Algèbre et géométrie

Joint work with Samaria MontenegroThe striking resemblance between the behaviour of pseudo-algebraically closed, pseudo real closed and pseudo p-adically fields has lead to numerous attempts at describing their properties in a unified manner. In this talk I will present another of these attempts: the class of pseudo-T-closed fields, where T is an enriched theory of fields. These fields verify a “local-global” principle with respect to models of T for the existence of points on varieties. Although it very much resembles previous such attempts, our approach is more model theoretic in […]

An open problem in theoretical computer science asks to characterize tameness for hereditary classes of finite structures. The notion of bounded twin-width was proposed and studied recently by Bonnet, Geniet, Kim, Thommasé and Watrignant. Classes of graphs of bounded twin-width have many desirable properties. In particular, they are monadically NIP (remain NIP after naming arbitrary unary predicates). In joint work with Szymon Torunczyk we show the converse for classes of ordered graphs. We then obtain a very clear dichotomy between tame (slow growth, monadically NIP, algorithmically simple ...) and wild […]

I will discuss recent work in collaboration with Edward Bierstone on transformation of a mapping to monomial form (with respect to local coordinates) by simple modifications of the source and target. Our techniques apply in a uniform way to the algebraic and analytic categories, as well as to classes of infinitely differentiable real functions that are quasianalytic or definable in an o-minimal structure. Our results in the real cases are best possible. The talk will focus on real phenomena and on an application to quantifier elimination of certain o-minimal polynomially […]

Shelah's conjecture predicts that any infinite NIP field iseither separably closed, real closed or admits a non-trivial henselianvaluation. Recently, Johnson proved that Shelah's conjecture holds forfields of finite dp-rank, also known as dp-finite fields. The aim of these two talks is to give an introduction to dp-rank in some algebraic structures and an overview of Johnson's work.In the first talk, we define dp-rank (which is a notion of rank in NIP theories) and give examples of dp-finite structures. In particular, we discuss the dp-rank of ordered abelian groups and use […]

Hensel minimality is a new axiomatic framework for doing tame geometry in non-Archimedean fields, aimed to mimic o-minimality. It is designed to be broadly applicable while having strong consequences. We will give a general overview of the theory of Hensel minimality. Afterwards, we discuss arithmetic applications to counting rational points on definable sets in valued fields. This is partially joint work with R. Cluckers, I. Halupczok and S. Rideau-Kikuchi, and partially with V. Cantoral-Farfan and K. Huu Nguyen.

We discuss new decidability and undecidability results for mixed characteristic henselian fields, whose proof goes via reduction to positive characteristic. The reduction uses extensively the theory of perfectoid fields and also the earlier Krasner-Kazhdan-Deligne principle. Our main results will be: (1) A relative decidability theorem for perfectoid fields. Using this, we obtain decidability of certain tame fields of mixed characteristic. (2) An undecidability result for the asymptotic theory of all finite extensions of ℚ_p (fixed p) with cross-section. We will also discuss a tentative step towards understanding the underlying model […]