Poisson Algebra, Combinatorics and Representations of Surface Groups

Explore a fascinating mathematical colloquium on Poisson algebras, combinatorics, and surface group representations. Delve into the fundamental concepts of Poisson algebras, starting with the basic example of polynomial algebras in two variables. Discover the connections between Poisson algebras, Hamiltonian dynamics, and quantum mechanics. Examine Bill Goldman's innovative combinatorial construction that imparts a Lie algebra structure to the vector space generated by loops on a surface. Investigate the motivation behind Goldman's work, rooted in the Poisson algebra of regular functions on the character variety of fundamental groups. Advance to the deformation space of Anosov representations and uncover more natural functions beyond regular ones. Explore how their Poisson structure can be described using a combinatorial object called the "Ghost Bracket." Gain insights from this joint work by Francois Labourie and Martin Bridgeman, presented in a 54-minute talk at the University of Chicago Department of Mathematics.