Summer School on Motivic Integration


Monday Tuesday Wednesday Thursday Friday

Franziska Jahnke

Johannes Nicaise

Johannes Nicaise

Johannes Nicaise

Registration and Coffee
10:30-11:00 Questions and Coffee Questions and Coffee Questions and Coffee Questions and Coffee

Franziska Jahnke

Wim Veys

Franziska Jahnke

Wim Veys

Tom Scanlon








Wim Veys

Raf Cluckers

Arthur Forey

Nero Budur

Julien Sebag

15:00-15:15 Questions Questions Free Afternoon Questions

Enrica Mazzon

Itay Glazer

Juan Viu-Sos

16:15-16:45 Questions and Coffee Questions and Coffee Questions and Coffee

David Bourqui

Ana Reguera

Dimitri Wyss

19:00-... Conference

Abstracts of Mini-courses

Franziska Jahnke:

Title: Introduction to (the model theory of) valued fields

Abstract: The aim of these lectures is to provide an introduction to valued fields and semi-algebraic sets, with a particular view towards model-theoretic methods.

Here are notes of the mini-course.

Tent-Ziegler: A Course in Model Theory
van den Dries: Lectures on the Model Theory of Valued Fields
Jahnke: An Introduction to Valued Fields. In: Lectures in Model Theory; editors: Jahnke, Palacin, Tent
Hils: Model Theory of Valued Fields. In: Lectures in Model Theory; editors: Jahnke, Palacin, Tent
Prestel-Roquette: Formally \(p\)-adic fields
Hils-Loeser: A first journey through logic
Marker: Model Theory: An Introduction

Wim Veys:

Title: \(p\)-adic Igusa zeta functions

Abstract: We provide an elementary introduction to \(p\)-adic Igusa zeta functions; in particular we establish their link with polynomial congruences, and we show how geometrical techniques are helpful in their study.

Here are notes of the mini-course (with more technical details than in the talks).

Johannes Nicaise:

Title: Hrushovski-Kazhdan motivic integration and applications

Abstract: The theory of motivic integration of Hrushovski-Kazhdan attaches a geometric notion of volume to any semi-algebraic set over the field of Puiseux series. I will explain this construction from the perspective of algebraic geometry, the connection with tropical geometry, and a few applications in singularity theory and birational geometry.

Here is a survey article whose first part roughly corresponds to the minicoiurse.

Abstracts of Research talks

David Bourqui:

Title: Degenerations of families of arcs and torus actions

Abstract: Arcs on an algebraic variety are naturally organized into families according to the valuation they define through their contact order along regular functions. To understand how these families degenerate in general is a natural yet difficult question, which has strong connections with the famous Nash problem. I will report on recent progress obtained in the case of varieties equipped with a torus action of complexity one (joint work with K. Langlois and H. Mourtada)

Nero Budur:

Title: Contact loci of arcs

Abstract: Contact loci are sets of arcs on a smooth variety with prescribed contact order along a fixed hypersurface. They form the building blocks of motivic integration. Motivic zeta functions are generating series for classes of contact loci in appropriate Grothendieck groups of varieties. In this talk we give an overview of recent results on some parts of the topology of contact loci, namely irreducible components and cohomology, which cannot be seen from the Grothendieck groups alone.

Raf Cluckers:

Title: A partial overview of motivic and uniform \(p\)-adic integration

Abstract: I will give a partial overview on the theory of motivic integration with a focus on uniform \(p\)-adic integration. I will also describe some open questions about descent (that is, when passing from a larger \(p\)-adic field to a \(p\)-adic subfield) and about motivic Mellin transforms.

Arthur Forey:

Title: Motivic integration with pseudo-finite residue field and motives

Abstract: Cluckers-Loeser theory of motivic constructible functions can be specialized to a pseudo-finite residue field. Combining this with a construction of Denef-Loeser, this can be used to construct a ring of constructible functions with values in the Grothendieck group of motives which is stable by integration, satisfies a Fubini theorem, and can be specialized to local fields. Such a ring is a central ingredient towards proving a motivic version of the fundamental lemma. This is joint work in progress with François Loeser and Dimitri Wyss.

