Teichmuller Theory and Quantum Topology

本计画旨在研究Teichmuller理论与纽结及三流形的量子不变量之间的关系。Teichmuller空间，定义为曲面上的双曲度量空间，是理解几何和低维流形拓扑的关键对象。它的量子化，产生了所谓的量子泰希穆勒空间，它起源于数学物理学，更具体地说，起源于2+1量子引力的研究。为了更深入地理解这一联系，将采取两种方法。第一种方法将包括在研究的可能性，构建一个有限维模块函子相关的表示理论的量子Teichmuller空间。如果这种方法是成功的，它将导致家庭的建设表示的映射类组的打孔表面，并最终建设的拓扑量子场论。第二种方法将包括研究量子Teichmuller空间和最近由PI和合作者开发的弧和链接的螺旋代数之间的关系。这两种观点之间可能的联系，与SU（2）-TQFT的几何方法和组合方法之间的关系是相似的。三维流形的研究本质上是对宇宙可能形状的研究，因此自然地处于数学和理论物理的交叉点。在过去的30年里，这个问题经历了几次重大突破，遵循两个先验无关的方法：一个来自于对三元流形所允许的可能几何的研究，另一个来自于数学物理和量子不变量的概念。近年来，许多成果和建议旨在调和这两种观点。PI打算进一步研究这些联系，目的是将方法从一种方法转换为另一种方法。

This project aims at studying the relationship between Teichmuller theory and quantum invariants of knots and three-manifolds. The Teichmuller space, defined as the space of hyperbolic metrics on a surface up to isotopy, is a key object in understanding the geometry and ultimately the topology of low-dimensional manifolds. Its quantization, giving rise to the so-called quantum Teichmuller space, finds its origin in mathematical physics, more specifically in the study of 2+1 quantum gravity. Two approaches will be developed in understanding this connection more deeply. The first approach will consist in studying the possibility of constructing a finite dimensional modular functor associated to the representation theory of the quantum Teichmuller space. If this approach is successful, it would lead to the construction of a family of representations of the mapping class groups of punctured surfaces and ultimately to the construction of a topological quantum field theory. The second approach will consist in studying the relationship between the quantum Teichmuller space and the skein algebra of arcs and links developed recently by the PI and a collaborator. The possible connections between the two points of view is expected to be similar to the relationship between the geometrical and the combinatorial approach to SU(2)-TQFT.The study of three-manifolds is in essence the study of the possible shapes of the universe and as such sits naturally at the intersection of mathematics and theoretical physics. The subject experienced several major breakthrough in the past thirty years following two a priori unrelated approaches: one coming from the study of the possible geometries a three-manifold admits and the other coming from mathematical physics and the notion of quantum invariants. In recent years, numerous results and conjectures have aimed at reconciling the two points of view. The PI intends on studying these connections further with the goal of translating methods from one to the other.