# AGRUME meeting in Colmar

The 2021 meeting of the ANR Project AGRUME (link) will take place from September 1st to September 4th at the Hôtel Beauséjour (link) in Colmar. The address is 25, rue du Ladhof.

The meeting is organized by Gianluca Basso, Itaï Ben Yaacov, Tomás Ibarlucía and Todor Tsankov.

## Schedule

### Wednesday

 15:30 - 16:00 Welcome coffee 16:00 - 16:45 François Le Maître A characterization of high transitivity for groups acting on trees abstract 17:00 - 17:45 Alessandro Vignati Uniform Roe algebras of uniformly locally finite metric spaces abstract

### Thursday

 09:00 - 09:45 Martin Bays Density of compressibility abstract 09:45 - 10:15 Coffee break 10:15 - 10:45 Jorge Muñoz Carvajal Universal Skolem sort for randomizations abstract 11:00 - 11:30 Antonio Scielzo $$L_1$$ Banach lattices with a distinguished automorphism and their types abstract 11:30 - 14:00 Lunch 14:00 - 14:45 Tomás Ibarlucía Approximate isomorphism and categoricity: new examples and counterexamples abstract 15:00 - 15:45 Gianluca Basso Topological dynamics beyond Polish groups abstract

### Friday

 09:00 - 09:45 Zoé Chatzidakis TBA 09:45 - 10:15 Coffee break 10:15 - 11:00 Rosario Mennuni The domination monoid in henselian valued fields abstract 11:15 - 12:00 Alessandro Carderi Ultraproducts of mm-spaces are (probably) the future abstract 12:00 - 14:00 Lunch 14:00 - 14:45 Zou Tingxiang Geometric random graphs abstract 14:45 - 15:15 Coffee break 15:15 Open questions and discussion

## Abstracts

### Topological dynamics beyond Polish groups (45 minutes)

Abstract. When $$G$$ is a Polish group, one way of knowing that it has nice dynamics is to show that $$M(G)$$, the universal minimal flow of $$G$$, is metrizable. For non-Polish groups, this is not the relevant dividing line: the universal minimal flow of the symmetric group of a set of cardinality $$\kappa$$ is the space of linear orders on $$\kappa$$–not a metrizable space, but still nice–, for example. In this talk, we present a set of equivalent properties of topological groups which characterize having nice dynamics. We show that the class of groups satisfying such properties is closed under some topological operations and use this to compute the universal minimal flows of some concrete groups, like $$\mathrm{Homeo}(\omega_{1})$$. This is joint work with Andy Zucker.

### Density of compressibility (45 minutes)

Abstract. Joint work with Itay Kaplan and Pierre Simon. Compressibility is a certain isolation notion suited to NIP theories. A theory is distal if and only if every type is compressible. I will discuss some good properties of this notion and their consequences, in particular the existence of "compressibly atomic" models over arbitrary sets in countable NIP theories, and uniform honest definitions for an NIP formula.

### Ultraproducts of mm-spaces are (probably) the future (45 minutes)

Abstract. In this talk, I'm going to present a notion of ultraproduct for (extended) metric measure spaces. This construction is interesting for two reasons, on the one hand, the triviality of such an ultraproduct is equivalent to the concentration of measure and on the other, one can use it to obtain ultraproducts of actions of locally compact groups. I'm going to briefly review how the latter notion can be used to obtain asymptotic invariants of lattices and we will discuss (and hopefully solve!) many questions about properties and other possible uses of the notion.

### Approximate isomorphism and categoricity: new examples and counterexamples (45 minutes)

Abstract. I will discuss some recent results related to approximate isomorphism and approximate $$\aleph_0$$-categoricity in different contexts of continuous logic. On the one hand, I will present examples and counterexamples arising from theories of beautiful pairs of randomizations (joint work with J. Hanson). On the other hand, I will present two examples of metrically generic (thus approximately conjugate) measure-preserving actions of the free group on a Lebesgue space, and discuss their continuous theory (joint work with A. Berenstein and W. Henson). The latter is the model completion of the theory of probability measure-preserving actions of the free group.

