Descriptive Set Theory in Turin
Date and place: September 6th to 8th, 2017 - Torino (Italy)
Organizers:
Alessandro Andretta, Gianluca Basso, Riccardo Camerlo, Vassilis Gregoriades, 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 A, ground floor. To reach Aula A, enter the building, turn right in
the first corridor, at the end which you will turn left. At the end of this corridor turn left again
and exit in the courtyard. In front of you is Aula A.
Useful information
Program
Day 1 - Wednesday, September 6
- 09:00 - Registration
- 09:30 to 10:20 - Gabriel Debs
The hyperspace of closed zero-dimensional subsets of a Polish space
Abstract.
The descriptive complexity of the set of all closed zero-dimensional subsets of a Polish Given a space $X$ we investigate the descriptive complexity class $\Gamma_X$ of the set $\mathcal F_0(X)$ of all its closed zero-dimensional subsets, viewed as a subset of the hyperspace $\mathcal F(X)$ of all closed subsets of $X$. We prove that
$\max \{ \Gamma_X; \ X \text{ analytic } \}=\boldsymbol{\Sigma}^1_2 $ and
$\sup \{ \Gamma_X; \ X \text{ Borel } \boldsymbol{\Pi}^0_ \xi\} \supseteq \Game\boldsymbol{\Sigma}^0_\xi$
for any countable ordinal $\xi$. (by Gabriel Debs and Jean Saint Raymond)
Hide abstract. - 10:30 to 11:00 - Coffee Break
- 11:00 to 11:25 - Dominique Lecomte
Universal and complete sets in martingale theory
Abstract.
The Doob convergence theorem implies that the set of divergence of any martingale has measure zero. Its definition shows that it is $G_{\delta \sigma}$. Louveau asked about the converse of this, namely is it true that any $G_{\delta \sigma}$ subset of the Cantor space with Lebesgue-measure zero can be represented as the set of divergence of some martingale? We study this question and its effective refinements. We provide some universal and complete sets for the whole projective hierarchy, via a general method, as well as some other complete sets for the classes $\Pi^1_1$ and $\Sigma^1_2$ in the theory of martingales.(joint work with Miroslav Zeleny)
Hide abstract. - 11:30 to 11:55 - Miroslav Zeleny
Convergence of a typical martingale (Slides)
Abstract.
We study convergence behavior of discrete martingales with
values in the interval $[0,1]$ from a measure theoretical point of view as
well as
from a topological one. We show that almost all martingales converge to
$0$ or $1$ almost everywhere. On the other hand, a typical martingale
diverges on a comeager set.
Hide abstract. - 12:00 to 13:30 - Lunch
- 13:30 - 14:30 Discussion Session: Contributions by Silvia Steila, Yann Pequignot, Itaï Ben Yacoov, and Jean Saint-Raymond.
- 14:30 to 14:55 - Sy-David Friedman
The Consistency of Uniformisation (Slides)
Abstract.
Hoffelner and I have shown the consistency of $\Pi^1_3$ uniformisation on Baire Space and $\Pi_1$ uniformisation on Generalised Baire Space, both relative to ZFC, answering a question of Gregoriades. My plan is to outline the proofs of these results.
Hide abstract. - 15:00 to 15:50 - Forte Shinko
Hyperfiniteness of boundary actions of cubulated hyperbolic groups (Slides)
Abstract.
A classical result of Dougherty, Jackson and Kechris states that the action of the free group on its Gromov boundary induces a hyperfinite equivalence relation. We will discuss a generalization of this result to a wider class of hyperbolic groups. Joint with Jingyin Huang and Marcin Sabok.
Hide abstract. - 16:00 to 16:30 - Coffee Break
- 16:30 to 16:55 - Maciej Malicki
Generic representations of countable groups
Abstract.
- 17:00 to 17:25 - Philipp Schlicht
Combinatorial Variants of Lebesgue's density theorem (Slides)
Abstract.
We introduce density points relative to an arbitrary sigma-ideal of Borel subsets of the Cantor or Baire space and define the density property for a sigma-ideal as a variant of Lebesgue’s density property. For the ideal of null sets with respect to the uniform measure on the Cantor space, these coincide with density points with respect to the measure up to a null set. The main results show that many well-known ccc ideals have this property and that it fails for many non-ccc ideals. This is part of a joint project with David Schrittesser, Sandra Uhlenbrock and Thilo Weinert.
Hide abstract.
Day 2 - Thursday, September 7
- 09:00 to 09:50 - Julien Melleray
Complexity of orbit equivalence of minimal homeomorphisms on a Cantor space
Abstract.
I'll (hopefully, as the proof hasn't been completely checked yet) explain how a recent result on sets of invariant measures of minimal homeomorphisms of Cantor space enables one to compute the complexity of orbit equivalence for these maps. All relevant notions will be recalled.
