Furstenberg's correspondence principle connects ergodic theory (the study of statistical properties of dynamical systems equipped with a measure) with number theory. We study how abstract dynamical methods can help with the "concrete" problem of finding arithmetic progressions in sets of integers - proving Roth's famous theorem. We start from the ground up with an introduction to ergodic theory.

This is an introduction to the study of growth in groups which lies at the interface of group theory, geometry, probability and combinatorics.We will first give an overview of classical results (i.e. Gromov’s polynomial growth theorem) and applications to geometry, dynamics and number theory. But our main goal will be recent advances and, in particular, the ‘product theorem’ in SL_n.

This seminar explores how tools from graph theory (notably expander graphs), probability, and algebra arise in interactive proofs—a framework where a verifier interacts with a prover to check a statement’s validity. We’ll introduce interactive proofs, cover the PCP theorem and its graph-theoretic proof, then touch on recent developments such as MIP* = RE and its implications for operator algebras.