Floris Vermeulen

This is the homepage for the Hrushovski-Kazhdan motivic integration reading group for 2023. The main references we will follow are

Hrushovski-Kazhdan: Integration in valued fields. See here or arxiv. This is one of the main reference for the reading course, but we will mostly follow the surveys below.
Nicaise: Motivic integration, tropical geometry, and rationality of algebraic varieties. See here. We will begin the reading group by following this reference.
Kamensky: Introduction to model theory of valued fields. See here. For some ideas of the proofs we will follow these notes.
Halupczok: An introduction to motivic integration. See here. This is an introduction to uniform p-adic integration from a model theoretic perspective.

Here is an outline and overview of the reading group. Below is the schedule of the talks.

16/10/23: Introduction to the reading group, by Floris Vermeulen. We discussed the generalization of p-adic integration to uniform p-adic integration to motivic integration. Here are the notes for the talk.
23/10/23: Semi-algebraic geometry in algebraically closed valued fields, by Mathias Stout. We discussed semi-algebraic geometry in Puiseux series fields, and the tropicalization and specialization maps. Here are the notes for the talk.
30/10/23: Various graded Grothendieck rings, by Art Waeterschoot. We introduced the Grothendieck ring of semi-algebraic sets in the valued field, the Grothendieck ring of varieties over the residue field, and the Grothendieck ring of polytopes in the value group. We saw the main result relating these three objects, via tropicalization and specialization.
06/11/23: Grothendieck rings of some structures, by Pierre Touchard. We introduced the Grothendieck ring of a model-theoretic structure, computed some examples, and studied it in detail for the structure C in the ring language.
13/11/23: Applications of motivic integration to birational geometry, by Simen Moe. We discussed some applications of the Hrushovski-Kazhdan framework to birational geometry, in particular about rationality and stable rationality of degenerations.
20/11/23: An introduction to model theory, by Mathias Stout. We introduced some basic notions from model theory: languages, formulas, sentences, theories, models, quantifier elimination, ... Here are the notes for this week.
27/11/23: Geometry in V-minimal theories, by Floris Vermeulen. We discussed V-minimality, the tameness notion used by Hrushovski-Kazhdan to control definable sets. We also discussed h-minimality and the Jacobian property. Here are the notes for this week.
04/12/23: Cell decomposition, by Floris Vermeulen. We discussed some more results about V-minimal geometry, in particular about cell decomposition. Here the notes for this week.
11/12/23: Grothendieck rings and reducing to RF and VG, by Pierre Touchard. We discussed some more Grothendieck rings of categories, and how to reduce from RV to RF and VG.
18/12/23: Reducing to RF and VG, by Pierre Touchard, and integration and RV-blowups, by Mathias Stout. We first finished up the reduction to RF and VG from last week, and then discussed the computation of the kernel via RV-blowups. Here are notes for the second part of today.

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