Blanc SIMI 1 - Blanc - SIMI 1 - Mathématiques et interactions

# Valuations, Combinatorics and Model Theory – ValCoMo

## Valuations, Combinatorics and Model Theory

The project is about fundamental research in model theory, a discipline of mathematical logic, and its recent applications in valuation theory, algebraic and arithmetic geometry and combinatorics. It aims to develop interactions between these mathematical disciplines and to foster real and concrete collaborations. Although the domains of application are quite distinct, the same ideas and notions from abstract model theory can be applied in these different contexts and form a unifying theme.

### From pure model theory to applications in valuation theory, algebraic and arithmetic geometry and combinatorics

Model theory is the analysis of classes of abstract mathematical structures by means of their first order properties. The early applications did not use much more than the compactness theorem, while the pure theory was very much concerned with syntactic questions. This changed with Morley's work on on uncountably categorical theories (1965) and Shelah’s development of classification theory, where he introduced a combinatorial notion of independence and dimension which turned out to be extremely powerful. This so-called stability theory was given a new orientation by Zilber and later Hrushovski, evolving into geometric stability theory, where combinatorial independence is used to obtain algebraic information (existence of groups or fields and their properties) out of geometric information (dependencie between realizations of types). Thus a deep and involved general structure theory became available, leading to Hrushovski’s well-known proofs of the Mordell-Lang and Manin-Mumford conjectures in the mid 90s, and starting a veritable back-and-forth between model theory and algebraic and arithmetic geometry.<br />In 2009 Hrushovski surprisingly generalized stability theoretic methods, in particular the so-called stabilizer theorem, to arbitrary theories. He applied this to additive combinatorics, showing that an approximate subgroup of a local group is close to a locally compact group. This was used by Breuillard, Green and Tao to completely classify finite approximate subgroups; their proof was then transformed back into model-theoretic terms by Hrushovski, who also extended his first result to approximate equivalence relations.<br />The aim of this project is to deepen this interaction and the intertwined development of pure model theory and its applications, and to foster collaborations between pure and applied model theorists, combinatorialists and geometers.

The scientific programme has been divided into four scientific tasks, each
1. Model theory of valued fields and Berkovich spaces.:
• Expansions of valued fields (separated pairs, expansions of ACVF by a multiplicative subgroup, imaginaries in enriched valued fields, Henselian valued fields).
• Berkovich spaces (tameness and real differential geometry via model theory).
• Generically stable measures in ACVF.
• Globally valued fields.
2. Applications to algebraic and arithmetic geometry.
• Height.
• Manin-Mumford on abelian surfaces.
• Exponential equations and the Zilber field.
• Differential fields.
• Difference fields.
3. Pseudofinite structures, measures and additive combinatorics.
• Applications of model theory to additive combinatorics (relatively compact approximate subgroups, almost flat manifolds, approximate homomorphisms).
• The model theory of pseudo-finite structures and measures (measures inducing independence, probability logic, definable envelopes, measurable structures, Larsen-Pink equality).
4. Model theoretic tools and neostability.:
• Complexity of forking (burden, Zilber trichotomy, ampleness hierarchy, canonical base property).
• Definability and type-definability (Elimination of hyperimaginaries and definable envelopes, definability of types, ultraimaginaries)
• Neostable groups and fields (chain conditions, stabilizer theorem, C-minimal groups, neostable fields).
All four scientific tasks involve members of all three partners.

