Character Varieties and Quantum Invariants

在为物理世界的实际问题建模时，数学家和物理学家经常使用矩阵，即数字的方形网格。特别地，一个给定问题的对称性通常用矩阵族来描述，这些矩阵族被称为李群。最近，量子物理学和统计力学的需要导致了李群变形的发展，这被称为量子群。在二十世纪的最后二十年里，通过引入非欧几里得（双曲）几何的技术，拓扑学取得了重大突破。非欧几里得（双曲）几何基于行列式为1的2 × 2矩阵的李群。几乎与此同时，量子群的发展为分析三维空间中曲线的打结提供了新的工具，其中由2 × 2矩阵变形产生的量子群发挥了特别重要的作用。这些项目研究了这些群体与2维和3维空间几何相互作用的各种问题。特别是，它利用为2 × 2矩阵开发的见解来处理涉及n × 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曲面的基本群到李群（如矩阵群）的同态来表示。例如，在2000年前后的几十年里，双曲几何的重大突破涉及到对特殊线性群SL_2中这种同态的分析。类似地，几乎同时发展的结和低维流形的量子不变量是基于称为量子群的李群的变形。特别地，结点的琼斯多项式不变量可以用变形李群sl_2的量子群U_q（sl_2）来表示。这项研究的主旨是建立在对这些低维李群的见解的基础上，以解决更高秩的李群和量子群。该项目的重点是由表面群到李群的同态组成的字符变体。它的两个主要主题建立在数学的两个不同领域之间的相互作用。第一个主题处理所谓的Hitchin同态的几何和动态性质，在分裂实代数群（如特殊线性群）中赋值，以及在这种同态的模空间上。PI和他的学生将研究Hitchin同态空间的泊松几何，从谱网络的角度研究Hitchin同态，并将研究仿射建筑上作为这种同态退化而出现的群作用。第二个主题研究了基于量子群U_q（sl_n）的结量子不变量理论中出现的串结代数及其在量子参数为单位根时的代数性质。PI将研究他在U_q（sl_2）的早期工作中发现的“神奇消去”的一般情况的推广。该项目包括一系列独立的问题，可以容纳研究生的博士工作，同时增强PI研究小组初级教师的博士后经验。这个项目的视觉方面让本科生参与其中，同时也让K-12和本科生对数学感到兴奋。

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.