Algèbre et géométrie

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 […]

Let G be a connected locally compact group with a left Haar measure μ, and let A,B ⊆ G be nonempty and compact. Assume further that G is unimodular, i.e., μ is also the right Haar measure; this holds, e.g., when G is compact, a nilpotent Lie group, or a semisimple Lie group. In 1964, Kemperman showed that μ(AB) ≥ min {μ(A)+μ(B), μ(G)} . The Kemperman inverse problem (proposed by Griesmer, Kemperman, and Tao) asks when the equality happens or nearly happens. I will discuss the recent solution of this […]

This talk describes the connection between /strong minimality/ of the differential equation satisfied by an complex analytic function and the real and imaginary parts of the function being /Pfaffian/. The talk will not assume the audience knows these notions previously, and will attempt to motivate why each of them are important notions in various areas. The connection we give, combined with a theorem of Freitag and Scanlon (2017) provides the answer to a question of Binyamini and Novikov (2017). We also answer a question of Bianconi (2016). We give what […]