Character theory of symmetric groups, analysis of long relators, and random walks.
Cogrowth and spectral gap of generic groups
The cogrowth exponent of a group controls the random walk spectrum. We prove that for a generic group (in the density model) this exponent is arbitrarily close to that of a free group. Moreover, this exponent is stable under random quotients of torsion-free hyperbolic groups.
Common summands in partitions