Krzysztof Krupiński: Topological dynamics and Borel cardinalities in model theory

Set Theory and Topology seminar (BGU)

Time: Tuesday, April 14, 12:15-13:50.
Place: Seminar room -101, Math building 58.
Speaker: Krzysztof Krupiński (UWr).
Title: Topological dynamics and Borel cardinalities in model theory.
Abstract: Newelski introduced methods and ideas from topological dynamics to the context of definable groups.

I will recall some fundamental issues concerning this approach, and I will present a few deeper results from my joined paper with Anand Pillay written last year, which relate the so called generalized Bohr compactification of the given definable group to its model-theoretic connected components. Then I will discuss more recent (analogous) results for the group of automorphisms of the monster model, relating notions from topological dynamics to various Galois groups of the theory in question. As an application, I will present a general theorem concerning Borel cardinalities of Borel, bounded equivalence relations, which gives answers to some questions of Kaplan and Miller and of Rzepecki and myself. This theorem was not accessible by the methods used so far in the study of Borel cardinalities of Borel, bounded equivalence relations (by Kaplan, Miller, Pillay, Simon, Solecki, Rzepecki and myself). The topological dynamics for the group of automorphisms of the monster model and its applications to Borel cardinalities are planned to be contained in my future joint paper with Anand Pillay and Tomasz Rzepecki.

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