Classical and quantum hyperbolic geometry

在过去的三十年中，我们对三维拓扑的理解的大部分进展都是基于双曲（非欧几里得）几何和拓扑量子场论。然而，这两个数学分支长期以来是并行发展的，彼此之间没有太多的互动。这两个领域之间的假想桥梁现在开始出现。其中之一是体积猜想，通过一些计算实验验证，它将结的琼斯多项式的渐近增长与其补的标准双曲度规的体积联系起来。该项目的目标是开发一个结合这两种观点的概念框架。该项目有一个2维的组成部分，专注于表面的量子Teichmuller空间的表示理论，由物理学家引入，以模拟2+1维度的量子引力。项目的第二部分建立在第2维度中获得的洞察力的基础上，以开发与体积猜想中出现的对象密切相关的双曲三维流形的不变量。该项目旨在更好地理解三维几何，例如确定空间中的两条打结曲线何时可以相互变形的问题。解决这个问题的一种传统方法是对与这些曲线的图形描述相关的多项式进行代数处理。另一种强大的技术，广泛使用的软件实现，使用双曲非欧几里得几何。实验证据表明，这两种观点之间存在着意想不到的联系。该项目的目标是开发技术和概念工具来证实（或反驳）这种联系，并更好地理解所涉及的现象。

In the last thirty years, much of the progress in our understanding of 3-dimensional topology has been grounded in hyperbolic (non- euclidean) geometry and in topological quantum field theory. However, these two branches of mathematics have long evolved in parallel, without much interaction with each other. Hypothetical bridges between these two fields are now beginning to emerge. One of them is the Volume Conjecture, experimentally verified by a few computations, which connects the asymptotic growth of the Jones polynomials of a knot to the volume of the canonical hyperbolic metric of its complement. The goal of the Project is to develop a conceptual framework combining the two points of view. The Project has a 2- dimensional component, focussed on the representation theory of the quantum Teichmuller space of a surface, as introduced by physicists to model quantum gravity in dimension 2+1. A second part of the Project builds on the insight gained in dimension 2 to develop invariants of hyperbolic 3-dimensional manifolds, closely related to the objects appearing in the Volume Conjecture.The Project aims at gaining a better understanding of 3-dimensional geometry, such as the problem of deciding when two knotted curves in space can be deformed to each other. One traditional approach to this problem involves the algebraic manipulation of certain polynomials associated to pictorial descriptions of these curves. Another powerful technique, with widely used software implementation, uses hyperbolic non-euclidean geometry. Experimental evidence suggests an unexpected connection between these two points of view. The goal of the Project is to develop technical and conceptual tools to confirm (or disprove) this connection, and to better understand the phenomena involved.