Topological Quantum Field Theory and Geometric Structures in Low Dimensional Topology

低维拓扑中的拓扑量子场论和几何结构

• 批准号: 2304033
• 负责人: Efstratia Kalfagianni
• 金额: $ 37.75万
• 依托单位: Michigan State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-09-01 至 2026-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2304033&HistoricalAwards=false
• 关键词: Topological; Quantum; Field; Theory; Geometric

The study of three-dimensional spaces, and knotted strings in them, is essential for our understanding of several aspects about the shape of the universe. To classify such spaces we need to mathematically understand the possible shapes they can take as well as their rigidity and flexibility properties. Such properties are called invariants of the spaces and they arise from a variety of mathematical considerations, often with crucial input from physics. The proof of Thurston's Geometrization Conjecture established that certain three-dimensional spaces, called manifolds, decompose into pieces with nice geometric properties. In the last few decades, ideas from quantum physics have led mathematicians to discover a variety of subtle invariants and structures of three-manifolds.There are several open conjectures, both in physics and in mathematics, that predict deep relations between quantum structures and geometries of three-manifolds. This project will study the relations of these quantum invariants to the geometric structures arising from Thurston's picture, with an eye towards developing tools to tackle open conjectures in topology and physics. The project includes topics for graduate student research and contributes to training and professional development by providing critical support through mentoring and conference travel.One direction of the project will study relations between asymptotic aspects of Topological Quantum Field Theories, the coarse geometry of Teichmuller spaces and volumes of fibered 3-manifolds.Another direction will study the asymptotic growth of the Turaev-Viro three-manifold invariants aiming to prove that it detects the existence of hyperbolic pieces in the geometric decompositions of three-manifolds. A third direction will continue the study of relations between the colored Jones knot polynomials and incompressible surfaces in link complements. The PI will further develop this framework and, in particular, study the strong slope conjecture, and its applications, within it. The PI will also use quantum invariants to understand classical invariants such as crossing numbers and crosscap numbers of knots.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.

对三维空间以及其中打结的弦的研究对于我们对宇宙形状的几个方面的理解至关重要。为了对此类空间进行分类，我们需要数学上了解它们可以采取的可能形状以及它们的刚性和灵活性。这种特性称为空间的不变性，它们源于多种数学考虑，通常具有物理学的关键输入。瑟斯顿（Thurston）的几何化猜想的证据表明，某些三维空间（称为歧管）将分解成具有不错的几何特性的碎片。在过去的几十年中，量子物理学的想法导致数学家发现了三个manifold的各种微妙的不变性和结构。在物理和数学方面，有几种开放式猜想，这些猜想预测了三个manifolds的量子结构和几何形状之间的深厚关系。该项目将研究这些量子不变的关系与瑟斯顿（Thurston）图片引起的几何结构的关系，并着眼于开发工具来解决拓扑和物理学中的开放猜想。该项目包括研究生研究的主题，并通过指导和会议旅行提供关键的支持，为培训和专业发展做出贡献。项目的一个方向将研究拓扑量子量子理论的渐近方面之间的关系，Teichmuller空间的粗糙几何形状和纤维化的3个manifolds的粗略几何形状将研究Traaevt的纤维增长。检测到三序列的几何分解中的双曲线碎片的存在。第三个方向将继续研究有色琼斯结的多项式和不可压缩的表面之间的关系。 PI将进一步开发此框架，尤其是在其中研究强烈的斜率猜想及其应用。 PI还将使用量子不变性来理解经典的不变性，例如交叉数字和打结的交叉盖数。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子和更广泛影响的评估审查标准来通过评估来获得支持的。