# Descriptive Set Theory Day in Torino

## Date and place: June 12th, 2018 - Torino (Italy)

#### Organizers:

Alessandro Andretta, Filippo Calderoni, Luca Motto Ros and Matteo Viale

#### Location:

Department of Mathematics
"Giuseppe Peano",

Palazzo Campana,
via Carlo Alberto 10, Torino.

All talks will take place in Aula C, on the ground floor.

Useful information

## Program

- 09:15 - 10:05 - Sławomir Solecki
###### Polishable Borel equivalence relations (**Slides**)

Abstract

We introduce the notion of Polishable equivalence relations. This class of equivalence relations contains all orbit equivalence relations induced by Polish group actions and is contained in the class of idealistic equivalence relations of Kechris and Louveau. We show that each orbit equivalence relation induced by a Polish group action admits a canonical transfinite sequence of Polishable equivalence relations approximating it. The proof involves establishing a lemma, which may be of independent interest, on stabilization of increasing $\omega_1$-sequences of completely metrizable topologies.

Hide - 10:15 – 10:40 - Coffee Break
- 10:40 - 11:30 - Gianluca Basso
###### A Lelek-like disconnected compact metric space (**Slides**)

Abstract

Given a class of compact metric spaces and continuous surjections, the theory of projective Fraïssé limits reduces the problem of finding a universal and approximately homogeneous space for said class to a combinatorial problem: checking whether amalgamation holds for a class of finite structures and epimorphisms.
We reverse such approach by proving that an interesting class of finite structures is Fraïssé and characterizing the compact metric space to which it gives rise. Such space, while disconnected, shares some similarities with the Lelek fan. We then attempt to find the optimal class of spaces and maps for which it is universal and approximately homogeneous. This is joint ongoing work with R. Camerlo.

Hide - 11:40 - 12.30 - Anush Tserunyan
###### A pointwise ergodic theorem for quasi-pmp graphs (**Slides**)

Abstract

We prove a pointwise ergodic theorem for locally countable quasi-pmp (probability measure preserving) graphs, which states that ergodicity (indecomposability) of a graph amounts to locally approximating global averages of $L^1$-functions via increasing subgraphs with finite connected components. The pmp version of this theorem was first proven by R. Tucker-Drob using probabilistic methods. Our proof is different: it is constructive and applies more generally to quasi-pmp graphs. It involves introducing a graph invariant, a packedness condition for finite Borel subequivalence relations, and an easy method of exploiting nonamenability. The non-pmp setting additionally requires a new gadget for analyzing the interplay between the underlying cocycle and the graph.

Hide - 12:30 – 14:00 - Lunch
- 14:00 - 14:50 - Gianluca Paolini
###### On the Admissibility of a Polish Group Topology (**Slides**)

Abstract

In [Sh771] Shelah rediscovered an old result of Dudley on the non-admissibility of a Polish group topology on an uncountable free group. Crucial to his proof is a so-called Compactness (or Completeness) Lemma for Polish groups, concerning satisfaction of algebraic equations for certain sequences of group elements converging to the identity element of the group. In joint work with Shelah we proved that the techniques of [Sh771] apply to many other groups and group constructions, most notably the combinatorial construction known as graph product of groups, a construction that generalizes simultaneously the notion of direct sum and free product of groups. In my talk I will survey our numerous results on the topic and mention the major questions left open by this work.

Hide - 15:00 - 15:50 - Marcin Sabok
###### Equidecompositions in ergodic actions of amenable groups

Abstract

I will discuss the problem of essential equidecomposability of sets with respect to ergodic pmp actions of amenable groups. During the talk I will give a combinatorial conditions equivalent to essential equidecomposability when the acting group is $\mathbb{Z}$.

Hide - 16:00 – 16:30 - Coffee Break

## Sponsors

The workshop is generously funded by

- The Department of Mathematics "Giuseppe Peano"