Motivic Integration and its Interactions with Model Theory and Non-Archimedean Geometry

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Motivic Integration and its Interactions with Model Theory and Non-Archimedean Geometry

 12 - 17 May 2008
 ICMS, 14, India Street, Edinburgh

Organiser

Name
Institution
Cluckers, RafUniversité Lille 1/University of Leuven
Macintyre, AngusQueen Mary University of London
Nicaise, JohannesUniversité Lille 1
Sebag, JulienUniversité Bordeaux 1

Short Report

 

The objective of this workshop was to realize progress on Motivic Integration by gathering some of the leading specialists active in and around this domain and neighbouring domains such as Rigid Geometry and Model Theory. Since its creation by M. Kontsevich in 1995, Motivic Integration has been a rapidly developing subject connected to Algebraic Geometry, Singularity Theory, Model Theory and Number Theory. Because of the many challenges related to the development of the theory and its applications and the rapid evolution of the subject, this meeting should yield substantial new developments and open up many new challenges.

The talks were divided into two parts:

  • Short courses. Their aim was to give an introduction to the following subjects: Non-Archimedean Geometry (Bosch (2×1h), Berkovich (2×1h), Huber (2×1h)), Model Theory (Macintyre (1h), Chatzidakis (1h), Cluckers (1h)), Motivic Integration (Denef (1h), Nicaise (2×1h), Scanlon (1h), Loeser (1h));
  • Specialized talks (50 minutes), with a special emphasis on applications in singularity theory (Merle, Temkin, Veys) and the Langlands program (Fargues, Hales, Kaletha).

The short courses were of particular importance because they offered a survey of some recent developments in domains in full expansion. In particular, it was to our knowledge the first time that all major approaches to non-Archimedean geometry (rigid varieties, formal schemes, analytic spaces, adic spaces, Zariski-Riemann spaces) were presented and explained at the same conference; the short course on motivic integration covered the various formalisms developed since Kontsevich’ original work (geometric motivic integration on algebraic varieties, formal schemes and non-Archimedean spaces; model-theoretic framework of Cluckers-Loeser and Hrushovski-Kazhdan). Since these developments are not cumulative and each of the theories presents its own merits in particular applications, it was quite fruitful to introduce them all and to allow for a comparison of their specific characteristics.

 Download the pdf file of the full report

Arrangements

Participation
Participation is by invitation only. The workshop will begin with registration at 09.00 on Monday 12 May and finish on the morning of Saturday 17 May 2008.

UK Visas
If you are travelling from overseas you may require an entry visa. A European visa does not guarantee entry to the UK. Please use this link to the UK Visas site to find out if you need a visa and if so how to apply for one. If you do require a visa, ICMS can provide a signed invitation letter.

Venue
The workshop will take place at the head-quarters of ICMS, 14 India Street, Edinburgh. This house is the birthplace of James Clerk Maxwell and is situated in the historic New Town of Edinburgh, near the city centre.

The ICMS travel pages contain advice on how to travel to Edinburgh. For local information the finding ICMS page shows the location of ICMS and contains useful maps of the city centre.

The seminar room at ICMS has whiteboards, 2 overhead projectors, a data projector and laptop.

Wireless access is available throughout the ICMS building. There are also 7 public PCs which may be used at any time for internet access and to check email.

Accommodation
ICMS will arrange single en-suite rooms in local guest houses for those who require it. Accommodation is typically about 15 to 30 minutes walk from ICMS. Participants are also free to make their own arrangements and may claim back the cost, with receipts, up to a maximum of £45.00 per night bed and breakfast. A list of Edinburgh accommodation of various sorts and prices is available here . Sections 1-3 are particularly relevant.

Meals and Refreshments
A sandwich lunch will be provided on the first day of the workshop, Monday 12 May. For the remainder of the days, participants are free to go out for lunch and explore the many cafes, restaurants, sandwich shops and bars in the surrounding area. On arrival we will provide you with a ‘welcome’ pack which will contain information about eating places nearby.

