Teichmuller Theory and Quantum Topology

本项目旨在研究TeichMuller理论与纽结和三维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，直到等距，它是理解低维流形的几何和最终拓扑的关键对象。它的量子化产生了所谓的量子泰希米勒空间，它起源于数学物理，更具体地说，是在对2+1量子引力的研究中找到的。为了更深入地理解这一联系，将开发两种方法。第一种方法是研究构造一个有限维模函子的可能性，该函子与量子Teichmuller空间的表示理论有关。如果这种方法是成功的，它将导致构造一族被穿透表面的映射类群的表示，并最终导致拓扑量子场论的构造。第二种方法将包括研究量子Teichmuller空间与圆弧和链环的skein代数之间的关系，该代数是由PI和一位合作者最近开发的。这两种观点之间的可能联系预计将类似于SU(2)-TQFT的几何和组合方法之间的关系。对三种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，这门学科经历了几次重大突破，遵循了两种先验的无关方法：一种来自三维流形承认的可能几何的研究，另一种来自数学物理和量子不变量的概念。近年来，许多结果和猜测都旨在调和这两种观点。国际和平研究所打算进一步研究这些联系，目的是将方法从一种翻译成另一种。

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.