Abstract: In this talk we will discuss some recent work by Guihéneuf and Lefeuvre who prove that shadowing is generic for Lebesgue measure preserving transformations of the 2-torus. We will spend most of our time motivating the problem, discussing the history of such questions- specifically touching upon earlier work of Oxtoby, Ulam, Alpern and Prasad and some general techniques used in the area. Time permitting, we will discuss the recent proof given by Guihéneuf and Lefeuvre.
We would like present several results in descriptive set theory involving definable equivalence relations on Polish spaces.
Given an equivalence relation E on a polish space X, we would like to study the classification problem of determining whether two objects x,y in X are E-related.