Morning and afternoon refreshments will be provided throughout the workshop.

There will be an informal wine reception after the close of lectures on Monday 12 May.

On Tuesday 13 May you are invited to attend an informal supper at Nargile Turkish Restaurant. The workshop dinner will take place on the evening of Thursday 15 May.

Registration
Registration will take place between 10.30 and 12.00 on Monday 12 May. The talks will start at 12.00.

Financial Arrangements
Unless otherwise specified in your invitation letter, the workshop grant will cover the cost of your bed and breakfast accommodation, tea/coffee throughout the workshop, lunch on the first day, the wine reception, the informal supper on Tuesday and the Workshop Dinner on Thursday evening.

If we have agreed to pay some of your travel costs, you will be informed by email. Reimbursement will take place after the workshop. At Registration you will be given an expenses claim form and this should be submitted to ICMS, with receipts. Please note that we cannot reimburse any item without a receipt.

Under the terms of our EPSRC funding we are required to charge a 30.00 GBP registration fee to cover costs not admissible under the grant. The fee will be payable on arrival at the workshop payment may be by cash, sterling cheque or credit/debit card. If you anticipate any difficulty covering the fee, please let me know.

 

Programme

Mini courses:

A. Non-archimedean Geometry (6h): Bosch (2h), Berkovich (2h), Huber (2h)

B. Model Theory (3h): Macintyre, Chatzidakis, Cluckers

C. Motivic integration (5h): Denef, Nicaise (2h), Scanlon, Loeser

 

Monday 12 May

09.00 - 10.00

Registration

10.00 - 11.00

Siegfried Bosch (Westfälische Wilhelms - Universität Münster)
Introduction to formal and rigid geometry (A 1) Link to pdf file of slides

11.00 - 11.30

Coffee/Tea

11.30 - 12.30

Siegfried Bosch (Westfälische Wilhelms - Universität Münster)
Introduction to formal and rigid geometry (A 2) Link to pdf file of slides

12.30 - 14.30

Lunch

14.30 - 15.30

Zoé Chatzidakis (CNRS - Université Paris 7)
The basic model theory of valued fields (B 1) Link to pdf file of slides

15.40 - 16.40

Angus Macintyre (University of London)
Quantifier elimination in the p-adics and their algebraic closure, and issues of uniformity (B 2)

16.40 - 17.10

Coffee/Tea

17.10 - 18.00

Françoise Delon (CNRS - Université Paris 7)
C-minimal structures

18.00 - 19.00

Wine reception at ICMS, 14 India Street

 

Tuesday 13 May

09.00 - 09.50

Emmanuel Kowalski (ETH Zürich)
Algebraic exponential sums and their uses in analytic number theory

10.00 - 11.00

Vladimir Berkovich (Weizmann Institute of Science)
Non-Archimedean analytic spaces (A 3)

11.00 - 11.30

Coffee/Tea

11.30 - 12.30

Vladimir Berkovich (Weizmann Institute of Science)
Non-Archimedean analytic spaces (A 4)

12.30 - 14.30

Lunch

14.30 - 15.30

Raf Cluckers (École Normale Supérieure, Paris)
Integration through cell decomposition and b-minimality (B 3)

15.40 - 16.30

Immanuel Halupczok (École Normale Supérieure, Paris)
Trees of definable sets in Z_p Link to pdf file of related slides - Link to related preprint

16.30 - 17.00

Coffee/Tea

17.00 - 17.50

Yan Soibelman (Kansas State University)
Motivic Donaldson-Thomas invariants

19.00

Informal group supper at Nargile Turkish Restaurant, 73 Hanover Street

 

Wednesday 14 May

09.00 - 10.00

Jan Denef (University of Leuven)
Course on motivic integration I (C 1)

