Classical and quantum homomorphisms from discrete groups to Lie groups

从离散群到李群的经典和量子同态

• 批准号: 1406559
• 负责人: Francis Bonahon
• 金额: $ 31.4万
• 依托单位: University of Southern California
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-07-01 至 2018-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1406559&HistoricalAwards=false
• 关键词: Classical; quantum; homomorphisms; discrete; groups

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.

各种物理问题的现代公式涉及称为李群的矩阵族（即数表）。经典力学的基础就是如此，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双曲几何分析打结的曲线在空间中。从量子物理学中获得灵感的其他发展，提供了基于李群（称为量子群）的某些变形来解决同样的打结问题的工具。该项目研究了几个问题，涉及经典李群和它们的量子群变形，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数学的几个不同分支，包括几何，拓扑，代数和动力系统。 该项目是沿着两个主题，这是非常不同的性质，但联合的事实，双曲几何可以作为一个智力指南，在他们每一个沿着。该项目也有一个强大的教育组成部分，因为它的研究主题的设计，使他们可以培养几个研究生的博士工作，并提供广泛的博士后培训初级教师的主要研究者（PI）的研究小组。 该项目的第一个主题是集中在从一个表面的基本群李群同态的经典几何。当李群是分裂的真实的，例如对于特殊的线性群SL（n，R），所谓的希钦同态满足许多重要的几何和动力学性质。第一个目标的建议是开发一个微分的频谱下希钦同态的简单的封闭曲线的表面上的图像。这包括一个参数化的希钦同态，是非常适合这样的微积分空间的发展。该项目还将使用这些方法来研究希钦同态空间在无穷远处的边界。移动到复杂的设置，PI将调查的几何同态价值在复杂的李群，但接近真实的希钦同态。 该项目的第二个主题涉及量子群不变量的结在三维流形。PI将继续他的调查表示的考夫曼绞代数在一个表面上，被认为是点的量化空间的同态从基本群的表面李群SL（2，C）。然后，他将建立在这项研究中开发的结果和工具，以及Kashaev-Baseilhac-Benedetti的早期工作，建立一个（2+1）维拓扑量子场论，混合量子拓扑和双曲几何。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3-空间中结的某些量子不变量的渐近行为与其补的双曲体积之间的精确关系。Kauffman skein代数提供的技术比早期的方法更内在，并且应该特别有用。