Hide abstract. - 10:00 to 10:25 - Mirna Dzamonja
Descriptive and combinatorial set theory at singular cardinals and their successors (Slides)
Abstract.
We consider some classical set-theoretical invariants in the context of the space $\kappa^\kappa$ where $\kappa$ is singular or at $\kappa^+$ for such a $\kappa$.
Hide abstract. - 10:30 to 11:00 - Coffee Break
- 11:00 to 11:25 - Donát Nagy
Simple witnesses of Haar null sets (Slides)
Abstract.
In Polish groups the sigma-ideal of Haar null sets (in the sense of Christensen) is a notion of `small sets' that allows us to formalize that something is true for `almost every' element of a group. A Borel set is said to be Haar null if there is a a Borel probability measure (which is called a witness measure) that assigns measure zero to all translates of our set. The focus of this talk will be an interesting idea which allows us to prove the existence of witness measures that are `simple' (in an informal sense) in the frequently studied Polish group $\mathbb{Z}^\omega$ of countable sequences of integers. At the end of the talk I will also briefly mention of the usage of this result.
Hide abstract. - 11:30 to 11:55 - Márk Poór
Cardinal invariants of Haar null sets (Slides)
Abstract.
We will prove that the cofinality of the Borel Haar-null ideal is continuum for every non-locally compact Polish group admitting an invariant metric. Furthermore we determine the non and cov (relatively to some elements of the Cichon diagram) in some special cases.
Hide abstract. - 12:00 to 13:30 - Lunch
- 13:30 - 14:30 Discussion Session: Contributions by Dorottya Sziraki, Riccardo Camerlo, Louis Vuilleumier, and Petr
Holicky.
- 14:30 to 14:55 - Jacques Duparc
On the Exponentiation inside the Wadge Hierarchy (Slides)
Abstract.
In 1977, in his PhD thesis "Determinateness and Subsystems of Analysis" John Steel introduced an operation on subsets of the Baire space that
behaves with regards to their ranks, like the ordinal multiplication. It has been an open problem since to find an operation that behaves like the exponentiation (joint work with Riccardo Camerlo).
Hide abstract. - 15:00 to 15:25 - Vladimir Kanovei
Definable minimal collapse functions at arbitrary projective levels (Slides)
Abstract.
Using a non-Laver modification of Uri Abraham's minimal lightface $\Delta^1_3$ collapse function, we define a generic extension $L[a]$ by a real $a$, in which, for a given $n$ bigger than 2, $a$ is a $\Pi^1_n$ singleton, $a$ effectively codes a cofinal map from $\omega$ to $\omega^L_1$, minimal over L, while every $\Sigma^1_n$ real is still constructible.
Hide abstract. - 15:30 to 15:55 - Pandelis Dodos
Uniformity norms and their weaker versions
Abstract.
We shall discuss some properties of Gowers uniformity norms,
a family of norms which are very useful in order to accurately count
the number of copies of certain "patterns" in subsets of discrete
structures. In particular, we will show that, under some mild hypotheses,
the Gowers uniformity norms are essentially equivalent to certain
weaker norms which are easier to understand. We will also discuss
applications of this equivalence. This is joint work with V. Kanellopoulos.
Hide abstract. - 16:00 to 16:30 - Coffee Break
- 16:30 to 16:55 - Raphael Carroy
Projective homogeneous spaces and Wadge theory
Abstract.
Fons van Engelen used the description of Wadge degrees of Borel sets to analyze Borel homogeneous spaces. I will explain the first steps we have made with Andrea Medini and Sandra Uhlenbrock towards the generalization of van Engelen's results in the projective hierarchy.
Hide abstract. - 17:00 to 17:25 - Lionel Nguyen Van Thé
On a remark on Glasner's problem (Slides)
Abstract.
A problem of Glasner, now known as "Glasner's problem", asks whether every minimally almost periodic, monothetic, Polish groups is extremely amenable. The purpose of this talk is to observe that a positive answer is obtained under the additional assumption that the universal minimal flow is metrizable.
Hide abstract.
- Social Dinner at Ristorante Quadre (map)
Day 3 - Friday, September 8
- 09:00 to 09:50 - Stephen Jackson
Undecidability of the graph homomorphism problem for action of $\mathbb Z^2$ (Slides)
Abstract.
Given a finite graph $G$, one can ask whehter there is a continuous graph homomorphism from the Cayley graphing of the free part of the shift action of $\mathbb Z^2$ to that graph. There is an interesting and complicated structure behind this question, and in fact it is complicated enough that we can show the question is undecidable. This is joint work with Gao, Krohne, and Seward.
Hide abstract. - 10:00 to 10:25 - Michal Doucha
Generic unitary representations (Slides)
Abstract.
For a fixed countable group G, we investigate the Polish space of all unitary representations of G in an infinite-dimensional separable Hilbert space. We are mainly interested in the unitary equivalence relation.