10.10 - 11.10

Roland Huber (Bergische Universität Wuppertal)
Analytic adic spaces (A 5)

11.10 - 11.30

Coffee/Tea

11.30 - 12.30

Roland Huber (Bergische Universität Wuppertal)
Analytic adic spaces (A 6)

12.30

Lunch
Free afternoon

 

Thursday 15 May

09.00 - 09.50

Fumiharu Kato (Kyoto University)
Topological rings in rigid geometry Link to pdf file of slides

10.00 - 11.00

Johannes Nicaise (Université Lille 1)
Non-archimedean geometry and complex singularities (C 2) Link to pdf file of slides

11.00 - 11.30

Coffee/Tea

11.30 - 12.30

Johannes Nicaise (Université Lille 1)
Non-archimedean geometry and complex singularities (C 3) Link to pdf file of slides

12.30 - 14.30

Lunch

14.30 - 15.30

Thomas Scanlon (University of California, Berkeley)
Integration in valued fields (after Hrushovski and Kazhdan) (C 4)

15.40 - 16.30

Willem Veys (Katholieke Universiteit Leuven)
Smallest poles of motivic zeta functions Link to related paper

16.30 - 17.00

Coffee/Tea

17.00 – 18:00

Problem Session

19.00

Workshop Dinner at First Coast Restaurant, 99-101 Dalry Road, Edinburgh

 

Friday 16 May

09.00 - 10.00

François Loeser (École Normale Supérieure)
Motivic integration via cell decomposition and applications (C 5)

10.10 - 11.00

Michael Temkin (University of Pennsylvania)
Desingularization of quasi-excellent schemes over Q Link to pdf file of slides

11.00 - 11.30

Coffee/Tea

11.30 - 12.20

Antoine Chambert-Loir (Université Rennes 1)
Igusa zeta functions in Diophantine geometry Link to pdf file of slides

12.30 - 14.30

Lunch

14.30 - 15.20

Tasho Statev Kaletha (University of Chicago)
The stabilization of the trace formula and the fundamental lemma

15.30 - 16.20

Laurent Fargues (CNRS - Université Paris-Sud)
On some results about the étale cohomology of rigid analytic spaces linked to the local Langlands correspondence

16.20 - 17.00

Coffee/Tea

17.00 - 17.50

Thomas Hales (University of Pittsburgh)
Transfer principle for the fundamental lemma Link to pdf file of slides

 

Saturday 17 May

09.00 - 09.50

Antoine Ducros (Université de Nice – Sophia Antipolis)
The image of a flat map in non-Archimedean geometry: description via the quantifyers elimination for algebraically closed valued fields

10.00 - 10.50

Michel Merle (Université de Nice – Sophia Antipolis)
Nearby cycles, convolution and composition with a two variable function Link to pdf file of slides

10.50 - 11.30

Coffee/Tea

11.30 - 12.20

Karl Rökaeus (Stockholm University)
A version of geometric motivic integration that specializes to p-adic integration Link to pdf file of related slides

 

