SoSe2022 seminar run by Aleksandra Kwiatkowska, Katrin Tent, and Martin Bays.

Mondays 10-12 SR1d, starting 04.04.

If you are interested in participating, please contact Martin Bays <baysm@wwu.de> or Aleksandra Kwiatkowska <kwiatkoa@uni-muenster.de>.

Topological dynamics studies continuous actions of topological groups. Key concepts include (extreme) amenability, universal minimal flows, and the Ellis semigroup. The relevance of topological dynamics to model theory has become increasingly clear over the last two decades. The connection arises both via the automorphism group of a structure and via groups definable in a structure. There is now a substantial literature on the fruitful interactions between the two subjects, with highlights including reinterpretations of the Kechris-Pestov-Todorcevic correspondence, which connects universal minimal flows and Ramsey theory, and the correspondences between model-theoretic stability and weakly almost periodic functions, and between NIP and tameness of a dynamical system. Moreover, there are applications to model-theoretic connected components of definable groups and Lascar's Galois group of a theory, which itself has applications in the study of approximate subgroups. This seminar will explore all these connections.

The seminar will be suitable for both bachelor and master's students, with a first course in model theory as the only prerequisite.

- Fundamentals of topological dynamics: amenability, minimal flows, Ellis semigroup.
- Kechris-Pestov-Todorčević correspondences between Ramsey properties of a structure and dynamical properties of its automorphism group [KP19].
- Model-theoretic interpretation of the Roelcke compactification. The theory of a structure is stable if and only if every Roelcke uniformly continuous function on the automorphism group is weakly almost periodic [BYT16].
- Correspondence between NIP and tameness in topological dynamics [Iba16].
- The connected components G^{00} and G^{000}, and applications to approximate subgroups [MW15,HKP22].
- Definability patterns and the Lascar group [Hru19].

- [BYT16] Itaï Ben Yaacov and Todor Tsankov. Weakly almost periodic functions, model-theoretic stability, and minimality of topological groups. Trans. Amer. Math. Soc., 368(11):8267--8294, 2016.
- [GL12] Yonatan Gutman, Hanfeng Li. A new short proof for the uniqueness of the universal minimal space
- [HKP22] Ehud Hrushovski, Krzysztof Krupinski, and Anand Pillay. Amenability, connected components, and definable actions. Selecta Math. (N.S.), 28(1):Paper No. 16, 56, 2022.
- [Hru19] Ehud Hrushovski. Definability patterns and their symmetries. 2019.
- [Hru20] Ehud Hrushovski. Beyond the lascar group. 2020.
- [Iba16] Tomás Ibarlucía. The dynamical hierarchy for Roelcke precompact Polish groups. Israel J. Math., 215(2):965--1009, 2016.
- [KP19] Krzysztof Krupinski and Anand Pillay. Amenability, definable groups, and automorphism groups. Adv. Math., 345:1253--1299, 2019.
- [KP21] Krzysztof Krupinski and Anand Pillay. On the topological dynamics of automorphism groups; a first-order perspective. Preprint.
- [MW15] Jean-Cyrille Massicot and Frank O. Wagner. Approximate subgroups. J. Éc. polytech. Math., 2:55--64, 2015.
- [Sim] Pierre Simon. Notes on Hrushovski's "Definability patterns and their symmetries". Unpublished note.

(Plan subject to change)

- 1. 04.04 Universal G-ambits [KP21]; Ola Kwiatkowska
- 2. 11.04 The KPT correspondence [KP21]; Maximilian Strohmeier
- 3. 25.04 The universal minimal flow of Aut(M) [KP21]; Blaise Boissonneau
- 4. 02.05 Weak almost periodicity and stability I [BYT16]; Alessandro Codenotti
- 5. 09.05 Uniqueness of UMFs [GL12]; Ola Kwiatkowska
- 6. 16.05 Weak almost periodicity and stability [BYT16]; Rob Sullivan
- 7. 23.05, 30.05 Definably amenable approximate subgroups [MW15]; Zhengqing He
- 8a. 13.06 NIP and topological dynamics I [Iba16]: NIP in continuous logic; Martin Hils
- 8b. 13.06 NIP and topological dynamics I [Iba16]: Tame functions; Ola Kwiatkowska
- 9. 27.06 NIP and topological dynamics II [Iba16]; Alessandro Codenotti
- 10. 04.07 Quasihomomorphisms and approximate subgroups [HKP22]; Martin Bays
- 11. 11.07 Definability patterns and the Lascar group [Hru19, Hru20, Sim]; Tingxiang Zou