# Research interests

My main interest is probability measures on algebraic structures with main
attention

on real and p-adic Lie groups, various classes of groups such as
Moore, MAP,

groups of polynomial growth, etc., and Gelfand pairs and
hypergroups.

I consider the following:

Action of automorphism groups on the space of probability measures
on groups which

leads to the study of decomposable measures, stable and
semi-stable probability measures, limit theorems;

Questions in dynamical systems related distal, ergodic and
proximal, etc., ;

Questions related to convolution powers of probability measures on
various algebraic structures;

Arithmetic of probability measures on Gelfand pairs, especially
Khinchin theorems;

Existence of recurrent random walks on groups and sufficient
conditions for random walks to be recurrent;

The Choquet-Deny theorem, the non-existence of non-constant
bounded continuous harmonic functions

on groups and Poisson boundary of
probability measures on locally compact groups;

Convergence of averages of unitary representations of various
classes of groups and related questions;

Various convergence of averages of finite-dimensional representations;

Ergodic theory on groups and algebraic dynamical systems;

Structure theory of Lie groups and algebraic groups over real and
p-adic fields.

I have general interests in abstract harmonic analysis on locally compact groups,
especially

unitary representations, abelian groups and Gelfand pairs.