Presentations

Berkovich, Vladimir
Non-Archimedean analytic spaces
View Abstract
In this series of two talks I intend to explain the notion of non-Archimedean analytic spaces, their relation to rigid analytic spaces, basic ideas of etale cohomology for them, and some of their applications.
Bosch, Siegfried
Introduction to formal and rigid geometry
View Abstract
We will give an introduction to the theory of classical rigid spaces, as invented by Tate. In addition, we will cover the approach by Raynaud via formal schemes.
Chambert-Loir, Antoine
Igusa zeta functions in Diophantine geometry
View Abstract
Recent work on Manin's conjecture concerning the number of points of bounded height in algebraic varieties and its asymptotic behaviour have shown the importance of a geometric analogue, namely the asymptotic behaviour of the volume of adelic subspaces defined by some kind of height inequalities. This behaviour has been mostly studied in the framework of algebraic groups. We explain how the analytic behaviour of analogues of Igusa zeta functions, together with Tauberian theorems, allow to recover these properties, proving them in a very general geometric situation. This is joint work with Yuri Tschinkel.
Chatzidakis, Zoé
The basic model theory of valued fields
View Abstract
I present for nonlogicians the basic concepts of the model theory of valued fields, including several of the formalisms one may use.
Cluckers, Raf
Integration through cell decomposition and b-minimality
View Abstract
I will present an introduction to cell decomposition and what it can do for p-adic and motivic integrals, and what it can not yet do for such integrals. Next I will present an axiomatic framework, which is joint work with F. Loeser, and which is built up around cell decomposition. I will comment on how it is directed towards integration and sketch some (new) open problems.
Delon, Françoise
C-minimal structures
View Abstract
A C-relation is a ternary relation first-order interpreting a tree which is a meet-semi-lattice, the domain of the C-relation being then a covering set of branches with no isolated branch. A set M, equipped with a C-relation and possibly an additional structure, is called C-minimal if any definable subset of M is definable without quantifiers in the pure language of C, and if the same holds in any elementarily equivalent structure. A C-minimal structure is algebraically bounded in the sense that finite uniformly definable sets have a bounded size. On the other hand the algebraic closure needs not satisfy the exchange principle. C-minimal structures with exchange are “geometric” in the sense of Zilber. In this context, we may ask the question of the trichotomy: Is it possible to define a group in modular non trivial structures? To define a field in non locally modular structures? To classify some of these structures?
Denef, Jan
Course on motivic integration I
View Abstract
We will look at the prehistory of motivic integration, from finite counting questions raised by Borevich - Shafarevich, Serre - Oesterlé, to p-adic integrals and to (p-adic and topological) zeta-functions. We will indicate how questions on Betti numbers of Calabi - Yau varieties led to the first approach to motivic integration by Kontsevich and how it was implemented by Denef - Loeser.
Ducros, Antoine
The image of a flat map in non-Archimedean geometry: description via the quantifyers elimination for algebraically closed valued fields
View Abstract
By transposing his (and Gruson's) scheme-theoretic flattening techniques to formal schemes, Raynaud proved that the image of a flat map between affinoid spaces is a union of affinoid domains. Answering a question by Berkovich, we will give a new proof of this result which doesn't use any formal model. It is based upon direct application of Raynaud and Gruson's dévissages methods in the context of (Berkovich) analytic geometry, Temkin's local study of analytic spaces through Riemann-Zariski spaces, and quantifyers elimination for algebraically closed valued fields.
Fargues, Laurent
On some results about the étale cohomology of rigid analytic spaces linked to the local Langlands correspondence
View Abstract
I will speak about three results on the étale cohomology of rigid analytic spaces linked to the geometric realization of the local Langlands correspondence in the cohomology of the Lubin-Tate and Drinfeld towers. The first one is the invariance under formal completion of the monodromy filtration of the l-adic vanishing cycles sheaves. The second one concerns an equivariant Poincaré duality theorem for the compactly supported l-adic cohomology of rigid analytic spaces, where duality has here to be understood in the sens of the Zelevinsky involution in the category of smooth representations of reductive p-adic groups. The last one deals with the comparison of the cohomology of the Lubin-Tate and Drinfeld tower with application a geometric form of the Jacquet-Langlands correspondence: a topos equivalence between some étale equivariant sheaves on the Drinfeld spaces and some étale equivariants sheaves on a rigid analyitic Severi-Brauer variety.
Hales, Thomas
Transfer principle for the fundamental lemma
View Abstract
This talk will explain how the identities of the fundamental lemma (arising in the theory of trace formula for automorphic representations) fall within the scope of the transfer principle, a general result that allows to transfer theorems about identities of p-adic integrals from one collection of fields to others. In particular, once the fundamental lemma has been established for one collection of fields (for example, fields of positive characteristic), it is also valid for others (fields of characteristic zero).
Halupczok, Immanuel
Trees of definable sets in Z_p
View Abstract
To a variety V defined over over the p-adic integers Z_p, one can naturally associate a tree: the nodes at depth k are the points with values in Z_p/(p^k Z_p), and the tree structure is given by the canonical maps Z_p/(p^(k+1) Z_p) -> Z_p/(p^k Z_p). A variant of this also permits to replace V by arbitrary definable subsets of Z_p^n.

