I will describe some recent advances in the study of
infinite and finite groups, related to probability,
growth and complexity.
I will start with the celebrated Tits alternative
for linear groups, and present extensions and variations,
including a joint work with Larsen on a probabilistic Tits alternative. This is related to the notion of probabilistic
identities, and related results and open problems will be
mentioned.
I will then discuss approximate subgroups, an important
result by Breuillard-Green-Tao and Pyber-Szabo, and
some new growth results with Liebeck and Schul.
Finally I will describe recent work of Gowers and Viola
on communication complexity, and present extensions and
solutions to some of their conjectures.
The talk will be accessible to a wide audience.