Character Varieties and Quantum Invariants

University Of Southern California, Los Angeles CA

Investigators

Abstract

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.