classification of phases of matter. New exotic phases of matter have direct applications to materials science and to quantum computing. This research project applies recent advances in abstract topology, the study of the gross structure of shapes, to this classification problem. The project also aims at broader foundational issues in geometric formulations of quantum field theory, as well as questions in pure geometry. Graduate students are involved in the research. This research involves both purely mathematical projects influenced by physics and projects that apply geometry and topology to concrete problems in quantum field theory, string theory, and condensed matter theory. As an example of the latter, the investigator aims to classify certain topological phases of matter using ideas and techniques from homotopy theory. Another set of projects aims to develop the general theoretical underpinnings of geometric models of quantum field theory, including issues of scale, unitarity, and long-range approximations. The investigator also plans to explore questions in pure geometry. One set of projects investigates invariants of three-dimensional manifolds using abelianization via spectral networks. This provides a new perspective on dilogarithm formulas for volumes of hyperbolic 3-manifolds. Another project investigates quadratic forms in topology with applications to special quantum field theories. Characteristic classes on smooth simplicial manifolds are also revisited, with an attempt to forge a relationship with central extensions of infinite dimensional groups.

这项研究项目是几何学和高能理论物理之间正在进行的积极互动的一部分。数学与其他科学的结合，以及科学与数学的结合，源源不断地产生富有成果的思想，并带来长期利益--我们目前的技术和经济依赖于过去几十年和几个世纪以来基础研究的基础。多年来，物理学与几何学的相互作用一直涉及量子场论和弦理论，现在又加入了凝聚态理论。后者的一个主要问题是物质相的分类。新的奇异物质相直接应用于材料科学和量子计算。这项研究项目将抽象拓扑学中的最新进展--研究形状的总体结构--应用于这一分类问题。该项目还致力于量子场论几何公式中更广泛的基础问题，以及纯几何中的问题。研究生参与了这项研究。这项研究既包括受物理学影响的纯数学项目，也包括将几何和拓扑应用于量子场论、弦理论和凝聚态理论中的具体问题的项目。作为后者的一个例子，研究者旨在利用同伦理论中的思想和技术对物质的某些拓扑相进行分类。另一组项目旨在发展量子场论几何模型的一般理论基础，包括尺度、一元性和长期近似问题。这位研究人员还计划探索纯几何问题。一组项目使用谱网络的离散化来研究三维流形的不变量。这为研究双曲三维流形体积的对数公式提供了一个新的视角。另一个项目研究拓扑学中的二次型及其在特殊量子场论中的应用。光滑单纯流形上的特征类也被重新讨论，试图与无限维群的中心扩张建立一种关系。