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.
The Wilson conjecture asks whether any locally nilpotent omega-categorical $p$-groups are nilpotent. In this talk, I will present solutions to a few cases (e.g. 4-Engel 5-groups, or groups of exponent 4), which use methods at the intersection of group theory, model theory and computer algebra.
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 this talk, I discuss some recent advances in the interchange of ideas between logic and the classification of C*-algebras, which are a specific subclass of operator algebras. Operator algebras are certain collections of bounded operators on Hilbert spaces. They give a strong foundation for understanding quantum mechanics, as well as non-commutative flavors of geometry, topology, and probability theory. A huge theme within operator algebras (and indeed in all mathematics) is that of classification, which asks the broad question of how we can tell objects apart or conclude they are the same.
In this talk, I will discuss a game-theoretic variant of the unital C*-classification theorem: we show that there is a transfer of strategies between Ehrenfeucht-Fraïssé games on classifiable C*-algebras and their invariants. The proof techniques are interesting in their own right: they involve an abstract descriptive set theoretic classification result, as well as a first-order language for functors.
This is based on joint work with Michał Szachniewicz and Mira Tartarotti.
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.
The global Zarankiewicz's problem for hypergraphs asks for an upper bound on the number of edges of a hypergraph, whose edge relation is induced by a fixed hypergraph $E$ that has no sub-hypergraphs of a given size. Basit-Chernikov-Starchenko-Tao-Tran (2021) obtained linear Zarankiewicz bounds in the case of a semilinear $E$, namely $E$ definable in a linear o-minimal structure. We extend this theorem to a broader range of "linear-like" structures, in o-minimal, Presburger arithmetic and stability theoretic settings. Some of the methods involved include (a) a reduction of the problem to the case of arbitrary subgroups $E$ of powers of groups, and (b) an abstract version of Zarankiewicz's problem in the saturated setting.
Joint work with Aris Papadopoulos.
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.
Thatcher and Wright showed that a property of trees of bounded degree is MSO-definable if and only if it is recognizable by a tree-automaton. In this talk we explore the question of when MSO-definability of a property of graphs is equivalent to the existence of a tree automata which, given a suitable expression encoding the input graph, recognizes the property. In this talk, I will survey the state of the art of the "definability equals recognizability" problem. For proving "definability equals recognizability" results the key step is to transduce a suitable tree-like decomposition of the input graph. I will present a new MSO-transduction which forms the core for transducing a particular type of graph decompositions.