Itay Glazer:

Title: Singularities of polynomial maps through integrability of pushforward measures

Abstract: Let \(f:X \rightarrow Y\) be a polynomial map between smooth varieties, and let \(\mu\) be a smooth, compactly supported measure on \(X(F)\), where \(F\) is a local field. An interesting phenomenon is that bad singularities of \(f\) manifest themselves in poor analytic behavior of the pushforward \(f_*(\mu)\) of \(\mu\) by \(f\). For example, when \(Y\) is an affine line, Igusa’s theory on exponential sums implies that the Fourier coefficients of \(f_*(\mu)\) have slow decay if \(f\) is very singular.

Instead of working with Fourier invariants, we introduce a singularity invariant \(\epsilon(f)\), which quantifies the integrability of pushforwards by \(f\), of smooth measures on \(X(F)\). When calculating \(\epsilon(f)\), it is enough to consider collections of measures \(\mu\) which are constructible, in the sense of Cluckers-Loeser. Therefore, motivic integration is a powerful tool when studying such problems. I will discuss some properties and estimates of \(\epsilon(f)\) and related results in motivic integration.

An advantage of \(\epsilon(f)\) over Fourier type invariants, is that it is both meaningful when \(Y=G\) is a non-commutative algebraic group and compatible with the group operation. This is important for group-theoretic applications, such as the study of random walks on compact \(p\)-adic and compact Lie groups. If time permits, I will mention some of these applications and some open questions.

Based on joint works with Yotam Hendel and Sasha Sodin, and with Yotam Hendel.

Enrica Mazzon:

Title: Non-archimedean SYZ fibrations for Calabi-Yau hypersurfaces

Abstract: The SYZ conjecture is a conjectural geometric explanation of mirror symmetry. Based on this, Kontsevich and Soibelman proposed a non-archimedean approach, which led to the construction of non-archimedean SYZ fibrations by Nicaise-Xu-Yu.

In this talk, I will focus on families of Calabi-Yau hypersurfaces in \(\mathbb{P}^n\). I will construct new types of non-archimedean retractions and solve a non-archimedean conjecture proposed by Li, which is the missing step to prove that classical SYZ fibrations exist on a large open region of CY hypersurfaces in \(\mathbb{P}^n\).

This is based on work with Léonard Pille-Schneider, and a work in progress with Jakob Hultgren, Mattias Jonsson and Nick McCleerey.

Ana Reguera:

Title: Small irreducible components of arc spaces in positive characteristic

Abstract: (Joint work with A. Benito and O. Piltant) In 1968, J. Nash initiated the study of the space of arcs \(X_\infty\) of a (singular) algebraic variety \(X\) over a field of characteristic zero, with the purpose of understanding the structure of the various resolutions of singularities of \(X\). His work was done shortly after Hironaka’s proof of Resolution of Singularities in characteristic zero. Nash proved, using Resolution of Singularities that the space of arcs \(X_\infty^{\operatorname{Sing}}\) centered in the singular locus of \(X\) has a finite number of irreducible components.

This Nash program extends, with some important differences, to perfect ground fields \(k\) of characteristic \(p > 0\). The first difference is that Resolution of Singularities is still an open problem if \(\operatorname{char} k = p > 0\) and \(\operatorname{dim} X \ge 4\). Another difference is that, in contrast with characteristic zero, \((\operatorname{Sing} X)_\infty\) may contain some of the irreducible components of \(X_\infty^{\operatorname{Sing}}\). Understanding these “small” components is the main purpose of our article [1].

