Character Varieties and Quantum Invariants

在模拟物理世界的实际问题时，数学家和物理学家经常使用矩阵，即数字的方格。特别地，给定问题的对称性通常由矩阵族来描述，这些矩阵族被称为李群。最近，量子物理和统计力学的需要导致李群的变形的发展，这被称为量子群。在二十世纪的最后二十年里，通过引入非欧几何（双曲几何）的技术，拓扑学取得了重大突破。非欧几何的基础是行列式为1的2乘2矩阵的李群。大约在同一时间，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二维和三维空间几何相互作用的各种问题。特别是，它利用了为2 × 2矩阵开发的见解来解决涉及n × n矩阵的高维组。 数学和数学物理中的许多经典问题可以用从基本曲面群到李群的同态来表示，例如矩阵群。例如，在2000年前后的几十年里，双曲几何学的重大突破就涉及到对特殊线性群SL_2中的这种同态的分析。类似地，纽结和低维流形的量子不变量大约在同一时间被发展出来，它们是基于李群的变形，称为量子群。特别地，纽结的琼斯多项式不变量可以用量子群U_q（sl_2）变形李群SL_2来表示。研究的重点是建立在这些低维李群的基础上，以解决更高秩的李群和量子群。该项目的重点是字符品种，包括从表面群的李群同态。它的两个主要主题建立在两个不同数学领域之间的相互作用。 第一个主题涉及的几何和动力学性质的所谓希钦同态，价值分裂真实的代数群，如特殊的线性群，并对模空间的这种同态。PI和他的学生将研究泊松几何空间的希钦同态，将调查希钦同态的角度来看，频谱网络，并将研究集团行动仿射建筑物出现退化等同态。 第二个主题是研究基于量子群U_q（sl_n）的纽结量子不变量理论中出现的skein代数，以及当量子参数是单位根时它们的代数性质。 PI将研究在U_q（sl_2）的早期工作中发现的“奇迹般的取消”的一般情况的扩展。该项目包括一系列单独的问题，可以容纳研究生的博士工作，同时提高初级教师在PI的研究小组的博士后经验。该项目的视觉方面有助于本科生的参与，以及旨在使数学令人兴奋的K-12和本科生的推广。

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.