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.
Abstract: In this talk, we introduce oriented abelian groups and present some tameness properties of these structures and their pairs. We show that, in certain theories of oriented abelian groups, the VC-density of formulas is bounded by the size of parameter variable. We further show that, for a specific pair structure, this bound becomes twice the size of parameter variable, and that these bounds are optimal. This is joint work with Ebru Nayir.
Abstract: After a brief historical survey I will describe how some axioms of the theory of Hardy fields can be generalized so as to cover the case of differential fields of germs at a non-principal cut in an o-minimal ordered field. I will sketch how this can be used to prove that Tressl's signature alternative holds in a large class of exponential o-minimal theories.
NOTES: unusual room, 2 hours seminar.
Abstract: F-V Kuhlmann's theory of tame (and separably tame) valued fields is one of the most general settings in which we have AKE principles. Such principles come in many flavours; in particular, we may constrain our attention to certain "subfragments" of the languages of valued fields/rings/ordered abelian groups. I will explain some of the underlying algebra, and show some recent work on such principles in expansions by sections of the residue map. This will touch on (separate) projects with Boissonneau and Fehm.
Abstract: Arising from extremal combinatorics, the global Zarankiewicz problem seeks an upper bound on the number of edges of a finite $r$-hypergraph where the edge relation is induced by some fixed hypergraph that does not contain the compete $r$-hypergraph $K_{k,\dots,k}$ as a subgraph for some $k$. Special cases of this problem, where the edge relation comes from a definable set in a particular structure, have been of interest to model theorists. This talk will give an introduction to the problem as well as some recent results.
NOTES: Unusual location.
Abstract: A pregeometric theory of fields is a theory T expanding ACF such that the model-theoretic algebraic closure acl satisfies the exchange property. Several important theories of fields satisfy this property, like ACF, RCF, ACVF, etc. Certain pregeometric fields have the additional property that we can expand the language with a generic derivation. We will discuss the structural implications of pregeometricness, and what we mean by a 'generic derivation'. Joint work with Antongiulio Fornasiero and Elliot Kaplan.
NOTES: Note the unusual Location.
Abstract: In this talk, I will introduce residually dominated types and then residually dominated groups in pure Henselian valued fields of equicharacteristic zero. These notions are analogous to stably dominated types and groups, which are central to the model theory of algebraically closed valued fields. I will first give an overview of the literature and then present the main results from our joint preprint with Paul Wang.
