# DESCRIPTIVE SET THEORY IN PARIS

## Titles and Abstracts of the talks

 Itai BEN YAACOV Recent results on metrisable universal minimal flows. Riccardo CAMERLO Lebesgue density and exceptional points Noé DE RANCOURT Ramsey determinacy of adversarial Gowers games Pandelis DODOS $L_p$ regular sparse hypergraphs Michal DOUCHA On universal Polish metric groups Sy-David FRIEDMAN Descriptive Set Theory and Absoluteness Gilles GODEFROY Convex hulls of measurable functions and the axioms. Vassilis GREGORIADES Characterizations of the Baire space up to HYP-isomorphism Takayuki KIHARA Some more results around decomposability of Borel functions. Aleksandra KWIATKOWSKA The Ramsey degree of the pre-pseudoarc François LE MAITRE More Polish full groups Maciej MALICKI Automorphism groups of homogeneous metric structures and consequences of the existence of ample generics Benjamin MILLER The open graph dichotomy Luca MOTTO ROS On isometry and isometric embeddability between (ultra)metric Polish spaces Christian ROSENDAL Geometries of groups Asger TORNQUIST Definable maximal discrete sets in forcing extensions Zoltan VIDNYANSZKY The size of conjugacy classes of automorphism groups

Itai BEN YAACOV                Recent results on metrisable universal minimal flows.

Melleray, Nguyen Van Thé and Tsankov gave a compact description of the universal minimal flow of a Polish group under the assumption that it is metrisable and admits a co-meagre orbit. In recent work with Melleray and Tsankov we showed that metrisability of the universal minimal flow implies the existence of a co-meagre orbit, giving a full description of metrisable universal minimal flows of Polish groups.

↑top of page

Riccardo CAMERLO                Lebesgue density and exceptional points

I will survey some classification results for sets arising from the study of the density function in a metric measure space.

↑top of page

Noé DE RANCOURT                Ramsey determinacy of adversarial Gowers games

We present a game-theoretic dichotomy in countable vector spaces which generalises, on one hand, Gowers' Ramsey type theorem in Banach spaces, and on the other hand, Borel determinacy of sets of reals.

↑top of page

Pandelis DODOS                $L_p$ regular sparse hypergraphs

We shall discuss structural properties of certain sparse weighted uniform hypergraphs which satisfy a mild pseudorandomness condition known as $L_p$ regularity. This is joint work with Vassilis Kanellopoulos and Thodoris Karageorgos.

↑top of page

Michal DOUCHA                On universal Polish metric groups

It is well-known that every Polish group, resp. second-countable Hausdorff topological group, can be equipped with a compatible left-invariant metric. There are several known examples of topologically universal Polish groups, however no example of a metrically universal Polish metric group; i.e. Polish metric group that would contain every Polish group equipped with any compatible left-invariant metric isometrically as a subgroup. We shall present a construction of such an example.

↑top of page

Sy-David FRIEDMAN                Descriptive Set Theory and Absoluteness

How absolute are the basic features of analytic and co-analytic equivalence relations? An analytic equivalence relation can have countably-many (small), uncountably-many but not perfectly-many (medium), or perfectly-many classes (large); in the last case it can be either Borel or non-Borel. Smallness, largeness and Borelness are persistent to outer models (by Shoenfield), but mediumness and non-Boreleness are not (using master codes). For orbit relations mediumness is persistent (using "tameness") but I don't know if non-Borelness is. Co-analytic equivalence relations behave like analytic equivalence relations, except they cannot be medium. The classes of an analytic equivalence relation can be countable (small) or contain a perfect set (large). For co-analytic equivalence relations they can also be uncountable with no perfect subset (medium). A large class can either be Borel or non-Borel. An analytic equivalence relation which is not large must have a large class. There is a medium analytic equivalence relation with no Borel class (explicit construction). In $L$, there is a co-analytic equivalence relation such that each class is medium (this uses ideas of Conley, Miller and Tˆrnquist to answer a question of Kechris' student William Chan). For analytic equivalence relations, having a small, large or Borel class is persistent (Shoenfield) but having a non-Borel class is not. Having at least one uncountable class is not persistent for co-analytic equivalence relations. There are many open questions beyond those mentioned above. For example is smallness for analytic equivalence relations $\Sigma^1_3$ complete?

↑top of page

Gilles GODEFROY                Convex hulls of measurable functions and the axioms.

Medial limits are non-zero linear forms on $l_\infty$ which vanish on $c_0$ and are universally measurable for the weak* topology. Such objects were shown to exist by G. Mokobodzki and J. P. R. Christensen in 1973, and P. Larson showed in 2009 that they fail to exist under the Filter Dichotomy. We will display some observations around these results, and related links between the topological and combinatorial properties of filters on $\omega$.

