Speaker:  Michael Magee (Durham University & IAS)

Title:  Random unitary representations

Abstract:  I'll discuss the notion of random unitary representations of discrete infinite groups. These give rise to simply stated, new, and challenging questions in random matrix theory. I'll focus on representations of free groups and surface groups, and highlight two of my contributions there. 

In the first, joint with Puder, we give an exact formula for the expected value of the trace of a  U(n)-valued word map --- or Wilson loop --- where the word/loop is in a free group, in terms of deep topological properties of the word/loop. 

In the second, we prove that for elements of surface groups, the expected value of the normalized trace of the  SU(n)-valued Wilson loop converges to zero as n tends to infinity if the loop is non-null-homotopic. This extends a classic result of Voiculescu from free groups to fundamental groups of closed orientable surfaces, and is also connected to quantum Yang-Mills theories.

Joint with Doron Puder.