DS10 - Défi des autres savoirs

Group Actions and Model Theory – AGRUME

Submission summary

The past few years have seen increasingly close connections between Model Theory, Dynamics (topological, measurable) and Combinatorics (Ramsey Theory), all revolving around dynamical systems of various kinds.
By this we refer to several striking results proved in one of these domains using techniques and tools from one of the others, as well as to striking similarities in existing parallel theories.
The time is ripe for a focused work at the interface of the aforementioned domains, by a consortium of researchers with complementary expertise and proven track-record.
We expect significant progress toward new applications of existing tools, as well as new tools and more unified and streamlined theories, on several related fronts:

- Relations between combinatorial (Ramsey) properties of a class of finite structures (a Fraïssé class) and dynamical properties of an associated Polish group (the automorphism group of the Fraïssé limit).
In particular, we hope for a useful sufficient condition for a Polish group to have a metrisable universal minimal flow.

- A model-theoretic approach to entropy in Ergodic Theory.
We aim to bring to light model-theoretic aspects of the existing theory, and use model-theoretic tools to push it further.

- Correspondences between model-theoretic properties (of structures, or formulas) and dynamical properties (of automorphism groups, or of functions defined thereon).
In particular, the extension of several existing such correspondences beyond the $\aleph_0$-categorical setting, or beyond the discrete setting.

- Study the normal subgroup structure of automorphism groups.
In particular, the study of the subgroup of ``bounded automorphisms'', and its subgroups, both in several specific cases and under general hypotheses.

Project coordination

Itaï BEN YAACOV (Institut Camille Jordan - UMR 5208)

The author of this summary is the project coordinator, who is responsible for the content of this summary. The ANR declines any responsibility as for its contents.

Partner

ICJ Institut Camille Jordan - UMR 5208
IMJ-PRG Institut de mathématiques de Jussieu - Paris Rive Gauche

Help of the ANR 322,616 euros
Beginning and duration of the scientific project: September 2017 - 48 Months

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