Under the hypothesis that the finite-dimensional unitary representations are dense in the unitary dual of G, we prove that if G is infinite and has the Haagerup property, then all equivalence classes are meager, while if G has the Kazhdan property, then there is a comeager equivalence class. The first case covers e.g. free groups and finitely generated residually finite amenable groups; the existence of an infinite group from the latter class is an open problem.
We also consider analogous results for representations of C* algebras. This is joint work with Maciej Malicki and Alain Valette.
Hide abstract. - 10:30 to 11:00 - Coffee Break
- 11:00 to 11:25 - Vojta Kovarik
Absolute F-Borel spaces (Slides)
Abstract.
If a space is $G_\delta$ in some compactification, it is necessarilly $G_\delta$ in every compactification, i.e. it is absolutely $G_\delta$. However, this is not true for $F_{\sigma\delta}$ or any higher $F$-Borel classes. I will demonstrate a technique which can be used to compute the "absolute" complexity of a space. As a corollary, we get the existence of spaces of given complexity and absolute complexity.
Hide abstract. - 11:30 to 11:55 - Filippo Calderoni
The bi-embeddability relation on countable torsion-free abelian groups (Slides)
Abstract.
Building on works by Louveau-Rosendal and Downey-Montalban we show that the embeddability relation on countable torsion-free abelian groups is a complete analytic quasi-order. In particular, it follows that the bi-embeddability relation on the space of countable torsion-free abelian groups is a complete analytic equivalence relation, thus is strictly more complex with respect to Borel reducibility than the isomorphism relation. This is joint work with Simon Thomas.
Hide abstract. - 12:00 to 13:30 - Lunch
- 13:30 - 14:30 Discussion Session: Contributions by Filippo Cavallari, Vibeke Quorning,
Giorgio Laguzzi, and Andrea Vaccaro.
- 14:30 to 14:55 - Asger Tornquist
Mad families, definability, and ideals, Part 1 (Slides)
Abstract.
- 15:00 to 15:25 - David Schrittesser
Mad families, definability, and ideals, Part 2 (Slides)
Abstract.
- 16:00 - Coffee and farewell
Registered Participants
Alessandro Andretta, Università di Torino.
Gianluca Basso, Université de Lausanne and Università di Torino.
Itaï Ben Yaacov, Université Claude Bernard Lyon 1.
Filippo Calderoni, Università di Torino.
Riccardo Camerlo, Polytechnic of Turin.
Raphael Carroy, Kurt Gödel Research Center.
Filippo Cavallari, University of Turin, University of Lausanne.
Gabriel Debs, Institut Mathématique de Jussieu.
Pandelis Dodos, Department of Mathematics, University of Athens.
Michal Doucha, Institute of Mathematics, Czech Academy of Sciences.
Jacques Duparc, University of Lausanne.
Mirna Dzamonja, University of East Anglia.
Sy-David Friedman, Kurt Gödel Research Center, U.Vienna.
Vassilios Gregoriades, University of Turin.
Petr Holicky, Charles University, Prague.
Stephen Jackson, University of North Texas.
Vladimir Kanovei, Institute for the Information Transmission Problems.
Vojta Kovarik, Charles University, Prague.
Giorgio Laguzzi, University of Freiburg.
Dominique Lecomte, Université Pierre et Marie Curie.
Maciej Malicki, Warsaw School of Economics.
Julien Melleray, Université Lyon 1.
Luca Motto Ros, University of Turin.
Donát Nagy, Eötvös Loránd University, Budapest.
Lionel Nguyen Van Thé, Aix-Marseille University.
Yann Pequignot, University of California, Los Angeles.
Márk Poór, Eötvös University, Budapest.
Vibeke Quorning, University of Copenhagen.
Jean Saint Raymond, Université Pierre et Marie Curie - Paris 6.
Philipp Schlicht, University of Bonn.
David Schrittesser, Kurt Gödel Research Center.
Forte Shinko, McGill University.
Silvia Steila, University of Bern.
Dorottya Sziraki, Alfred Renyi Institute of Mathematics, and Central European University.
Asger Tornquist, University of Copenhagen.
Andrea Vaccaro, Università di Pisa - York University.
Matteo Viale, Università di Torino.
Louis Vuilleumier, Université de Lausanne.
Domenico Zambella, Università di Torino.
Miroslav Zeleny, Faculty of mathematics and physics, Charles University, Prague, Czech Republic.
Sponsors
The workshop is generously funded by
- The Department of mathematics "Giuseppe Peano"
- Programma Giovani Ricercatori "Rita Levi Montalcini", "Nuovi sviluppi in teoria
descrittiva degli insiemi", (PI:Luca Motto Ros)
- PRIN 2012 "Modelli e insiemi" (PI: Carlo Toffalori)