Geometric analogue of the height function for equivariant compactifications of affine space.
Application of the model-theory of valued fields to the study of Berkovich spaces.
Study of separably closed fields of finite imperfection degree.
Eliminations of quantifiers for certain expansions of valued fields.
Simple model-theoretic proofs of various implications between tthe theorems of Manin-Mumford and Mordell-Lang.
Ax-Lindemann type theorem for products of Shimura curves.Development of the theory of generics in definably amenable NIP groups.
A complete answer to the relative conjecture of Manin-Mumford.
Geometric elimination of imaginaries for compact complex varieties with a generic automorphism, and canonical base property for types of finite dimension.
Links between combinatorics (Zarankiewich problem, Szemeredi regularity lemma, strong Erdös-Hajnal property) and neostability.
Existence of a locally compact definable quotient of a definably amenable approximate subgroup.
Existence of big abelian subgroups in a pseudofinite group with almost chain condition on centralizers.
Study of the model theory of right-angled buildings.
Study of various expansions of a supersimple stucture of SU-rank omega.
Characterisation of the bounded automorphisms of a field or almost centreless group in a simple structure.
Study of burden and related cardinal invariants.
Model-theoretic study of pseudo-real closed, pseudo-p-adically closed, and dp-minimal fields.
Study of definably amenable, NTP2, and almost Mc groups (generic types, definable envelopes, Fitting subgroup).
Study of enrichments of NIP theories (imaginaries, invariant or definable types).

Outstanding events of the first period:
Semester Model Theory, Arithmetic Geometry and Number Theory, MSRI Berkeley, January - May 2014, with three conferences (Introductory Workshop: Model Theory, Arithmetic Geometry and Number Theory, 3 – 7 February 2014 ; Connections for Women: Model Theory and Its Interactions with Number Theory and Arithmetic Geometry, 10/11 February 2014 ; Model Theory in Geometry and Arithmetic, 12 – 16 May 2014).
Conference Model Theory, Difference/Differential Equations and Applications, CIRM, 7 - 10 Aprill 2015.
Outstanding events of the second period:
Conference Neostability, BIRS, Oaxaca, 12 - 17 July 2015, centred on the theme of task 4.
Meeting Model theory of fields: Derivations, orders and valuations, Paris, 2/3 June 2016.
Interdisciplinary seminar Geometry and Model Theory.
Two lecture series by E. Hrushovski: Topics in pseudo-finite model theory (task 3) and Towards a model theory of global fields (task 1).
Perspectives: A trimester Model theory, combinatorics and tame valued fields at the IHP, Paris, 8 January - 5 April 2018, with an introductory workshop at the CIRM, 8 - 12 January 2018 and three conferences, one centred on the applications in combinatorics (task 3), one on valued fields (task 1), and a general conference.

Baudisch MARTIN-PIZARRO Ziegler A Model Theoretic Study of Right-Angled Buildings JEMS
Bays HILS Moosa Model Theory of Compact Complex Manifolds with an Automorphism TAMS
BENOIST BOUSCAREN Pillay On function field Mordell-Land and Manin-Mumford JML
BERTRAND Generalized jacobians and Pellian polynomial JThNB
- Kummer theory for abelian varieties over function fields OWR
BERTRAND Masser Pillay Zannier Relative Manin-Mumford for semi-abelian surfaces PEdMS
BERTRAND Pillay Galois theory, functional Lindemann-Weierstrass, and Manin maps PJM
BLOSSIER MARTIN-PIZARRO WAGNER A la recherche du tore perdu JSL
BLOSSIER Hardouin MARTIN PIZARRO Sur les automorphismes bornés de corps munis d'opérateurs MRL
Caycedo HILS Bad fields with torsion JSL
CHAMBERT-LOIR LOESER Motivic height zeta functions ALM
CHATZIDAKIS Harrison-Trainor Moosa Differential-algebraic jet spaces preserve internality to the constants JSL
CHATZIDAKIS Perera A criterion for p-Henselianity in characteristic p BBMS
CHERNIKOV Starchenko Regularity lemma for distal structures JEMS
Cubides-Kovacsics DELON Definable types over algebraically closed fields MLQ
DELON SIMONETTA Abelian C-minimal valued groups APAL
HEMPEL On n-dependent groups and fields MLQ
HEMPEL Onshuus Groups in NTP2 IJM
Jahnke SIMON Walsberg Dp-minimal valued fields JSL
MONTENGRO Pseudo real closed field, pseudo p-adic closed fields and NTP2 APAL
MASSICOT WAGNER Approximate subgroups JEP
Palacin WAGNER A Fitting Theorem for Simple Theories BLMS
RIDEAU Imaginaries and invariant types in existentially closed valued differential fields CRELLE
- Some properties of analytic difference valued fields JIMJ
RIDEAU SIMON Definable and invariant types in enrichments of NIP theories JSL
SIMON A Note on Regularity lemma for distal structures PAMS
- Rosenthal compacta and NIP formulas FM
- VC-sets and generic compact domination IJM
SIMON Walsberg Tame topology over dp-minimal structures NDJFL
WAGNER Plus ultra JML
- The right angle to look at orthogonal sets JSL

