**Title**: Quantitative multiple recurrence in ergodic theory

**Speaker**: Sebastian Donoso (Universidad de O’Higgins)

**Abstract**: In this talk I will survey recent development of the multiple recurrence problem in ergodic theory. For a probability space (X,,μ) and measure preserving transformations T1,…,Td, the problem is to study the largeness of the set of n∈ℕ such that

μ(A∩T−a1(n)1A∩⋯∩T−ad(n)dA)>F(μ(A))

where a1,…,ad take integer values on the integers and F is a suitable function. I will mention key results and comment on the problem for commuting transformations, linear and polynomial functions ai. I plan to provide some proofs that rely on combinatorial constructions.