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We consider the problem of isometric embedding of metric spaces into Banach spaces; and introduce and study the remarkable class of so-called linearly rigid metric spaces: these are the spaces that admit a unique, up to isometry, linearly dense isometric embedding into a Banach space. The first nontrivial example of such a space was given by R. Holmes; he… (More)

- Julien Melleray
- J. Symb. Log.
- 2010

We reformulate, in the context of continuous logic, an oscillation theorem proved by G. Hjorth and give a proof of the theorem in that setting which is similar to, but simpler than, Hjorth’s original one. The point of view presented here clarifies the relation between Hjorth’s theorem and first-order logic.

- Julien Melleray
- Math. Log. Q.
- 2007

We compute here the Borel complexity of the relation of isometry between separable Banach spaces, using results of Gao, Kechris [1] and Mayer-Wolf [4]. We show that this relation is Borel bireducible to the universal relation for Borel actions of Polish groups.

If G is a Polish group and Γ is a countable group, denote byHom(Γ,G) the space of all homomorphisms Γ → G. We study properties of the group π(Γ) for the generic π ∈ Hom(Γ,G), when Γ is abelian and G is one of the following three groups: the unitary group of an infinite-dimensional Hilbert space, the automorphism group of a standard probability space, and… (More)

We prove that if the universalminimal flow of a Polish groupG ismetrizable and contains a Gδ orbit G · x0, then it is isomorphic to the completion of the homogeneous space G/Gx0 and show how this result translates naturally in terms of structural Ramsey theory. We also investigate universal minimal proximal flows and describe concrete representations of… (More)

- Julien Melleray
- 2005

Building on earlier work of Katětov, Uspenskij proved in [9] that the group of isometries of Urysohn’s universal metric space U, endowed with the product topology, is a universal Polish group (i.e it contains an isomorphic copy of any Polish group). Answering a question of Gao and Kechris, we prove here the following, more precise result: for any Polish… (More)

- Julien Melleray
- 2005

This note is devoted to proving the following result: given a compact metrizable group G, there is a compact metric space K such that G is isomorphic (as a topological group) to the isometry group of K.

- Yair Glasner, D. Kitroser, Julien Melleray
- J. London Math. Society
- 2016

Let G be a countable group, Sub(G) be the (compact, metric) space of all subgroups of G with the Chabauty topology and Is(G) ⊆ Sub(G) be the collection of isolated points. We denote by X! the (Polish) group of all permutations of a countable set X. Then the following properties are equivalent: (i) Is(G) is dense in Sub(G); (ii) G admits a ‘generic… (More)

- Tomás Ibarlućıa, Julien Melleray
- 2017

We give a characterization of sets K of probability measures on a Cantor space X with the property that there exists a minimal homeomorphism g of X such that the set of g-invariant probability measures on X coincides with K. This extends theorems of Akin (corresponding to the case when K is a singleton) and Dahl (when K is finite-dimensional). Our argument… (More)

- Ward Henson, Paul Baginski, +9 authors Slawomir Solecki
- 2014

The first day We began Monday morning with a lecture prepared by the organizers and delivered by Ben Yaacov, presenting the background on continuous logic and metric structures that would be needed during the week. This was designed to elicit lots of questions and discussion, and the end result seemed pretty successful. This event lasted most of the morning… (More)