### A characterization of high transitivity for groups acting on trees (45 minutes)

Abstract. A countable group is called highly transitive when it admits a faithful action on a countable set which is n-transitive for every n. In this talk, I will present a joint work with Pierre Fima, Soyoung Moon and Yves Stalder where we characterize highly transitive groups among groups admitting a faithful minimal action of general type on a tree by establishing the converse of a recent result of Adrien Le Boudec and Nicolás Matte Bon.

### Double Fraïssé classes and limits (CANCELLED) (45 minutes)

Abstract. We introduce double-Fraïssé classes, where instead of embeddings between structures a couple of section-retraction are considered. In particular, we use this to present new properties of the Pełczyński's universal basis space. This work-in-progress is a joint work with Jamal K. Kawach and S. Todorcevic.

### The domination monoid in henselian valued fields (45 minutes)

Abstract. In their book on stable domination, Haskell, Hrushovski and Macpherson classified invariant types in algebraically closed valued fields up to domination-equivalence. This motivated a study of domination in unstable theories, and of how it interacts with the Morley product of invariant types. I will talk about recent joint work with Martin Hils concerning domination in henselian valued fields, expansions thereof, and other related structures.

### Universal Skolem sort for randomizations (30 minutes)

Abstract. Recently, Itaï Ben Yaacov defined the notion of universal Skolem sorts and gave a way to associate to each complete theory that admits such a sort a topological groupoid which allows to reconstruct the theory, modulo bi-interpretation. In this talk we show that if a theory admits a universal Skolem sort then so it does its randomization. We also show how to build the groupoid associated to the randomization from the groupoid of the original theory. Finally, we show how we expect to use the groupoids in order to prove some preservation results.

### $$L_1$$ Banach lattices with a distinguished automorphism and their types (30 minutes)

Abstract. In this talk, I will discuss the theory of $$L_1$$ Banach lattices with a distinguished automorphism. This theory happens to admit a model companion, denoted by $$T_A$$, which enjoys some useful properties, most notably stability and quantifier elimination. In the framework of continuous logic, the space of types $$S_x(T)$$ of a theory $$T$$ can be endowed with two different topologies: the initial topology with respect to the family of formulas and the metric topology induced by the distance between realisations of these types. For which points $$p \in S_x(T)$$ does the two topologies coincide at $$p$$? It turns out that for $$T_A$$, there is only one such point, the type of $$0$$. I will attempt to show the ideas behind such result.

### Uniform Roe algebras of uniformly locally finite metric spaces (45 minutes)

Abstract. Coarse geometry is the study of metric spaces when one forgets about the small scale structure and focuses only on large scales. Objects of interests are coarse spaces. The maps of interest are coarse equivalences, required to preserve the large scale geometry. Typical examples that are important for applications are finitely generated groups with word metrics, and discretizations of non-discrete spaces such as Riemannian manifolds. To a coarse space one associates, after Roe, several $$C^*$$-algebras capable of detecting algebraically the geometric properties of the spaces. Chief among them for applications is the uniform Roe algebra of $$X$$, $$C_u^*(X)$$. We show that if $$X$$ and $$Y$$ are uniformly locally finite metric spaces whose uniform Roe algebras are isomorphic, then $$X$$ and $$Y$$ are coarsely equivalent metric spaces.

### Geometric random graphs (45 minutes)

Abstract. Geometric random graphs are graphs on a countable dense set of some underlying metric space such that locally in any ball of radius one, it is a random graph. The geometric random graphs on $$\mathbb R^n$$ and on circles have been studied by probabilists and graph theorists. In this talk I will present some model theoretic views, the elementary equivalence, the dividing lines and so on. I will also talk about some geometric properties of the underlying metric space that can be recovered from the first order theory of graphs. This is joint work with Omer Ben-Neria and Itay Kaplan.

Last updated: 31 August 2021