Arithmetic of fields: model theory, valuations, and geometry / L'arithmétiques des corps : théories des modèles, valuations, et geometrie

Collection Arithmetic of fields: model theory, valuations, and geometry / L'arithmétiques des corps : théories des modèles, valuations, et geometrie

Organizer(s) Anscombe, Sylvy ; Bagayoko, Vincent ; Forey, Arthur ; Jahnke, Franziska

Date(s) 06/04/2026 - 10/04/2026

linked URL https://conferences.cirm-math.fr/3500.html

00:00:00 / 00:00:00

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Kontsevich-Zagier in valued fields

By Floris Vermeulen

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.

Information about the video

Date of recording 07/04/2026
Date of publication 30/04/2026
Institution CIRM
Licence CC BY NC ND
Language English
Audience Researchers, Graduate Students
Director(s) Luca Recanzone
Format MP4

Citation data

DOI 10.24350/CIRM.V.20467403
Cite this video Vermeulen, Floris (07/04/2026). Kontsevich-Zagier in valued fields. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.20467403
URL https://dx.doi.org/10.24350/CIRM.V.20467403

Domain(s)

Logic

Bibliography

KONTSEVICH, Maxim et ZAGIER, Don. Periods. In : Mathematics unlimited—2001 and beyond. Berlin, Heidelberg : Springer Berlin Heidelberg, 2001. p. 771-808. - https://doi.org/10.1007/978-3-642-56478-9_39
CLUCKERS, Raf et HALUPCZOK, Immanuel. A p-adic variant of Kontsevich–Zagier integral operation rules and of Hrushovski–Kazhdan style motivic integration. Journal für die reine und angewandte Mathematik (Crelles Journal), 2021, vol. 2021, no 779, p. 105-121. - https://doi.org/10.1515/crelle-2021-0042
CLUCKERS, Raf et LOESER, François. Constructible motivic functions and motivic integration. Inventiones mathematicae, 2008, vol. 173, no 1, p. 23-121. - https://doi.org/10.1007/s00222-008-0114-1
HRUSHOVSKI, Ehud et KAZHDAN, David. Integration in valued fields. In : Algebraic Geometry and Number Theory: In Honor of Vladimir Drinfeld's 50th Birthday. Boston, MA : Birkhäuser Boston, 2006. p. 261-405. - https://doi.org/10.1007/978-0-8176-4532-8_4

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