Notions of entropy in ergodic theory and representation theory

Entropy has its origins in thermodynamics and statistical mechanics.  It gained mathematical rigour in Shannon's work on the foundations of information theory, and quickly found striking applications to ergodic theory in work of Kolmogorov and Sinai. Many variants and other applications have appeared in pure mathematics since, connecting probability, combinatorics, dynamics and other areas.

I will survey a few recent developments in this story, with an emphasis on some of the basic ideas that they have in common.  I will focus largely on (i) Lewis Bowen's "sofic entropy", which helps us to study the dynamics of "large" groups such as free groups, and (ii) a cousin of sofic entropy in the world of unitary representations, which leads to new connections with random matrices.