# Trimester Seminar

Date: every Thursday, 11:00-12:30 (unless stated otherwise)
Venue: HIM lecture hall, Poppelsdorfer Allee 45

## Thursday, September 12

11:00 - 12:30 Itay Kaplan: AGL2(Q) is maximal
Abstract: We will show that AGL2(Q) is a countable maximal closed subgroup of the infinite permutation group, thus answering a question of Macpherson. Joint work with Pierre Simon.

## Monday, September 16

15:00 - 16:00 Welcome Meeting
Abstract: Welcome meeting with the participants and the organizers of the Trimester Program.

## Monday, September 23

11:00 - 12:30 Ulrich Kohlenbach: Proof-theoretic methods in ergodic theory and topological dynamics
Abstract: During the last two decades a systematic program of ‘proof mining’ emerged as a new applied form of proof theory and has successfully been applied to a number of areas of core mathematics. This program has its roots in Georg Kreisel’s pioneering ideas of ‘unwinding of proofs’ going back to the 1950’s who asked for a ‘shift of emphasis’ in proof theory away from issues of mere consistency of mathematical theories (‘Hilbert’s program’) to the question ‘What more do we know if we have proved a theorem by restricted means than if we merely know that it is true?’
We are primarily concerned with the extraction of hidden finitary and combinatorial content from proofs that make use of highly infinitary principles. The main logical tools for this extraction are novel forms and extensions of Kurt Gödel’s famous functional (‘Dialectica’) interpretation. Logical metatheorems based on such interpretations have been applied with particular success in the context of nonlinear analysis, ergodic theory and topological dynamics. The combinatorial content can manifest itself both in explicit effective bounds as well as uniformity results.
In this talk we will outline the general background of this novel form of applied proof theory and indicate some recent applications in the context of nonlinear ergodic theory.

## Thursday, September 26

11:00 - 12:30 Boris Zilber, Oxford: Hrushovski's construction, pseudo-analytic structures and number theory
Abstract: Hrushovski's construction (of new stable structures) is one of the most powerful tools introduced to model theory some 25 years ago and still not fully explored.  We seek to identify a geometric meaning of this construction. This leads to the class of pseudo-analytic structures, which are formal analogues of analytic objects defined in terms of classical transcendental functions. The study of these structures established deep connections with number theory, algebraic and analytic geometry.

## Wednesday, October 2

11:00 - 12:30 Lionel Nguyen Van Thé, Aix-Marseille: Structural Ramsey theory and topological dynamics for automorphism groups of homogeneous structures: a preamble
Abstract: In 2005, Kechris, Pestov, and Todorcevic established a striking connection between structural Ramsey theory and the topological dynamics certain automorphism groups. The purpose of this talk will be to present this connection, together with recent related results. This will also serve as a warm-up for the forthcoming mini-course that will be given by Todor Tsankov later in the semester. (Joint work with Todor Tsankov and Julien Melleray.)

## Thursday, October 10

11:00 - 12:30 Andre Nies, University of Auckland (NZ): The complexity of similarity relations for Polish metric spaces
Abstract: We consider the similarity relations of isometry and homeomorphism for Polish spaces. We survey known results on the complexity of such relations. For instance, Gao and Kechris showed that isometry is orbit complete, while Gromov proved that for compact spaces it is smooth. The exact complexity of homeomorphism for compact metric spaces is not known. Using a result of Camerlo and Gao, we show that, in the computable setting, homeomorphism is Sigma-1-1 complete for equivalence relations. Another interesting similarity relation is having Gromov-Hausdorff distance 0, which is related to the Scott rank in continuous logic.
Notes for the talk

## Tuesday, October 15

11:00 - 12:30 László Lovász: Graphons and their automorphisms
Abstract: Graphons are symmetric measurable functions from [0,1]2 to [0,1], which can serve as limit objects for sequences of finite graphs that are "convergent". An automorphism of a graphon is a measure preserving permutation of [0,1] that preserves the value of the function almost everywhere. It turns out that after some standardization, the automorphism group of every graphon will be compact. The orbits of the automorphism group can be characterized by certain generalized degrees.
The tools needed to prove these results include topologies on the set of points of a graphon, which in turn is related to the Regularity Lemmas of Szemeredi and Frieze-Kannan.
Viewing graphons as limit objects for graph sequences, some properties of the automorphism groups of these objects carry over to the limit; for example, the limit of graphs with node-transitive automorphism groups is a graphon with a point-transitive automorphism group. But this transition to the limit is not fully understood. This is joint work with Balazs Szegedy.

