Model Theory

We will discuss how Chow varieties can be used to control the large-scale behaviour of certain systems of exponential polynomials. This is a work-in-progress talk.

Recent developments in neostability theory have witnessed the proliferation of versions of Kim's lemma used to characterise ever-wider classes of unstable first-order theories, including NTP_2, NSOP_1, NSOP_4, NBTP, and beyond. These results highlight some limitations of Adler's framework for the study of abstract independence relations and suggest that new tools are at play underneath this diversity. In this talk, I will extend Adler's framework to study Kim's lemma and its consequences in their full generality. After briefly reviewing the role of independence relations in developing the theory of several neostability-theoretic properties, I will reinterpret Kim's lemma as a binary relation between independence relations under minimal assumptions. I will exemplify how this generalises several results from the literature, offer some new results that this framework proves, and mention previously known theorems which can be obtained in a semantic fashion using this technology.

Globally valued fields were introduced by Ben Yaacov and Hrushovski to capture the model theory of 'global fields with heights' in continuous logic. They subsequently proved that the algebraic closure of rational functions, with the usual projective heights, is existentially closed, while Szachniewicz proved the same for the field of algebraic numbers. I will give a minimal introduction to global fields and heights, keeping the presentation as close as possible to valuation theory, and to continuous logic, then explain what existential closedness means and how it relates to existing problems in Diophantine geometry. The talks are very loosely based on a small subsets of the 'foundations' paper by Ben Yaacov, Destic, Hrushovski, Szachniewicz and various other notes.

Globally valued fields were introduced by Ben Yaacov and Hrushovski to capture the model theory of 'global fields with heights' in continuous logic. They subsequently proved that the algebraic closure of rational functions, with the usual projective heights, is existentially closed, while Szachniewicz proved the same for the field of algebraic numbers. I will give a minimal introduction to global fields and heights, keeping the presentation as close as possible to valuation theory, and to continuous logic, then explain what existential closedness means and how it relates to existing problems in Diophantine geometry. The talks are very loosely based on a small subsets of the 'foundations' paper by Ben Yaacov, Destic, Hrushovski, Szachniewicz and various other notes.

Globally valued fields were introduced by Ben Yaacov and Hrushovski to capture the model theory of 'global fields with heights' in continuous logic. They subsequently proved that the algebraic closure of rational functions, with the usual projective heights, is existentially closed, while Szachniewicz proved the same for the field of algebraic numbers. I will give a minimal introduction to global fields and heights, keeping the presentation as close as possible to valuation theory, and to continuous logic, then explain what existential closedness means and how it relates to existing problems in Diophantine geometry. The talks are very loosely based on a small subsets of the 'foundations' paper by Ben Yaacov, Destic, Hrushovski, Szachniewicz and various other notes.

This is a short series of talks on the extended manuscript `Oligomorphic groups and tensor categories’ by Nate Harman and Andrew Snowden 2204.04526. Motivated by representation theory of finite groups and Deligne interpolation, the authors construct concrete pre-Tannakian categories. The methods go via, for certain omega-categorical structures M, constructing measures on Aut(M). I will attempt to give an overview of the work.

This is a short series of talks on the extended manuscript `Oligomorphic groups and tensor categories’ by Nate Harman and Andrew Snowden 2204.04526. Motivated by representation theory of finite groups and Deligne interpolation, the authors construct concrete pre-Tannakian categories. The methods go via, for certain omega-categorical structures M, constructing measures on Aut(M). I will attempt to give an overview of the work.

This is a short series of talks on the extended manuscript `Oligomorphic groups and tensor categories’ by Nate Harman and Andrew Snowden 2204.04526. Motivated by representation theory of finite groups and Deligne interpolation, the authors construct concrete pre-Tannakian categories. The methods go via, for certain omega-categorical structures M, constructing measures on Aut(M). I will attempt to give an overview of the work.

In the context of a 3-dimensional real analytic vector field at a singular point, Cano, Moussu and Sanz introduced and studied the notion of integral pencils of trajectories at that point in order to obtain informations on the possible dynamical behaviours. We extend this approach on the formal side, taking advantage of the computability of transseries (in particular they are grid-based in the sense of Ecalle - van der Hoeven) when solving differential equations.

More precisely, for a real formal planar vector field at 0, we introduce a notion of transserial trajectories and provide an explicit description of all the possible transserial pencils. This is meant to be a first step toward the same sort of description in dimension 3.

As a motivation, we expect these transserial trajectories to reflect tameness properties of actual solutions, in a way similar to that of differentially algebraic transseries for germs in some Hardy fields: cf the recent results of Aschenbrenner-van den Dries-van der Hoeven.

Joint work in progress with Daniel Panazzolo and Fernando Sanz.

Abstract: In this talk I will discuss homotopy groups definable in the o-minimal setting. After giving a brief overview of the classical theory of homotopy groups, I will talk about previous results which have been proven in the o-minimal field case, before discussing my own work in the o-minimal linear case.