Activities This Week

We apply the Tur\´an sieve and the simple sieve developed by Ram Murty and Yu-Ru Liu to study problems in random graph theory. More specifically, we obtain bounds on the probability of a graph having diameter 2 (or diameter 3 in the case of bipartite graphs). An interesting feature revealed in these results is that the Tur´an sieve and the simple sieve “almost completely” complement to each other. This is joint work with Yu-Ru Liu.

Bounded harmonic functions on groups are closely related to random walks on groups. It has long been known that all abelian groups, and more generally, virtually nilpotent groups are “Choquet-Deny groups”: these groups cannot support non-trivial bounded harmonic functions. Equivalently, their Furstenberg-Poisson boundary is trivial, for any random walk. I will present a recent result where we complete the classification of discrete countable Choquet-Deny groups, proving a conjuncture of Kaimanovich-Vershik. We show that any finitely generated group which is not virtually nilpotent, is not Choquet-Deny. Surprisingly, the key is not the growth rate of the group, but rather the algebraic infinite conjugacy class property (ICC).

This is joint work with Joshua Frisch, Omer Tamuz and Pooya Vahidi Ferdowsi.

Schur-Weyl Duality is a remarkable theorem giving an intimate link between the representation theories of the Symmetric group S_n, and the General Linear group GL(k). Such a link also holds between other objects, in particular the Brauer Algebra and the Orthogonal group, and the Walled Brauer algebra and GL(k). I will give an introduction to these relationships.