Geometric and Quantum Structures of 3-Manifolds

三流形的几何和量子结构

• 批准号: 2004155
• 负责人: Efstratia Kalfagianni
• 金额: $ 36.85万
• 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: 2020
• 资助国家: 美国
• 起止时间: 2020-06-15 至 2024-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2004155&HistoricalAwards=false
• 关键词: Geometric; Quantum; Structures; Manifolds; Geometric; Quantum; Structures; Manifolds

The study of three-dimensional spaces, and knotted curves in them, is essential to our understanding of large- and small-scale aspects of the universe. A classification of these spaces will rest on a mathematical understanding of the possible shapes they can take and the rigidity and flexibility properties they can have. These properties are known as invariants and they come from algebraic, analytic, and geometric considerations, often with crucial input from physics. The proof of Thurston's Geometrization Conjecture established that three-dimensional spaces, called manifolds, decompose into pieces that admit explicit geometries. In the last few decades, ideas from quantum physics have led mathematicians to the discovery of a variety of subtle invariants and structures of three-manifolds and the knotted curves contained in them. 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 investigate the relations of these quantum invariants to the geometric structures arising from Thurston's picture and explore the ramifications and applications of these connections to mathematics and physics. The project also provides topics for graduate student research.The project will combine geometric and quantum topology techniques to study the interplay of geometry, topological quantum field theories (TQFT), and combinatorial structures of three-manifolds, with an eye towards developing tools to tackle open conjectures in quantum topology. One part of the project will continue work around the Turaev-Viro invariants volume conjecture, and on the geometry of quantum representations of surface mapping class groups. The goal is to understand the extent to which asymptotic features of TQFT detect or determine the existence of hyperbolic pieces in the geometric decomposition of 3-manifolds. A second part of the project will study relations between the colored Jones knot polynomials, the topology of incompressible surfaces in link complements, and hyperbolic geometry. A third part will develop methods for recognizing geometric structures on three-manifolds from purely combinatorial input and derive estimates on geometric quantities from topological data. This includes the study of low-genus incompressible surfaces in certain link complements and the understanding of how knot diagrammatic properties and constraints affect the geometric structure of link complements.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曲线的研究对于我们对宇宙大小方面的理解至关重要。这些空间的分类将基于对它们可能采取的可能形状以及它们可能具有的刚性和灵活性的数学理解。这些特性称为不变性，它们来自代数，分析和几何考虑，通常具有物理学的关键输入。瑟斯顿的几何化猜想的证据表明，三维空间（称为歧管）被分解成吸收显式几何形状的碎片。在过去的几十年中，量子物理学的想法导致数学家发现了各种微妙的不变性和三个manifolds的结构以及其中包含的打结曲线。在物理和数学方面，都有几种开放猜想，这些猜想预测了量子结构与三个manifolds的几何形状之间的密切关系。该项目将研究这些量子不变的关系与瑟斯顿图片产生的几何结构的关系，并探索这些连接到数学和物理学的后果和应用。该项目还提供了研究生研究的主题。该项目将结合几何和量子拓扑技术，以研究几何形状，拓扑量子场理论（TQFT）的相互作用和三序列的组合结构，以期待开发工具以开发量子拓扑中开放猜想的工具。该项目的一部分将继续围绕Turaev-Viro不变式的猜想，以及表面映射类组的量子表示的几何形状。目的是了解TQFT检测或确定三序列分解中双曲线碎片的存在的程度。该项目的第二部分将研究有色琼斯结多项式，链接补充中不可压缩表面的拓扑与双曲线几何形状之间的关系。第三部分将开发出纯粹的组合输入中三序列上几何结构的方法，并从拓扑数据中得出对几何量的估计。这包括研究某些链接补充中低形状不可压缩表面的研究，以及对结的图形特性和约束如何影响链接补充的几何结构。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优点和更广泛影响的评估来通过评估来获得支持的。