What do we know about the eigenvalues of the interchange process?

The interchange process is a system of interacting particles defined as follows. Take a graph, put marbles on each vertex, all different, and then at each step choose one edge of the graph randomly and exchange the two marbles on its vertices. The result is a random walk on the symmetric group.

The talk will survey the structure of the eigenvalues of the (n! x n!) adjacency matrix of this process. This involves both their algebraic structure and analytic and probabilistic estimates on their sizes.