# Descriptive set theory conference

#### EPFL, CIB

December 18, 2018

#### A descriptive view of the density point property in measure theory

In this talk we will present some results on the complexity of sets of exceptional points with respect to some given measurable set. This is joint work with Riccardo Camerlo.

#### Complexity of the minimal ideal in the greatest ambit

The minimal ideal in the greatest ambit is the union of all minimal subflows of the greatest ambit. While the minimal subflows are always closed the minimal ideal is not Borel for any countable discrete group. We show that for a Polish group $G$ the minimal ideal is closed if and only if all minimal $G$-flows are metrizable.

This is a joint work with Andy Zucker.

#### A new space from a projective Fraïssé limit

I present a new example of a compact metrisable space obtained as a quotient of a projective Fraïssé limit, and discuss some of its properties.

This is a joint work in progress with G. Basso.

#### Unfriendly coloring

Given a graph, a red/blue coloring of its vertices is _unfriendly_ if every red vertex has at least as many blue neighbors as red neighbors, and vice-versa. Such colorings always exist for finite graphs, but for infinite graphs their existence quickly becomes quite subtle. We investigate certain descriptive set-theoretic analogs of these colorings in the measure-theoretic and Borel contexts. This is joint work with Omer Tamuz.

#### Some remarks on Baire’s grand theorem

This is joint work with Riccardo Camerlo. We provide a game theoretical proof of the act that if f is a function from a zero-dimensional Polish space to the Baire space that has a point of continuity when restricted to any non-empty compact subset, then f is of Baire class 1. We use this property of the restrictions to compact sets to give a generalization of Baire’s grand theorem for functions of any Baire class.

#### Descriptive set theoretic aspects of Haar null and Haar meagre sets

In 1972 Christensen introduced the notion of Haar null subsets of Polish groups, which is a well-behaved generalisation of the notion of Haar measure zero subsets of locally compact Polish groups. In 2013 Darji introduced the dual notion of Haar meagre sets.

In this talk we discuss various results concerning these notions that involve descriptive set theory, for example the calculation of the most well-known cardinal invariants of these small sets.

#### Projective Singletons

A real is a Pi-1-n singleton if it is the unique element of a set which is Pi-1-n definable. Assuming PD the Pi-1-n singletons for odd n are well-understood and are indeed pairwise comparable under the (strong) relation of Delta-1-n reducibility. But this is not the case for even n. In 1990 I showed that assuming the existence of 0#, there are Pi-1-2 singletons which are incomparable and strictly below 0# under L-reducibility. The natural conjecture has been that an analogous result holds for even n greater than 2 under PD. For n = 4 this states that there are Pi-1-4 singletons which are incomparable and strictly below M_2^# (the least "natural" nontrivial Pi-1-4 singleton) under M_2-reducibility. (x is M_2-reducible to y iff x belongs to M_2(y), the canonical, iterable inner model with 2 Woodin cardinals containing y.) In this talk I'll outline progress on this conjecture, which builds on my earlier work for the case n = 2 and a coding technique in the presence of Woodin cardinals, as well as fine-structural features of the model M_2(x) for a real x. This is joint work with Sandra Müller and Yizheng Zhu.

#### On Approximability in Orbit Equivalence and Applications to Bernoulli Percolation

Hyperfiniteness consists in approximating an object by finite pieces. It plays a central role in the orbit equivalence theory of probability measure preserving group actions. In a joint work with Robin Tucker-Drob we study a general similar notion of "approximation" of such actions by sub-equivalence relations. For non-amenable product groups, we identify circumstances where there exists no approximation at all. This result has consequences in Bernoulli percolation on Cayley graphs for these groups: the uniqueness threshold p_u doesn't belong to the uniqueness phase. I shall present these notions and give an overview of the subject.

#### Borel Combinatorics of Countable Abelian Group Actions

In this talk I will give a survey of recent development on Borel and continuous combinatorics of free actions of $\mathbb{Z}^n$. Our motivations are graph-theoretic questions such as chromatic numbers, matching, lining, and the existence of graph homomorphisms. We develop techniques to prove positive and negative results on the existence of Borel and continuous structuring on the Cayley graphs of free actions of $\mathbb{Z}^n$. This is joint work with Steve Jackson, Edward Krohne, and Brandon Seward.

#### What can hold exactly on N-th projective level

There exist generic models of set theory in which this or another property of reals and pointsets holds at a previously chosen level N of the projective hierarchy. The talk contains historical remarks, explanations related to the construction of models, and a review of related results.

#### On the comparison of analytic digraphs of uncountable Borel chromatic number

We study the class of analytic digraphs of uncountable Borel chromatic number. The main goal of the talk is to present new results, in collaboration with Miroslav Zeleny, comparing these digraphs with the notion of injective Borel homomorphism. We will also recall a number of older results involving some other notions of comparison.

#### The Separable Quotient Problem for Topological Groups

The famous Banach-Mazur problem, which asks if every infinite-dimensional Banach space has an infinite-dimensional separable quotient Banach space, has remained unsolved for 85 years, though it has been answered in the affirmative for all $C(K)$, where $K$ is a compact space; reflexive Banach spaces and even Banach spaces which are duals.
The similar problem for locally convex spaces has been answered in the negative,
but has been shown to be true for large classes of locally convex spaces including all non-normable Frechet spaces.
We investigated the analogous problem of existing of separable quotients for topological groups. There are four natural questions: Does every non-totally disconnected topological group have a separable quotient group which is
(i) non-trivial; (ii) infinite; (iii) metrizable; (iv) infinite metrizable.
We show in our work that even for abelian precompact groups all questions have the negative answer.
However, positive answers are given for important classes of topological groups including (a) all compact groups; (b) all locally compact abelian groups; (c) all $\sigma$-compact locally compact groups; (d) all abelian pro-Lie groups;
(e) all $\sigma$-compact pro-Lie groups; (f) all pseudocompact groups.
This is joint work with Sidney Morris, and Mikhail Tkachenko.

