Institute of Mathematics and Mathematical Modeling, Almaty

An algebraic structure M is said to be stable up to Δ in λ if for any complete Δ-type s over M and for any A⊆M of cardinality at most λ the number of complete types over A that are consistent with s is at most λ. A theory is stable up to Δ in λ if each its model is. If a theory is stable up to Δ and each formula in Δ is NIP, then the theory is NIP; this gives some characterization of NIP theories.

A complete 1-type over a model of a stable up to Δ theory is definable iff its restriction to Δ is definable. The most applications will be given for Δ = {x<y, y<x}, that is for o-stable theories. In the talk, I give some properties of o-stable ordered groups.

Changwon National University, Changwon

We present a general construction of NSOP₁ theories as limits of Fraïssé classes K satisfying a collection of hypotheses. We first describe the model-theoretic properties of the resulting theory T. We then focus on several modifications of the class K that preserve these hypotheses. These constructions correspond to certain generic expansions of T. We will also give some examples of NSOP₁ theories that can be obtained in this way.

Inner Mongolia University, Hohhot

It is a classical result in first-order model theory that Dedekind-cut many types lead to the independence property. We prove a similar result in the setting of PC classes.

Fudan University, Shanghai

We study dp-finite fields whose dp-rank is a prime number, which we call dp-prime fields. Our first theorem, valid for arbitrary dp-finite fields, shows that all rank minimal definable subsets of the field have the same dp-rank. Consequently, if dp-rk(K) = n, then every critical coordinate configuration consists of k coordinates of one common rank n/k, where k divides n.

For an unstable dp-prime field, only two cases remain: Cₙ, where all critical coordinates are dp-minimal, and C₁, where every infinite unary definable set has full dp-rank. We prove that dp-rank is definable in arbitrary definable families over every dp-prime field. We also prove that the dp-rank of a type is the minimum of the dp-ranks of its definable neighbourhoods. Finally, we study the canonical topology.

Under C₁, full dp-rank is equivalent to non-empty interior, definable functions are generically continuous, and cell decomposition holds. Under Cₙ, the same conclusions, with a local form of cell decomposition, follow from the unary strict frontier inequality SF₁.

National University of Singapore, Singapore

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)

Korea Institute for Advanced Study, Seoul

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 Fraïssé limits.

Nanjing University, Nanjing

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.

Fudan University, Shanghai

Let G be a definable group over p-adics and M an elementary extension of ℚ_p. We show that ℰ^M_G, the Ellis group of G over M, is isomorphic to the Ellis group of a definably amenable component of G. Hence, ℰ^M_G is independent of 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 ℰ^M_G is a topological group in the τ-topology. We also show that, as a topological group in the τ-topology, ℰ^M_G is also independent of the choice of M.

This is joint work with Ningyuan Yao.