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.