#### Strongly surjective linear orders: when embeddability and epimorphism agree

A linear order $L$ is strongly surjective if there exists an order preserving surjection from $L$ onto each of its suborders.

Our main result is that the set $\mathsf{StS}$ of countable strongly surjective linear orders is the union of an analytic and a coanalytic set, and is complete for the class of sets that can be expressed in this way. More in detail, we show that the countable strongly surjective linear orders which are scattered form a $\mathbf{\Pi}^1_1$-complete set, while the countable strongly surjective linear orders which are not scattered form a $\mathbf{\Sigma}^1_1$-complete set. Even if the study of the first two levels of the projective hierarchy is a long-standing topic, examples of sets that are true $\mathbf{\Delta}^1_2$ are very rare. In fact, as far as we know, $\mathsf{StS}$ is the first example of a "natural" set which is complete for the class of unions of an analytic and a coanalytic set.

If time allows I will discuss also the existence of uncountable strongly surjective linear orders. We showed that certain set-theoretic hypothesis imply their existence, and more recently Daniel Soukup showed that it is also consistent that they do not exist.

This is joint work with Riccardo Camerlo and Raphaël Carroy

#### Nonuniform universality for countable Borel equivalence relations

Inspired by Martin's conjecture, Slaman and Steel posed an open problem of whether Turing equivalence is hyper-recursively-finite. That is, can Turing equivalence be written as an increasing of countable Borel equivalence relations E_i, where every E_i class does not contain any infinite sequence which is uniformly recursive in any of its elements. We connect this question to the study of countable Borel equivalence relations by showing that if it has a positive answer, there is a universal countable Borel equivalence relation which is not uniformly universal. We also connect this problem to the study of homomorphisms from Turing equivalence to many-one equivalence which was initiated by Kihara and Montalban.

#### Polish metric spaces with fixed distance set

We study Polish spaces for which a set of possible distances $A \subseteq \mathbb{R}^+$ is fixed in advance. We determine, depending on the properties of $A$, the complexity of the collection of all Polish metric spaces with this property, and describe the properties that $A$ must have in order that all Polish spaces with distances in that set belong to a given class, such as zero-dimensional, locally compact, etc. These results lead us to give a fairly complete description of the complexity, with respect to Borel reducibility and again depending on the properties of $A$, of the relations of isometry and isometric embeddability between these Polish spaces.

#### Topological isomorphism for classes of closed subgroups of the group of permutations of N

The closed subgroups of the group of permutations of N coincide with the automorphism groups of structures with domain N. We consider Borel classes of such groups, such as being profinite (each orbit is finite), or being oligomorphic (for each k there are only finitely many k-orbits). For either class, work with A. Kechris and K. Tent (JSL, in press, independently Rosendal and Zielinski) shows that the topological isomorphism relation is Borel below graph isomorphism. For the profinite groups, we show that this bound is sharp. On the other hand, for oligomorphic groups, recent work with Schlicht and Tent shows that the isomorphism relation is below a Borel equivalence relation with only countable classes. A lower bound other than the identity on R remains open.

#### Finite versus infinite: An intricate shift

The Borel chromatic number – introduced by Kechris, Solecki, and Todorcevic (1999) – generalizes the chromatic number on finite graphs to definable graphs on topological spaces. While the $\mathbb{G}_0$-dichotomy states that there exists a minimal analytic graph with uncountable Borel chromatic number, it turns out that characterizing when a graph has infinite Borel chromatic number is far more intricate. Even in the case of graphs generated by a single function, our understanding is actually very poor.

The Shift Graph on the space of infinite subsets of natural numbers is generated by the function that removes the minimum element. It is acyclic but has infinite Borel chromatic number. Kechris, Solecki, and Todorcevic asked whether the Shift Graph is minimal among the graphs generated by a single Borel function that have infinite Borel chromatic number. Using a representation theorem for $\mathbf{\Sigma}^1_2$ sets due to Marcone, we answer this question negatively.

#### Lebesgue's density theorem for ideals

In recent work, we introduced a notion of density points relative to an arbitrary notion of smallness for subsets of the Cantor or Baire space. This generalizes Lebesgue's notion of density points and gives rise to density properties for various ideals.

In this talk, I will give an outline of results for several ideals induced by classical tree forcings that connect density properties with the ccc. This is joint work with Sandra Müller, David Schrittesser and Thilo Weinert.

#### Infinitary continuous logic and descriptive set theory

The descriptive set theoretic study of the complexity of the equivalence relation of isomorphism of various classes of countable structures is intimately related to the model theory of infinitary logic. Friedman, Hjorth, Kechris, Louveau, and others have established precise correspondences and developed a rather comprehensive theory in the case where isomorphism is Borel. We propose to extend this theory to metric structures using continuous logic. As a first step, we generalize a characterization, due to Hjorth and Kechris, of when isomorphism is an essentially countable equivalence relation from discrete to locally compact structures and recover some old results as special cases. This is joint work in progress with Michal Doucha and Maciej Malicki.