A gentle introduction to measurable cocycles, rigidity and recent advances

Measurable cocycles arise in different fields of mathematics, and cocycle rigidity is a self standing research topic that goes back to Zimmer's work. In the first part of this talk I will give a gentle introduction to cocycles, describing some explicit examples coming out in different situations such as orbit equivalences and measure equivalences. In the the second part I will describe a technique to investigate rigidity that involves bounded cohomology. A comparison with groupoids will be carried on and, time permitting, in the last part I will show how the above machinery might be described in the category of measured groupoids. This is the starting point of a joint work in progress with A. Savini about a theory of bounded cohomology for groupoids.