Geometric and Quantum Structures of 3-Manifolds

对三维空间和其中的弯曲曲线的研究，对于我们理解宇宙的大尺度和小尺度方面是至关重要的。对这些空间的分类将取决于对它们可能采取的形状以及它们可能具有的刚性和柔韧性的数学理解。这些性质被称为不变量，它们来自代数、解析和几何方面的考虑，通常还需要物理学的重要输入。瑟斯顿几何化猜想的证明证明了三维空间，称为流形，分解成承认显式几何的碎片。在过去的几十年里，量子物理学的思想使数学家们发现了各种微妙的不变量和三流形的结构以及其中包含的打结曲线。在物理学和数学中，有几个开放的猜想预测了量子结构和三流形几何之间的深层关系。该项目将研究这些量子不变量与瑟斯顿图像中产生的几何结构之间的关系，并探索这些联系在数学和物理中的分支和应用。该项目还为研究生提供了研究课题。该项目将结合几何和量子拓扑技术，研究几何、拓扑量子场论（TQFT）和三流形组合结构之间的相互作用，着眼于开发工具来解决量子拓扑中的开放猜想。该项目的一部分工作将继续围绕Turaev-Viro不变量体积猜想，以及表面映射类群的量子表示的几何。目标是了解TQFT的渐近特征在多大程度上检测或确定3流形几何分解中双曲块的存在。项目的第二部分将研究彩色琼斯结多项式、连杆补中不可压缩曲面的拓扑结构和双曲几何之间的关系。第三部分将开发从纯组合输入中识别三流形几何结构的方法，并从拓扑数据中导出几何量的估计。这包括对某些连杆补中的低属不可压缩曲面的研究，以及对结图性质和约束如何影响连杆补的几何结构的理解。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.