If G is a topological group, a G-flow X is a non-empty, compact, Hausdorff space on which G acts continuously; it is minimal if all G-orbits are dense. By a theorem of Ellis, there is a (unique) minimal G-flow M(G) which is universal: there is a continuous G-map to every other G-flow.

Here, we will be interested in the case where G = Aut(K) for some structure K, usually omega-categorical. Work of Kechris, Pestov and Todorcevic and others gives conditions on K under which structural Ramsey Theory (due to Nesetril - Rodl and others) can be used to compute M(G).

In the first part of the talk I will give a description of the above theory and when it applies (the 'tame case'). In the second part, I will describe joint work with J. Hubicka and J. Nesetril which shows that the omega-categorical structures constructed in the late 1980's by Hrushovski as counterexamples to Lachlan's conjecture are not tame and moreover, minimal flows of their automorphism groups have rather different properties to those in the tame case.

# Logic Seminar

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## Further Information:

joint work with Moshe Kamensky and Silvain Rideau

Let $p$ be a fixed prime number and let $SCVF_p$ be the theory of separably closed non-trivially valued fields of

characteristic $p$. In the talk, we will see that, in many ways, the step from $ACVF_{p,p}$ to $SCVF_p$ is not more

complicated than the one from $ACF_p$ to $SCF_p$.

At a basic level, this is true for quantifier elimination (Delon), for which it suffices to add parametrized $p$-coordinate

functions to any of the usual languages for valued fields. It follows that all completions are NIP.

At a more sophisticated level, in finite degree of imperfection, when a $p$-basis is named or when one just works with

Hasse derivations, the imaginaries of $SCVF_p$ are not more complicated than the ones in $ACVF_{p,p}$, i.e., they are

classified by the geometric sorts of Haskell-Hrushovski-Macpherson. The latter is proved using prolongations. One may

also use these to characterize the stable part and the stably dominated types in $SCVF_p$, and to show metastability.