↑top of page

Vassilis GREGORIADES                Characterizations of the Baire space up to HYP-isomorphism

It is known that uncountable recursive Polish spaces are not necessarily HYP-isomorphic to the Baire space. In this talk we present two characterizations of the Baire space up to HYP-isomorphism in terms of measures and the perfect kernel.

↑top of page

Takayuki KIHARA                Some more results around decomposability of Borel functions.

Gregoriades and the speaker have shown that by using the Louveau separation theorem and the Shore-Slaman join theorem, for every function $f$ between transfinite dimensional Polish spaces, if the preimage of a $\Sigma^0_m$ set under $f$ is always $\Sigma^0_n$, then $f$ is decomposable into countably many $\Sigma^0_{n-m+1}$-measurable functions. In this talk, I report that by modifying Kumabe-Slaman forcing, we can generalize the above result to any function from an analytic space into a separable metrizable space. We also see that the fine-structural analysis of Kumabe-Slaman forcing provides some weak decomposition results for C-measurable functions, etc.

↑top of page

Aleksandra KWIATKOWSKA                The Ramsey degree of the pre-pseudoarc

I will discuss a proof that the Ramsey degree of the family of finite reflexive linear graphs is infinite. This result together with a work due to Andy Zucker implies that the automorphism group of the pre-pseudoarc has a non-metrizable universal minimal flow. This is joint work with Dana Bartosova. The question whether the universal minimal flow of the homeomorphism group of the pseudoarc is metrizable is still open.

↑top of page

FranÁois LE MAITRE                 More Polish full groups

If $G$ is a Polish group acting on a standard probability space $(X,\mu)$ by measure-preserving transformations, the associated orbit full group is the group of all measure-preserving transformations which preserve the $G$-orbits. When $G$ is locally compact, it can be shown that this group completely remembers the partition of the space into orbits induced by the action, so that its properties should reflect those of the partition. I will explain how to endow this full group with a Polish group topology and give examples of how this topology "remembers" the partition of the space into orbits. This is joint work with Alessandro Carderi.

↑top of page

Maciej MALICKI                Automorphism groups of homogeneous metric structures and consequences of the existence of ample generics

We will define a simple criterion for a homogeneous, complete metric structure $X$ implying that its automorphism group satisfies all the main consequences of the existence of ample generics: the automatic continuity property, the small index property, and uncountable cofinality for non-open subgroups. It turns out that it holds for the Urysohn space, the Lebesgue probability measure algebra, and the Hilbert space. We also formulate a condition for $X$ implying that every homomorphism of its automorphism group into a separable group with a left-invariant, complete metric is trivial, and we will verify it for the Urysohn space, and the Hilbert space.

↑top of page

Benjamin MILLER                The open graph dichotomy

We will discuss consequences of the countably-infinite-dimensional analog of the open graph dichotomy.

↑top of page

Luca MOTTO ROS                On isometry and isometric embeddability between (ultra)metric Polish spaces

We determine the complexity with respect to Borel reducibility of various classification problems for (ultra)metric Polish spaces up to isometry and isometric bi-embeddability. This answers some questions of Gao-Kechris (2000), Clemens (2007) and Gao-Shao (2011). Joint work with Camerlo and Marcone.

↑top of page

Christian ROSENDAL                Geometries of groups

We investigate the various geometries at the small and large scale that may be defined on a topological group. Central in this are questions of metrisability and the associated concepts of minimal and maximal metrics.

↑top of page

Asger TORNQUIST                Definable maximal discrete sets in forcing extensions

I will discuss some recent results due to me and David Schrittesser, which show that (all lightface) $\Sigma^1_1$ binary relations admit $\Delta^1_2$ maximal discrete sets in the Sacks and Miller extensions of $L$. This should be compared with the Cohen and Random extension, where this fails badly. We apply this to obtain results about the definability of maximal orthogonal sets Borel probability measures.

↑top of page

Zoltan VIDNYANSZKY                The size of conjugacy classes of automorphism groups

The automorphism groups of FraissÈ limits are usually interesting objects both from group theoretic and set theoretic viewpoint. However, these groups are often non-locally compact, hence there is no natural translation invariant measure on them. Christensen introduced the notion of Haar null sets in non-locally compact Polish groups which is a well-behaved generalisation of the null ideal to such groups. In my talk I will present some new results concerning the size of the conjugacy classes of automorphism groups of Fraisse limits and other Polish groups with respect to this notion.

↑top of page