The general question that leads the course is: what can we deduce about a group when studying random walks on it.
Stationary dynamics is a branch of Ergodic theory that focuses on measurable group actions arising from random walks. The main object studied is the Furstenberg-Poisson boundary.
Applications of the theory can be found in rigidity theory, and recently connections with operator theory have been established
- Brief introduction to Ergodic theory: Borel spaces, factors, compact models. Probability measure preserving actions and measure class preserving actions. Stationary measures.
- Random Walks: Markov chains, Martingale convergence theorem, Random walks on groups, Furstenberg-Poisson boundary. Choquet-Deny theorem, amenable groups. Entropy. Realization of the Furstenberg Poisson boundary.
- Applications to Rigidity: Margulis’ Normal Subgroup Theorem, and Bader-Shalom’s theorem (IRS rigidity).