## Tuesday, October 22

11:00 - 12:30 Anatoly Vershik: Invariant measures for large groups
Abstract: Equivalence relations and invariant measures. We give examples of the smooth and non-smooth case. We discuss the dyadic filtration and our notion of standardness. As examples, we discuss

• Aldous's theorem about invariant measures on the space of tensors and its refinement;
• Invariant measure on the Urysohn space and on the space of universal graphs;
• Characters, C*-algebras and central measures on paths of Bratteli diagrams: such as the Pascal graph, Young graph and Hasse diagrams of distributive lattices.

Part of a series on Classification of invariant measures, for homogeneous and universal objects; 2nd lecture in the Workshop October 28; 3rd lecture in November.

## Thursday, November 7

11:00-12:30 Yu Manin (MPIM): Kolmogorov complexity: fractality and probability contexts

## Tuesday, November 12

14:00 - 15:00 Eli Glasner: On the canonical nil factor of order d

Abstract: Every compact minimal G-dynamical system admits a maximal d-nil factor. This "characteristic" factor determines many dynamical properties of the original system, like the behaviour of the ergodic sums in Furstenberg's multiple recurrence theorem. I'll outline a new short proof of the fact that the "d regionally proximal relation" is a closed equivalence relation, and thus yields the maximal d nil factor.

## Thursday, November 14

11:00-12:30 Adriane Kaïchouh (ICJ Lyon I): Amenability and Convex Ramsey Theory

Abstract: Moore recently characterized the amenability of automorphism groups of countable ultrahomogeneous structures by a Ramsey-type property (in the vein of the Kechris-Pestov-Todorcevic correspondence). We will present this property and give a generalization of Moore's result to automorphism groups of separable metric structures, which encompass all Polish groups. We will also discuss some nice consequences of this characterization.

## Thursday, November 14

14:00 - 15:00 Yonatan Gutman (IMPAN, Oxford): Nilspaces and Structure Theorems for Topological Dynamical Systems

Abstract: Nilsequences, namely sequences of complex numbers arising naturally from translations on nilmanifolds, introduced by Bergelson, Host & Kra, appear in different guises in several areas of mathematics: Topological Dynamics (maximal nilfactors), Ergodic Theory (convergence of multiple ergodic averages), Additive Number Theory (solving linear equations in primes) and Additive Combinatorics (generalizations of Szemerédi's Theorem). A flexible framework to investigate nilsequences and related concepts was introduced by Camarena and Szegedy. The main object, a nilspace, is a (compact) space which satisfy some straightforward axioms. A fundamental result is the representability of a nilspace as an inverse limit of nilmanifolds. In work in progress with Freddie Manners and Péter Varjú we give a new proof of this fundamental result. The new tools allow us to derive generalizations of the Host-Kra-Maass Structure Theorem for topological dynamical systems of finite order.

## Thursday, November 21

11:00-12:30 Michael Pinsker: Topological Birkhoff and reconstructing the random graph

Abstract: An algebra with countably infinite domain is called oligomorphic iff the set of its term functions contains an oligomorphic permutation group. We show that a classical theorem about finite algebras due to Birkhoff, called the finite HSP theorem, can be generalized to oligomorphic algebras. Birkhoff's HSP theorem states that a finite algebra B satisfies all equations that hold in a finite algebra A of the same signature if and only if B is a homomorphic image of a subalgebra of a finite power of A. We prove that if A and B are oligomorphic, then the mapping which sends each function from A to the corresponding function in B preserves equations *and is continuous* if and only if B is a homomorphic image of a subalgebra of a finite power of A.

A model-theoretic corollary is that two omega-categorical structures are primitive positive bi-interpretable if and only if their polymorphism clones are isomorphic as topological clones. In complexity theory, our result implies that the complexity of the constraint satisfaction problem of an omega-categorical structure only depends on its topological polymorphism clone.

Together with last week's work on reconstruction of the topology of polymorphism clones, we obtain that if a structure has a polymorphism clone which satisfies the same equations as the polymorphism clone of the random graph, then this structure is primitive positive biinterpretable with the random graph.

