Tobias Kaiser (Universität Passau)
Tame Definable Analysis

We consider three central concepts in Analysis: Differential Calculus, Differential Equations and Integral Calculus. We study them in the context of first order logic. And we concentrate here on tameness, both model theoretically and geometrically: we ask whether the objects are definable in an o-minimal structure.

Differentiability is a first order property. Hence the derivative of a differentiable function definable in a structure expanding the real field is again definable. In the o-minimal setting, we have additional tameness properties: a unary definable function is differentiable outside a finite set, and much more.

Solving differential equations and integration is not a first order property. One can in general only ask whether the solution of a differential equation or the antiderivative of a given function is definable in an o-minimal expansion.

In the case of scalar differential equations the following structural result holds: there is an o-minimal expansion such that the solutions of scalar differential equations with raw data definable in the given o-minimal structure are definable in the expansion. For systems of definable differential equations interesting things can happen: there is an o-minimal expansion such that each solution is definable therein; each solution is definable in an o-minimal expansion, but these expansions are not compatible; no solution is o-minimal.

In integration theory we consider parameterized integrals. We motivate this point of view by features in Analysis and in Number Theory. We present the known results on o-minimality of parameterized integrals in the restricted analytic setting and show the difficulties in the exponential case.

We also discuss the realization of the above three concepts in nonstandard setting. And we have an eye on model theoretic properties of the structures involved. Good model theoretic properties lead to nice descriptions of definable sets and functions which are important for applications of o-minimality.

Eran Alouf (Hebrew University)
Towards classifying dp-minimal expansions of the group of integers

Roughly speaking, every known dp-minimal proper expansion of the group of integers falls into one of three families, loosely defined by the following representatives: (1) The usual order, (2) cyclic orders induced by embeddings into the unit circle, and (3) generalized valuations. Are these really the only options, or are we yet to find examples of a completely new kind? In a previous work, I showed that the first family is characterized by an a-priori much weaker model-theoretic condition. I will present a new result, doing the same for the second family.

Yayi Fu (University of Notre Dame)
Erdős-Hajnal property for dp-minimal graphs

Jakub Gogolok (Uniwersytet Wrocławski)
Variants of existential closedness for fields with operators

There are various interesting classes of "big" fields in algebra, such as (pseudo)algebraically closed, separably closed or large fields. In recent years, differential variants of those classes were introduced and studied, leading to very interesting results. In this talk we want to show how to treat all these notions in a uniform way, as variants of existential closedness in appropriate categories, for operators more general than derivations. Under certain natural assumptions we prove that some classes of "generalized existentially closed fields with operators" are elementary classes and give nice (uniform) geometric axioms for them. This unifies and generalizes many results from the literature, and also answers some open questions.

Paolo Marimon (Imperial College London)
Measures in Homogeneous 3-hypergraphs

We study invariant Keisler measures and related notions of measures in ternary homogeneous structures. Most known tools to study invariant Keisler measures do not adequately generalise to this context. However, this is because there are structures exhibiting very interesting behaviour, which requires new tools to be understood.

We focus on three simple homogeneous uniform 3-hypergraphs. We prove that the universal homogeneous two-graph has a unique invariant Keisler measure in spite of having no invariant types. Then, we show that the generic tetrahedron-free 3-hypergraph is not MS-measurable providing the first example of a supersimple one-based ω-categorical structure which is not MS-measurable. We conclude by showing that the universal homogeneous 3-hypergraph has an extremely large space of invariant Keisler measures disproving a conjecture of Ensley. In order to do this, we show a correspondence between a subclass of its invariant Keisler measures, Cameron measures on the random graph, and S_∞-invariant measures on Struc_E concentrating on graphs.

David Meretzky (University of Notre Dame)
Picard-Vessiot Extensions, Linear Differential Algebraic Groups and their Torsors

I will talk about recent joint work with Anand Pillay generalizing the following theorem: A differential field K is algebraically closed and Picard-Vessiot closed iff the differential Galois cohomology is trivial for linear differential algebraic groups defined over K. We refine the idea that much of the cohomological information for linear differential algebraic groups lives in the iterated Picard-Vessiot closure of a differential field: For finite dimensional linear differential algebraic groups, points on torsors in a fixed differential closure all live in the iterated PV closure. Moreover, outside the finite dimensional case, the iterated PV points on torsors are Kolchin dense. Relatedly, we propose some definitions of boundedness for differential fields to give a common generalization of the original theorem and a theorem of Serre in the differential context. I will introduce and define the main concepts.

Aris Papadopoulos (University of Leeds)
Generalised indiscernibles in products of structures

My talk will be about Ramsey’s theorem and compactness. Perhaps at this point, dear reader, you're thinking that you can safely skip this talk, as you are indeed a model theorist and you know all about Ramsey’s theorem and compactness. Be that as it may, I will try to take a more general approach, and talk about generalised indiscernibles and structural Ramsey Theory (compactness is still compactness, of course) and how these tools allow us to understand the model theoretic properties of (lexicographic/full) products of structures. More precisely, many dividing lines can be defined through (generalised) indiscernible sequences, and in recent joint work with N. Meir and P. Touchard we were able to completely characterise generalised indiscernible sequences in the product of structures in terms of generalised indiscernible sequences in the structures themselves, allowing us to obtain transfer principles for dividing lines such as (k-)NIP, monadic NIP, and distality. I will survey the basic definitions and discuss our motivation and results.

Adrián Portillo (Uniwersytet Wrocławski)
Maximal stable quotients of type-definable sets in NIP theories

The main result of the paper “On maximal stable quotients of definable groups in NIP theories” by M. Haskel and A. Pillay says that for a group G definable in an NIP theory there exists a smallest type-definable subgroup G^st such that the quotient G/G^st is stable. We start by recalling some results of the aforementioned paper. Then, we present several characterizations of stability of hyperdefinable sets and a result says that if G is a type-definable group in a distal theory, then G^st = G⁰⁰ (where G⁰⁰ is the smallest type-definable subgroup of bounded index). In order to get it, we prove that distality is preserved under passing from T to the hyperimaginary expansion T^heq (contained in the paper “On stable quotients” by K. Krupiński, P.). The main focus of the talk is a counterpart of the main result of Haskel and Pillay in the context of invariant types (contained in the paper “Maximal stable quotients of invariant types in NIP theories” by K. Krupiński, P.). Namely, for a NIP theory T, a sufficiently saturated model ℭ of T, and an invariant (over some small subset of ℭ) global type p ∈ S(ℭ), we prove that there exists a finest relatively type-definable over a small set of parameters from ℭ equivalence relation on the set of realisations of p (in a bigger monster model) which has stable quotient. Our proof is a non-trivial adaptation of the ideas from the proof of the main result of Haskel and Pillay, using relatively type-definable subsets of the group of automorphisms of the monster model as defined in the paper “On first order amenability” by E. Hrushovski, K. Krupiński, and A. Pillay.

Benedikt Stock (Oxford University)
On Fields Elementarily Characterised by their Absolute Galois Groups

Galois groups contain a lot of arithmetic information about their base fields. In this talk, I will report on fields that are elementarily completely characterised by their absolute Galois groups. We will also hint at some recent attempts and approaches to eliminate certain edge cases that appear in the classification of these fields.

Michał Szachniewicz (Oxford University)
Globally valued fields - existential closedness and properties of the arithmetic intersection product

Intersection products play important role in proving existential closedness of some globally valued fields (abbreviated GVF). I will talk about this in the context of Arakelov intersection product and if time permits I will outline how adelic equidistribution theorem may lead to uniqueness of some GVF extensions.

Informal workshop dinner