This project aims to strengthen the interactions between model theory, on one hand, and valuation theory, combinatorics and geometry on the other hand. Modern model theory analyses the category of definable sets in a first-order structure or class of structures. A combinatorial notion of independence for certain classes of structures called stable, which had been introduced by Shelah, was given a geometric interpretation by Zilber and Hrushovski and used to derive algebraic information. The theoretical machinery developed was then applied to algebraic and arithmetic geometry, most spectacularly in Hrushovski’s proofs of the Mordell-Lang and Manin-Mumford conjectures, via a detailed study of certain theories of fields: separably closed fields, differentially closed fields, and existentially closed difference fields.

In the last five years, three important developments have significantly extended this picture, one on the pure side, one on the applied side, and one concerning applications of model theory. The first one is the study of independence first in dependent (or NIP) theories and then in NTP2 theories, which generalize the basic machinery of stability theory to a very wide class of structures containing all previously considered classes. The second one is the detailed analysis of certain classes of valued fields with an operator which are NTP2. It can be expected that their understanding will again lead to significant geometric applications. The third development is Hrushovski’s transfer of stability-theoretic ideas to additive combinatorics via a dimension- and measure-theoretic study of pseudofinite structures, culminating in the recent characterisation of approximate subgroups by Breuillard, Green and Tao.

Consequently, our proposal is divided into four main scientific tasks:
1.The model theory of valued fields and Berkovich spaces,
2.Applications to algebraic and arithmetic geometry,
3.Pseudofinite structures, measures and additive combinatorics, and
4.Model-theoretic methods and neostability (the study of stability-theoretic phenomena in unstable contexts).

The fourth task lays the common theoretical foundations: we shall study NTP2 theories, important subclasses such as dependent or simple theories, and extensions to infinitary or continuous logic. The second task is the most established topic, but important questions still remain open. We shall approach them in the light of the recent theoretical advances on dependent and NTP2 theories. The first task aims to develop the model theory of valued fields with additional structure, for instance via relative quantifier elimination or the use of continuous logic, and to understand the relationship between the definable category in algebraically closed valued fields and Berkovich spaces. The third task is the most innovating and audacious one, since the use of model theory in additive combinatorics is very recent. We aim to study pseudofinite structures, measures and dimensions in general from a neostability-theoretic point of view, aiming for asymptotical applications in additive combinatorics.

All four tasks are highly interrelated: algebraic objects also have a geometric flavour, extensions of the logic developed for their pure aspects can and will be used in all three applied tasks, measures and dimensions permeate all four topics. The aim thus is to bring together the pure model theory expertise of the Lyon group with the applied model theory knowledge of the Paris and Orsay logicians, as well as interested algebraists, geometers and combinatorialists in a common project which will pursue research in the four scientific tasks transversally: concrete examples will be distilled into abstract principles leading the way for theoretical generalizations, which in turn yield new tools for the study of specific theories, quite possibly in a different area of application.

## Project coordination

Frank Wagner (Institut Camille Jordan UMR5208) – wagner@math.univ-lyon1.fr

The author of this summary is the project coordinator, who is responsible for the content of this summary. The ANR declines any responsibility as for its contents.

## Partner

PSUD-LMO Université Paris-Sud - Laboratoire de Mathématiques UMR8628
IMJ Institut de Mathématiques de Jussieu UMR 7586
ICJ Institut Camille Jordan UMR5208

Help of the ANR 226,924 euros
Beginning and duration of the scientific project: December 2013 - 48 Months