Character Varieties and Quantum Invariants

在对物理世界的实际问题进行建模时，数学家和物理学家经常使用矩阵，即数字的正方形网格。特别地，给定问题的对称性通常由被称为李群的矩阵族来描述。最近，量子物理和统计力学的需要导致了李群的变形的发展，这被称为量子群。在二十世纪的最后二十年里，通过引入非欧几里德(双曲)几何的技术，拓扑学取得了巨大的突破。非欧几里德(双曲)几何的技术是基于具有行列式1的2乘2矩阵的李群。同时，量子群的发展为分析三维空间中的曲线打结提供了新的工具，其中由2乘2矩阵的变形产生的量子群起着特别重要的作用。这些项目研究了这些群与2维和3维空间的几何相互作用的各种问题。特别是，它利用为2乘2矩阵开发的见解来处理涉及n乘n矩阵的高维群。数学和数学物理中的许多经典问题都可以用从基本曲面群到李群(如矩阵群)的同态来表示。例如，发生在2000年前后几十年的双曲几何的重大突破涉及到对特殊线性群SL_2中的这种同态的分析。类似地，几乎同时发展起来的纽结和低维流形的量子不变量是基于称为量子群的李群的形变。特别地，纽结的Jones多项式不变量可以表示为使Lie群SL_2变形的量子群U_q(Sl_2)。研究的重点是建立在这些低维Lie群的基础上，以解决高阶Lie群和量子群。该项目专注于由从表面群到李群的同态组成的特征标变体。它的两个主要主题建立在两个不同的数学领域之间的相互作用上。第一个主题是关于所谓的Hitchin同态的几何和动力学性质，它取值于一个分裂的实代数群，如特殊的线性群，以及这种同态的模空间。PI和他的学生将学习Hitchin同态空间的泊松几何，将从谱网络的角度研究Hitchin同态，并将研究作为此类同态退化而出现的仿射建筑物上的群作用。第二个主题研究了基于量子群U_q(Sl_N)的纽结的量子不变量理论中的skein代数，以及当量子参数为单位根时它们的代数性质。PI将调查他在U_q(Sl_2)的早期工作中发现的“奇迹取消”的推广到这一一般情况。该项目包括一系列独立的问题，以适应研究生的博士工作，同时加强PI研究组初级教员的博士后经验。该项目的视觉方面有助于本科生的参与，以及旨在使K-12和本科生对数学感兴趣的外联活动。

When modeling practical problems of the physical world, mathematicians and physicists often use matrices, namely square grids of numbers. In particular, the symmetries of a given problem are often described by families of matrices, which are called Lie groups. More recently, the needs of quantum physics and statistical mechanics have led to the development of deformations of Lie groups, which are called quantum groups. In the last two decades of the Twentieth Century great breakthroughs in topology were obtained through the introduction of techniques from non-euclidean (hyperbolic) geometry, which is based on the Lie group of 2-by-2 matrices with determinant 1. At about the same time, the development of quantum groups provided new tools to analyze the knotting of curves in 3-dimensional space, with the quantum group arising from deformations of 2-by-2 matrices playing a particularly important role. The projects investigates various problems where these groups interact with the geometry of spaces of dimension 2 and 3. In particular, it takes advantage of the insights developed for 2-by-2 matrices to address higher dimensional groups, involving n-by-n matrices. Many classical problems in mathematics and in mathematical physics can be expressed in terms of homomorphisms from fundamental groups of surfaces to Lie groups, such as groups of matrices. For instance, the great breakthroughs of hyperbolic geometry that occurred in the decades surrounding the year 2000 have involved the analysis of such homomorphisms valued in the special linear group SL_2. Similarly, the quantum invariants of knots and low-dimensional manifolds that were developed at about the same time are based on the deformations of Lie groups called quantum groups. In particular, the Jones polynomial invariant of knots can be expressed in terms of the quantum group U_q(sl_2) deforming the Lie group SL_2. The thrust of the research is to build on the insights developed for these low-dimensional Lie groups in order to address higher rank Lie groups and quantum groups. The project is focused on the character varieties that consist of homomorphisms from surface groups to Lie groups. Its two main themes build on the interaction between two different areas of mathematics. The first topic deals with the geometric and dynamic properties of the so-called Hitchin homomorphisms, valued in a split real algebraic group such as the special linear group, and on the moduli spaces of such homomorphisms. The PI and his students will study the Poisson geometry of the space of Hitchin homomorphisms, will investigate Hitchin homomorphisms from the point of view of spectral networks, and will study the group actions on affine buildings that arise as degenerations of such homomorphisms. The second theme studies the skein algebras that occur in the theory of quantum invariants of knots based on the quantum group U_q(sl_n), and their algebraic properties when the quantum parameter is a root of unity. The PI will investigate the extension to this general case of the "miraculous cancellations" that he discovered in earlier work for U_q(sl_2). The project includes a series of separate problems that can accommodate the doctoral work of graduate students, while enhancing the postdoctoral experience of the junior faculty in the research group of the PI. The visual aspects of the project lend themselves to the involvement of undergraduate students, as well as to outreach aimed at making mathematics exciting for K-12 and undergraduate students.