57:43
published on April 30, 2026
Order-topological characterization of groups and fields with automatic $\emptyset$-definability
By Anna De Mase
Kontsevich-Zagier in valued fields
Appears in collection : Arithmetic of fields: model theory, valuations, and geometry / L'arithmétiques des corps : théories des modèles, valuations, et geometrie
A period is a real number which is the measure of a semialgebraic set defined over the rationals. The ring of periods was formally introduced by Kontsevich and Zagier in 2001, and remains a rather mysterious object. Kontsevich and Zagier conjectured that every equality between periods can already be deduced from simple integration rules such as additivity, change of variables, and the fundamental theorem of calculus. While this conjecture remains completely open, a p-adic analogue was proven by Cluckers-Halupczok. In joint work with Mathias Stout, we use motivic integration and h-minimality to prove a general version of the Kontsevich-Zagier conjecture in henselian valued fields. In this talk I will introduce periods in the reals and discuss the p-adic analogue due to Cluckers-Halupczok. Afterwards I will introduce motivic integration and discuss the period conjecture in this generality. Time permitting, I will give some ideas of the proof.