In this talk, we will propose some questions which would have an affirmative answer if a resolution of singularities existed :
Q1 : Has \(X_\infty^{\operatorname{Sing}}\) a finite number of irreducible components ?
Q2 : Given a variety X, does there exist a proper and birational morphism \(Y \to X\) such that \(Y_\infty\) is irreducible ?
We will give partial answers and explain the status of these problems.

[1] A. Benito, O. Piltant, A.J. Reguera, Small irreducible components of arc spaces in positive characteristic, J. Pure Appl. Algebra 226 (2022)

Tom Scanlon:

Title: The perfectoid tilting correspondence as a bi-interpretation

Abstract: A perfectoid field of characteristic zero is a valued field \((K,v)\) for which the characteristic of the residue field \(Kv\) is \(p > 0\), \(\lim_{n \to \infty}\) \(v(p^n) = \infty\), \(K\) is complete, and \(\mathcal{O}_{K,v} / p \mathcal{O}_{K,v}\) is perfect in the sense that the map \(x \mapsto x^p\) is surjective. To such a field one may attach the tilt \(K^\flat\) which is a perfect, complete valued field of characteristic \(p\). The tilting functor is known to allow from strong transfer results between mixed and positive characteristic. For example, \(K\) and \(K^\flat\) have the same absolute Galois groups. Scholze famously extended the equivalence to certain categories of spaces over \(K\) and \(K^\flat\), respectively, and used the correspondence to solve a long-standing open problem by transferring it from positive to mixed characteristic using a certain approximation lemma.

We offer a model theoretic explanation of the tilting correspondence as a bi-interpretation between a perfectoid field and its tilt in either continuous logic or a multisorted first-order language. Equivalences of adic spaces then arise through homeomorphisms of type spaces and Scholze’s approximation lemma is an instance of a general result about definability in bi-interprertable structutes.

This is a report on joint work with Silvain Rideau-Kikuchi and Pierre Simon.

Julien Sebag: (cancelled)

Title: A functorial formula of Davison-Meinhardt

Abstract : In this talk, I will explain how the virtual motives associated with nearby cycles by motivic integration can actually be understood as piecewise avatars of non-virtual motives. As a consequence, I will explain how to obtain a (non-virtual) motivic formula which in particular provides, by specialization, a positive answer to the original conjectures of Behrend-Bryan-Szendrői and Davison-Meinhardt for virtual motives. (The main parts of this talk are based on joint works with Joseph Ayoub and Florian Ivorra.)

Juan Viu-Sos:

Title: Zeta functions, orbifold motivic measures and \(\mathbb{Q}\)-resolutions of singularities

Abstract: The Monodromy Conjecture is usually approached by computing embedded resolutions of \(f\): any exceptional divisor gives a "candidate pole" of the motivic or topological zeta function coming from the associated numerical data. However, most of them are usually not poles and disappear in the final expression.

In this talk, we introduce some recent techniques that we have developed for the study of these zeta functions for \(\mathbb{Q}\)-divisors in \(\mathbb{Q}\)-Gorenstein varieties, together with a change of variables formula depending on the relative canonical divisor.

By using toric partial resolutions, we reduce this study to the local case of normal crossing divisors where the ambient space is an abelian quotient singularity, obtaining models with less exceptional divisors and providing a closed formula for zeta functions.

Joint work with Edwin LEON-CARDENAL (UNIZAR), Jorge MARTIN-MORALES (UNIZAR) and Wim VEYS (KU Leuven).

Dimitri Wyss:

Title: Applications of the orbifold measure

Abstract: Motivated by work of Batyrev on the McKay correspondence, Denef and Loeser defined a motivic measure on algebraic varieties with quotient singularities, the orbifold measure. This construction has proven to be useful in many geometric situations and in my talk I’ll give an overview over some of these applications, in particular to moduli spaces of Higgs bundles. If time permits I will also talk about possible extensions of the orbifold measure to non-quotient singularities. This is joint work (in progress) with Arthur Forey, Michael Groechenig, François Loeser and Paul Zielger.