3:00–4:20 pm Eckhart Hall, Room 202
Poisson Algebra, Combinatorics and Representations of Surface Groups
Abstract
In this talk, I will start by recalling what a Poisson algebra is by first concentrating on the basic example of the algebra of polynomials in 2 variables. I will then briefly recall the relationship of Poisson algebras with Hamiltonian and Quantum Dynamics. Then I will explain a beautiful combinatorial construction due to Bill Goldman giving the structure of a Lie algebra on the vector space formally generated by loops on a surface $S$. The motivation underlying Goldman's construction is motivated by the Poisson algebra of regular functions on the character variety of $\pi_1(S)$ in a Lie group $G$. From that I will move to the deformation space of Anosov representations of $\pi_1(S)$ in a non-compact Lie group $G$ and show that there are more natural functions than regular ones (think of length or cross-ratio functions on Teichmüller space) and show that their Poisson Structure can also be described by a combinatorial object called « Ghost Bracket ». This is a joint work with Martin Bridgeman.