There are old results on the number of nodes of these trees at each depth. A natural question (posed to me by Loeser) is whether these results can be strenghened to result about the structure of the trees. The goal of this talk is to present a conjecture which gives such a description. The conjecture is true for curves and for arbitrary definable subsets of Z_p^2.
Huber, Roland
Analytic adic spaces
View Abstract
We will explain the definition of analytic adic spaces. The topological spaces underlying affinoid analytic adic spaces are spaces of continuous valuations of topological rings. We will explain the relation of analytic adic spaces to classical rigid geometry and some aspects of the etale cohomology of analytic adic spaces.
Kaletha, Tasho Statev
The stabilization of the trace formula and the fundamental lemma
View Abstract
The trace formula is an important tool in the study of automorphic representations. Many applications of the trace formula, like the comparison of automorphic representations for inner forms, require that it be stabilized, and the process of stabilization inevitably leads to the need for establishing a certain identity of p-adic integrals, called the fundamental lemma, which, despite its name, had remained an open problem for multiple decades. This talk will be an overview of the stabilization process and the fundamental lemma.
Kato, Fumiharu
Topological rings in rigid geometry
View Abstract
I would like to explain our attempt to give an "umbrella notion" for topological rings in rigid geometry, which affords classical ones (type (V) and type (N)) and assists developing "absolute" notion of rigid spaces. Joint-work with Kazuhiro Fujiwara.
Kowalski, Emmanuel
Algebraic exponential sums and their uses in analytic number theory
View Abstract
The talk will be a fairly informal survey of some of the results of analytic number theory where exponential sums over finite fields or rings arise naturally, together with some of the problems and methods suggested by these. This will include, for instance, Kloosterman sums in the circle method, sums of Kloosterman sums, exponential sums over definable subsets of finite fields, and counting problems used in sieve methods.
Loeser, François
Motivic integration via cell decomposition and applications
View Abstract
In this talk we shall present our joint work with Raf Cluckers on constructing motivic integration in the definable setting using Denef-Pas cell decomposition. We shall define constructible motivic functions and consider exponentials. We shall end by giving, as an application, our general transfer principle for identities between functions defined by non archimedean integrals.
Macintyre, Angus
Quantifier elimination in the p-adics and their algebraic closure, and issues of uniformity
View Abstract
I sketch the basic idea of such eliminations, in preparation for Cluckers's lecture on cell-decomposition.
Merle, Michel
Nearby cycles, convolution and composition with a two variable function
View Abstract
We consider a two variable polynomial function and we want to study its Milnor fibre and monodromy, Hodge spectrum,... More generally, we want to compute the nearby cycles of the composed map with a mapping to the affine plane. This problem was first addressed by Némethi (1991), and by Némethi-Steenbrink (1994-95). Joint work with Gil Guibert and François Loeser.
Nicaise, Johannes
Non-archimedean geometry and complex singularities
View Abstract
We give an introduction to the theory of motivic integration on formal schemes and rigid varieties, and its applications to motivic zeta functions. We explain how non-archimedean geometry can be used to study complex hypersurface singularities.
Rökaeus, Karl
A version of geometric motivic integration that specializes to p-adic integration
View Abstract
We give a version of geometric motivic integration, valid over any complete DVR, and with the property that when we integrate with respect to an affine space over the p-adic numbers, the integral specializes to p-adic integration by counting F_p-points on it.
The theory develops along the same lines as geometric motivic integration. The main difference is that when the geometric motivic measure takes values in a localization of the Grothendieck ring of varieties, completed with respect to the dimension filtration, we have to complete it with respect to a stronger topology. The reason for this is that we need the counting homomorphism to be continuous.
We use this to explain the phenomena that certain p-adic integrals, that we are interested in, are rational functions in p, where the rational function is independent of p. We do this by computing the corresponding motivic integral to see that it is a rational function in L, where L is the Lefschetz class.
Scanlon, Thomas
Integration in valued fields (after Hrushovski and Kazhdan)
View Abstract
I will report on the work of Hrushovski and Kazhdan developing motivic integration over algebraically closed valued fields and related structures.
Soibelman, Yan
Motivic Donaldson-Thomas invariants
View Abstract
This is a joint work with Maxim Kontsevich devoted to the counting problem of stable objects in 3d Calabi-Yau categories
Temkin, Michael
Desingularization of quasi-excellent schemes over Q
View Abstract
Grothendieck proved in EGA IV that if any integral scheme of finite type over a locally noetherian scheme X admits a desingularization, then X is quasi-excellent, and conjectured that the converse is probably true. We prove this conjecture for noetherian schemes of characteristic zero. Namely, starting with the resolution of singularities for algebraic varieties of characteristic zero, we prove the resolution of singularities for noetherian quasi-excellent schemes over Q.
Veys, Willem
Smallest poles of motivic zeta functions
View Abstract
Let f be a regular function on an nonsingular complex algebraic variety. The space of n-jets satisfying f=0 can be partitioned into locally closed subsets which are isomorphic to a cartesian product of some variety with an affine space of appropriate dimension. This implies a divisibility property for this space of n-jets in the Grothendieck ring of varieties, and a lower bound for the smallest poles of the motivic zeta function of f.

