# Summer School on Motivic Integration

12.09.2022-16.09.2022

Conference of the GRK2240, funded by the DFG in Düsseldorf, Germany

## About Motivic Integration

Motivic integration is an abstractly defined integration theory, inspired by usual (Lebesgue) integration in the \(p\)-adic numbers, but which exists over fields which have no reasonable Lebesgue measure, like the field \(\mathbb{C}((t))\) of formal Laurant series. On the one hand, it satisfies various properties known from usual integration, like a change of variable formula. On the other hand, the value of a motivic measure is an element of some kind of Grothendieck ring of varieties, which contains a lot of geometric information about the set that has been measured. This combination of properties makes it an interesting tool in algebraic geometry, e.g. for the study of additive invariants or of singularities. Since its invention by Kontsevich in 1995, many such applications have been found, and the formalism has also been improved and refined several times, notably by Denef-Loeser, Cluckers-Loeser and Hrushovski-Kazhdan.

Modern versions of motivic integrations can also be used to integrate in \(\mathbb{Q}_p\) “for all \(p\) simultaneously”. This turns it into a useful tool to make results relying on \(p\)-adic integration uniform in \(p\). Concretely, Hales suggested to apply this to (complex) representation theory of \(p\)-adic groups. While one is still far from making this entire representation theory uniform in \(p\), this approach already had several impressive consequences, in particular the transfer of the Fundamental Lemma of the Langlands Program (which was an open conjecture for a long time) from positive to mixed characteristic.

## About the Summer School

The goal of this summer school is to introduce the participants to motivic integration and some of its applications. It is mainly aimed at PhD students and Postdocs who do work in an area around algebra and/or geometry, but who have not yet had any prior contact to this motivic integration. There will be three minicoures, which will explain the necessary tools and background material, give down-to-earth motivation and examples, and which will end with some interesting applications. In addition, there will be around ten one-hour talks about current research related to motivic integration. The summer school is part of the program of the DGF-funded research training group.

**GRK2240**: **Algebro-geometric Methods in Algebra, Arithmetic and Topology.**

It will take place at the Heinrich Heine Universität Düsseldorf (see Arrangements for more details about the venue).

Support for travel expenses and accommodation is available for a limited number of PhD students and early career researchers (see Support) for more details).

## Speakers

### Mini-courses

- Franziska Jahnke: Introduction to model-theoretic background
- Wim Veys: \(p\)-adic Igusa zeta functions
- Johannes Nicaise: Hrushovski-Kazhdan motivic integration and applications

## Invited Speakers

- Nero Budur
- Raf Cluckers
- Arthur Forey
- Itay Glazer
- Enrica Mazzon
- Ana Reguera
- Tom Scanlon
- Julien Sebag
- Juan Viu-Sos
- Dimitri Wyss

## Organizers

- Immanuel Halupczok
- Thor Wittich
- Pablo Cubides Kovacsics
- Florian Felix
- Hamed Khalilian
- Saba Aliyari
- David Bradly-Williams
- Zeynep Kisakürek