For more information, please visit https://sites.google.com/view/sklinoshome/beijing-model-theory-day/beijing-model-theory-day-2025.

Speaker: Javier de la nuez González (Korea Institute for Advanced Study,Seoul)

Title: Sharply k-transitive actions on relational structures

Abstract: We say that an action by isomorphisms of a group G on a relational structure M is (structurally) sharply k-transitive if for any two tuples of distinct elements of M, a and a', which are in the same orbit under the diagonal action of Aut(M) there is exactly one element g in G mapping a to a'. I will discuss recent work joint with Rob Sullivan in which we establish sufficient conditions for the existence of sharply k-transitive actions of finitely generated virtually free groups on relational Fräissé limits.

Speaker: Jinhe Ye (Nanjing University,Nanjing)

Title: Beautiful Pairs Re-Revisited

Abstract: We investigate beautiful pairs in the context of unstable theories.

By classical results of Poizat, the theory of beautiful pairs of models of a stable theory T is “meaningful” precisely when the set of all (definable) types in T is strict pro-definable. This holds if and only if T does not have the finite cover property. Together with Cuibides Kovacsics and Hils, we establish Ax–Kochen–Ershov principles for various questions concerning beautiful pairs of valued fields, providing powerful tools for analyzing spaces of definable types. We show that the theory of beautiful pairs of models of ACVF is indeed “meaningful,” and from this we deduce the strict pro-definability of several geometrically motivated spaces of definable types in ACVF.

More recently,with Hils,Hrushovski,and Zou,we initiated the study of beautiful pairs of difference fields (possibly with additional structures). In particular,we will discuss an extension of Hrushovski’s non-standard Frobenius to the setting of beautiful pairs of difference fields.

Speaker: Chieu-Minh Tran (National University of Singapore,Singapore)

Title: Large implies henselian

Abstract: In the classic paper"Henselian implies large",Pop showed that a field is large if it is elementarily equivalent to the fraction field of a proper Henselian local domain. We show that it can,in fact,be improved to an"if and only if"statement.

We will explain how this result arises from the effort to introduce  canonical topologies over a field,namely,a result comparing the etale-open topology with a newly introduced finite-closed topology. (Joint with Will Johnson,Erik Walsberg,and Jinhe Ye)

Speaker: Zhentao Zhang (Fudan University,Shanghai)

Title: On Ellis groups for definable groups over $p$-adics

Abstract: Let $G$ a definable groups over $p$-adics and $M$ an elementary extension of $\mathbb{Q}_p$. We show that $\mathcal{E}_G^M$,the Ellis group of $G$ over $M$,is isomorphic to the Ellis group of a definably amenable component of $G$. Hence,$\mathcal{E}_G^M$ is independent on the choice of $M$. We also show that Newelski's conjecture holds on $G$ iff $G$ is definably amenable.

Moreover,we prove the revised Newelski's conjecture for $G$ on $M$ and then $\mathcal{E}^M_G$ is a topological group in $\tau$-topology. We also show that,as a topological group in $\tau$-topology,$\mathcal{E}_G^M$ is also independent on the choice of $M$.

This is a joint work with Ningyuan Yao.