## Thursday, November 28

11:00-12:30 Matti Rubin (Ben Gurion University):Undecidability of the complete theories of some locally moving groups

Let $X$ be a regular space and $G$ be a group of auto-homeomorphisms of$X$.$G$ is said to be a locally moving group for $X$, if for every nonempty open $U \subseteq X$, there is $g \in G \widesetminus$\{${Id}$} such that $g \restriction (X \setminus U) = \rfs{Id}$. A group $H$ is called a locally moving group, if for some regular space $X$ and a group $G$ which is locally moving for $X$, $H \cong G$.

Examples: The following groups are locally moving.
(1) The group of auto-homeomorphisms of $\mathbb{R}^2$.
(2) The Thompson Group $/$ The group of all piecewise linear homeomorphisms $g$ of $[0,1]$ such that every slope of $g$ is an integral power of 2, and every break-point of $g$ is a dyadic number.
(3) The automorphism group of the binary tree.
(4) The group of measure preserving automorphisms of the quotient of the Borel field of $[0,1]$ over the ideal of measure 0 sets.

Conjecture The complete theory (in the language of groups) of every locally moving group is undecidabe.

The conjecture is true for a large subclass of the class of locally moving groups. And this subclass contains Examples (1) - (4). Here is the definition of this subclass.

Definition Let $X$ be a regular space without isolated points, and $G$ be a group of auto-homeomorphisms of $X$. $G$ is flexible for $X$, if for every nonempty open subset $U$ of $X$ there is a nonempty open subset $V$ of $U$ such that for every nonempty open $W \subseteq V$,

$V$ $\cap\ %$$\bigcup\kern1mm \setm${${g(W)}|{g \in G \mbox{ and } g \restriction (X$\ $U) = \rfs{Id}}}$}

is dense in $V$.

Note that flexibility implies local movability,

and that local movability implies the non-existence of isolated points.

Theorem Let $G$ be a group which is a flexible group for some regular space$X$ without isolated points. Then the complete theory of $G$ is undecidable.

## Wednesday, December 4

10:00 - 11:30 Jaroslav Nesetril (Charles University in Prague): Homomorphisms, Ramsey classes and structural limits

Abstract: We explain the fruitful relationship of the notions in the title, and illustrate this on some particular examples.

## Thursday, December 12

11:00 - 12:30 Valentin Ferenczi: Twisted Hilbert Spaces and Interpolation

Abstract: Twistings appear in different forms in the isomorphic or isometric theory of Banach spaces: for example push-out constructions of universal spaces,  twisted representations of groups, or twisted spaces induced by interpolation. After general comments on these different aspects, we shall concentrate on the last, showing how the study of interpolation scales induces new twistings of the Hilbert space. This is joint work with Jesús M. Castillo and Manuel Gonzalez.

## Thursday, December 12

14:00- 15:00 Michal Doucha: Universal abelian metric group and group structures on the Urysohn space

Abstract: I will present a construction of a metrically universal abelian separable metric group, i.e. every other separable abelian metric group embeds by an isometric homomorphism. This answers the question and extends the results of Shkarin and Niemiec, who constructed an abelian separable metric group which is only topologically universal (for the class of abelian Polish groups). I will also discuss group structures on the Urysohn universal space. Cameron and Vershik were the first who found a structure of an abelian group on the Urysohn space. Recently, Niemiec proved that the topologically universal group (constructed independently by him and Shkarin) is also isometric to the Urysohn space. However, it is still unclear whether this group is any different from the Cameron-Vershik group(s). I will show that the metrically universal group is topologically isomorphic neither to the Shkarin/Niemiec group nor to the Cameron-Vershik group(s). I will also present a non-abelian metric group which is also isometric to the Urysohn space, with a conjecture that this group is a universal Polish group admitting two-sided invariant metric.

## Wednesday, December 18

11:00 - 12:30 Jan Hubicka: Bowtie-free graphs have a Ramsey expansion

Abstract: A bowtie graph is a graph consisting of two triangles with one vertex identified. We consider the class of graphs not containing bowtie as a (non-induced) subgraph. It was shown by Komjath that there is a universal countable graph for this class. A more general result was shown by Cherlin, Shelah and Shi that also gives existence of an omega-categorical universal graph.
We show an explicit construction of this graph by means of a Fraisse limit of expansions of bowtie-free graphs, and show that this class is Ramsey. This is what we believe to be the first application of the partite construction on a class with non-trivial algebraic closure. We will also show how this method relates to other similar classes.

This is joint work with Jaroslav Nesetril.