Abstract: 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. A natural approach to this problem is to search for complete (and simple) invariants that can be assigned to the members of X and determine their E-classes. More generally, we can try to reduce the classification problem of E to a simpler one, by constructing a Borel map f : X --> X' which reduces the equivalence relation E on X to a simpler equivalence relation E' on X'. The existence of such a reduction is denoted by E <=_B E' (E is Borel reducible to E'). We will present several results concerning the order <=_B and their applications to classification problems.
Tue, 18/12/2018 - 12:00 to 13:00