This seminar will be presented in hybrid mode.  The speaker will deliver his talk in person.

Title: Coincidence, equivalence and singularity of harmonic measures

Abstract: In the absence of measures fully invariant with respect to a group action,
this role can be to a certain extent played by the measures “invariant on
average”, with respect to a certain fixed distribution on the group. These
measures are called stationary, and they naturally arise as harmonic
measures of random walks.

The simplest geometrical instance of this situation is the action of a
Fuchsian group (in particular, of the fundamental group of a closed
surface) on the boundary circle of the hyperbolic plane. The general
singularity conjecture (currently proved in a number of particular cases)
is that the harmonic measure of any random walk with a finitely supported
step distribution is singular to the Lebesgue measure.

I will provide several partial answers to the question about the
dependence of harmonic measures on the underlying step distributions on
the group and discuss counterexamples related to the Minkowski and Denjoy
measure classes on the boundary of the classical modular group.

The talk is based on joint work with Behrang Forghani.

The talk will be followed by refreshments in the Huxley Common Room at 5pm.