Participants

Name
Institution
Matthias, AschenbrennerUniversity of California, Los Angeles
Chetan, BalweUniversity of Pittsburgh
Vladimir, BerkovichWeizmann Institute of Science
Manuel, BlickleUniversity of Essen
Siegfried, BoschWestfälische Wilhelms - Universität Münster
Antoine, Chambert-LoirUniversité Rennes 1
Zoé, ChatzidakisCNRS/Université Paris Diderot - Paris 7
Raf, CluckersUniversité Lille 1/University of Leuven
Georges, ComteUniversité de Nice - Sophia Antipolis
Françoise, DelonCNRS/Université Paris 7
Jan, DenefUniversity of Leuven
Antoine, DucrosUniversité de Nice – Sophia Antipolis
Laurent, FarguesCNRS - Université Paris-Sud
Goulwen, FichouUniversité Rennes 1, France
Thomas, HalesUniversity of Pittsburgh
Immanuel, HalupczokUniverstiy of Leeds
Roland, HuberBergische Universität Wuppertal
Tasho Statev, KalethaUniversity of Chicago
Christian, KappenWestfälische Wilhelms - Universität Münster
Fumiharu, KatoKyoto University
Emmanuel, KowalskiETH Zürich
François, LoeserÉcole Normale Supérieure
Angus, MacintyreQueen Mary University of London
Dugald, MacphersonUniversity of Leeds
Michel, MerleUniversité de Nice - Sophia Antipolis
Matthew, MorrowUniversity of Nottingham
Johannes, NicaiseUniversité Lille 1
Anand, PillayUniversity of Leeds
Michel, RaibautUniversité de Nice - Sophia Antipolis
Karl, RökaeusStockholm University
Thomas, ScanlonUniversity of California, Berkeley
Julien, SebagUniversité Bordeaux 1
Yan, SoibelmanKansas State University
Michael, TemkinUniversity of Pennsylvania
Lou, van den DriesUniversity of Illinois
Willem, VeysKatholieke Universiteit Leuven
Christian, WahleWestfälische Wilhelms - Universität Münster