Classical and quantum homomorphisms from discrete groups to Lie groups

各种物理问题的现代表述涉及称为李群的矩阵族（即数字表）。经典力学的基础是如此，广义相对论和量子物理学更是如此。由于这个原因，在过去的150年里，许多数学都是在这个框架下发展起来的。在20世纪80年代，一些突破强调了特定李群在各种几何问题中的作用。这包括令人惊讶的有效使用非欧几里得双曲几何来分析空间中曲线的打结。从量子物理学中获得灵感的其他发展，已经提供了基于称为量子群的李群的某些变形来解决相同打结问题的工具。该项目研究了几个涉及经典李群及其量子群变形的问题，以及它们在结和三维空间研究中的应用。它的动机和技术工具来自几个不同的数学分支，包括几何、拓扑、代数和动力系统。该项目围绕着两个本质上非常不同的主题进行阐述，但双曲几何可以作为每一个主题的智力指导，这一事实将它们结合在一起。该项目还具有很强的教育成分，因为其研究主题的设计可以培养几名研究生的博士工作，并为首席研究员（PI）研究小组的初级教师提供广泛的博士后培训。项目的第一个主题是关注从曲面的基群到李群的同态的经典几何。当李群是分裂实的，例如对于特殊的线性群SL(n，R)，所谓的Hitchin同态满足许多重要的几何和动力学性质。本文的第一个目标是发展表面上简单闭合曲线在希钦同态下的图像光谱的微分演算。这包括希钦同态空间的参数化的发展，它很好地适应于这样的微积分。本项目也将使用这些方法来研究希钦同态空间的无穷边界。转到复结构，PI将研究在复李群中值的同态的几何，但接近于真实的希钦同态。项目的第二个主题涉及三维流形中结的量子群不变量。PI将继续研究表面上的Kauffman skein代数的表示，将其视为从表面的基本群到李群SL（2，C）的同态空间的量子化点。然后，他将在这项研究的结果和工具的基础上，以及Kashaev-Baseilhac-Benedetti的早期工作的基础上，建立一个（2+1）维拓扑量子场论，将量子拓扑和双曲几何混合在一起。这项工作的长期目标是提供概念和技术工具来攻击体积猜想，该猜想预测了三维空间中结的某些量子不变量的渐近行为与其补体的双曲体积之间的精确关系。Kauffman skein代数提供的技术比以前的方法更具有内在性，并且应该特别有用。

The modern formulation of various physical problems involves families of matrices (that is tables of numbers) called Lie groups. This is true of the foundations of classical mechanics, and even more so for general relativity and quantum physics. For this reason much mathematics has been developed in this framework for the past 150 years. In the 1980s, several breakthroughs have emphasized the role of specific Lie groups in various geometric problems. This includes the surprisingly effective use of non-euclidean hyperbolic geometry to analyze the knotting of curves in space. Other developments, drawing their inspiration from quantum physics, have provided tools to attack the same knotting problems based on certain deformations of Lie groups called quantum groups. The Project investigates several problems involving classical Lie groups and their quantum group deformations, and their applications to the study of knots and 3-dimensional spaces. It draws its motivation and technical tools from several different branches of mathematics, including geometry, topology, algebra and dynamical systems. The Project is articulated along two themes that are very different in nature, but united by the fact that hyperbolic geometry can be used as an intellectual guide in each of them. The Project also has a strong educational component, as its research themes are designed so that they can nurture the doctoral work of several graduate students, and provide a broad postdoctoral training to junior faculty in the research group of the Principal Investigator (PI). The first theme of the Project is focused on the classical geometry of homomorphisms from the fundamental group of a surface to a Lie group. When the Lie group is split real, for instance for the special linear group SL(n,R), the so-called Hitchin homomorphisms satisfy many important geometric and dynamical properties. A first goal of the proposal is to develop a differential calculus for the spectrum of the images under Hitchin homomorphisms of simple closed curves on the surface. This includes the development of a parametrization of the space of Hitchin homomorphisms that is well-adapted to such a calculus. The Project will also use these methods to investigate the boundary at infinity of the space of Hitchin homomorphisms. Moving to the complex set-up, the PI will investigate the geometry of homomorphisms valued in complex Lie groups but close to real Hitchin homomorphisms. The second theme of the Project involves quantum group invariants of knots in 3-dimensional manifolds. The PI will continue his investigation of representations of the Kauffman skein algebra on a surface, considered as points of a quantization of the space of homomorphism from the fundamental group of the surface to the Lie group SL(2,C). He will then build on the results and tools developed in this investigation, and on earlier work of Kashaev-Baseilhac-Benedetti, to build a (2+1)-dimensional topological quantum field theory that mixes quantum topology and hyperbolic geometry. The long term goals of this work is to provide conceptual and technical tools to attack the Volume Conjecture, which predicts a precise relationship between the asymptotic behavior of certain quantum invariants of a knot in 3-space and the hyperbolic volume of its complement. The technology provided by the Kauffman skein algebra is more intrinsic than earlier approaches, and should be particularly useful.