Topological Quantum Field Theory and Geometric Structures in Low Dimensional Topology

研究三维空间和其中打结的弦，对于我们理解关于宇宙形状的几个方面是必不可少的。为了对这种空间进行分类，我们需要从数学上了解它们可能采取的形状以及它们的刚性和灵活性属性。这样的性质被称为空间的不变量，它们产生于各种数学考虑，通常有来自物理学的关键输入。瑟斯顿几何化猜想的证明证明了某些三维空间，称为流形，可以分解成具有良好几何性质的碎片。在过去的几十年里，量子物理的思想使数学家们发现了三维流形的各种微妙的不变量和结构。在物理学和数学中，有几个公开的猜想预测了量子结构和三维流形的几何之间的深层联系。这个项目将研究这些量子不变量与瑟斯顿图像所产生的几何结构的关系，着眼于开发工具来解决拓扑学和物理学中的公开猜想。该项目包括研究生研究课题，通过指导和会议旅行提供关键支持，有助于培训和专业发展。项目的一个方向将研究拓扑量子场论的渐近方面、TeichMuller空间的粗几何和纤维三维流形的体积之间的关系。另一个方向将研究Turaev-Viro三维流形不变量的渐近增长，旨在证明它在三维流形的几何分解中检测到双曲线段的存在。第三个方向将继续研究有色Jones纽结多项式与环补中不可压缩曲面之间的关系。PI将进一步发展这一框架，特别是研究强斜率猜想及其在其中的应用。PI还将使用量子不变量来理解经典不变量，如交叉数和交叉节的交叉数。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势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 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.