Schedule

 

	Monday	Tuesday	Wednesday	Thursday	Friday
10:00-11:00	Ludomir Newelski	Sergei Starchenko	Todor Tsankov	Franz-Viktor Kuhlmann	Jamshid Derakhshan
11:00-11:30	Coffee Break	Coffee Break	Coffee Break	Coffee Break	Coffee Break
11:30-12:00	Philipp Hieronymi	Artem Chernikov	Rizos Sklinos	Ehud Hrushovski	Pierre Simon
12:00-12:30			Silvain Rideau
12:30-14:00	Lunch break	Lunch break	Lunch break (until 13:45)	Lunch break	Lunch break
14:00-14:30	Katrin Tent	Itaï Ben Yaacov	Excursion	Predrag Tanović	Daniel Hoffmann
14:30-15:00					Ningyuan Yao
15:00-15:30	Alf Onshuus	Dugald Macpherson		Itay Kaplan	Coffee break
15:30-16:00					Lou van den Dries
16:00-16:30	Coffee break	Coffee break		Coffee break
16:30-17:00	Özlem Beyarslan	Samaria Montenegro		Jana Maříková
17:00-17:30	Gareth Boxall	Poster session		Daniel Palacín
17:30-18:00
19:00-∞	Reception (bonfire)	Dinner	Dinner	Conference dinner	Dinner

Itaï Ben-Yaacov, Université Claude Bernard – Lyon 1

Amalgamation in Globally Valued Field

Globally Valued Fields (GVFs), studied jointly with E. Hrushovski, are fields equipped with many valuations, in which the “sum formula” holds: the sum (or rather, integral) of all valuations of a non-zero element always vanishes. As such, they serve as an abstraction of certain aspects of global fields. The class of GVFs is elementary in continuous logic, but the question of finding its model companion is rather more complicated than for many other familiar classes of fields with additional structure.

I shall try to give some idea of how GVFs behave through the discussion of amalgamation of GVF extensions. In particular, I will show that (under some reasonable hypotheses), if a GVF is an amalgamation base, then it satisfies an additional axiom we call “fullness”. By a recent result, the converse is also true: a full GVF is an amalgamation base. Within time limits, I might give some ideas about the proof of the converse.

Whether or not full GVFs are existentially closed (i.e., whether or not fullness axiomatises the model companion) is yet to be determined.

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Özlem Beyarslan, Boğaziçi University

Geometric Representation in Pseudo-finite Fields

Groups which are "attached" to theories of fields, appearing in models of the theory as the automorphism groups of intermediate fields fixing an elementary submodel are called geometrically represented.

We will discuss the concept “geometric representation” in the case of pseudo finite fields. Then will show that any group which is geometrically represented in a complete theory of a pseudo-finite field must be abelian.

This result also generalizes to bounded PAC fields. This is joint work with Zoe Chatzidakis.

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Gareth Boxall, Stellenbosch University

The definable (p,q)-conjecture

The definable (p,q)-conjecture asserts the following. For an NIP structure M, a tuple b and a formula φ(x,y) such that φ(x,b) does not divide over M, there is some ψ(y) ∈ tp(b/M) such that {φ(x,c) : M ⊧ ψ(c)} is consistent.

It has long been known for stable M. I shall discuss the currently known non-stable special cases due to Simon, Simon-Starchenko and Boxall-Kestner.

Artem Chernikov, University of California, Los Angeles

Model Theory and Combinatorial Geometry, II

Zarankiewicz’s problem in graph theory asks to determine the largest possible number of edges |I| in a bipartite graph G = (U, V, I) with the parts U and V containing m and n vertices, respectively, and such that G contains no complete bipartite subgraphs on k vertices. We show how the distal cutting lemma can be used to obtain asymptotic bounds for Zarankiewicz's problem for graphs definable in distal structures, in particular providing generalizations of Szemeredi-Trotter and related incidence counting bounds. We then discuss how the exponent appearing in these bounds is connected to VC-density estimates and the trichotomy principle, and present some results on the locally modular case.

This is a joint work with David Galvin and Sergei Starchenko.

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Jamshid Derakhshan, University of Oxford

Definable sets, Euler products of p-adic integrals, and zeta functions

I will talk about Euler products, over primes p, of p-adic integrals over definable sets. By work of Denef, each such p-adic integral is a rational function of p^-s (where s is a complex variable), and I will prove that the Euler product admits meromorphic continuation beyond its abscissa of convergence (as a function of s).

These Euler products are global versions of the local p-adic integrals. It turns out that they cover a wide range of Dirichlet series in algebra and number theory including subgroup growth and representation growth zeta functions of groups. The methods of proof can be made to work also for the height zeta functions of adelic points of algebraic varieties and adelic points of orbits of group actions on varieties.

We apply this to give asymptotic formulas for the number of conjugacy classes in algebraic groups over the rationals, the number of subgroups and representations of nilpotent groups, and the number of rational points of bounded height on algebraic varieties and on orbits of group actions on varieties over number fields (with their equidistribution).

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Philipp Hieronymi, University of Illinois, Urbana-Champaign

On continuous functions definable in expansions of the ordered real additive group

For every expansion of the ordered real additive group one of the following holds: every continuous definable function [0,1] → R is C² on an open dense subset of [0,1], or every definable C² function [0,1] → R is affine, or every continuous function [0,1] → R is definable. The first case holds for any NTP₂ expansion of the the ordered real additive group, more generally for any expansion that does not interpret the monadic second order theory of one successor. This is joint work with Erik Walsberg.

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Daniel Hoffmann, Uniwersytet Wrocławski

Model theoretic dynamics in a Galois fashion

Start with your favourite L-theory T, which has a stable model companion T^mc and such that T^mc allows quantifier elimination and elimination of imaginaries. You may be interested in adding some dynamics to models of the theory T. To do this in our “Galois fashion”, you need to fix a group G and extend the language L to L^G:=L ∪ {σ_g}_{g ∈ G}, where each σ_g is a unary function symbol. The first thing which we do (and which is quite obvious) is to consider the L^G-theory T_G which models are exactly L^G-structures (M,(σ_g)_g∈G) such that M⊧T and G∋g↦σ_g∈Aut_L(M) is a group homomorphism.

However, there are no reasons for T_G to have any interesting new properties. Therefore it is better to deal with a model companion of T_G, which we denote T_G^mc. One minor problem is that T_G^mc may not exist, but we are working on that... Assume that T_G^mc exists. It turns out that T_G^mc is simple if models of T_G^mc are bounded as L-structures. Moreover, T_G^mc allows geometric elimination of imaginaries and some variant of elimination of quantifiers. We have also a description of algebraic closure and forking independence in T_G^mc.

To prove that our work can be interesting we give one nice corollary, which is an output from our work. Consider the language of rings in the place of L and the theory of fields in the place of T. In this situation, T^mc is equal to ACF, hence satisfies all required assumptions. Assume that T_G^mc exists and let (K,(σ_g)_{g∈ G}) ⊧ T_G^mc (it may happen that K⊭ACF). The following are equivalent.

1. The theory of K in the language L is simple.
2. The field K is bounded.
3. The theory of (K,(σ_g)_{g∈ G}) in the language L^G is simple.

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Ehud Hrushovski, Hebrew University of Jerusalem, University of Oxford

Elementary consequences of the product formula

The local fields obtained by completing a number field are entirely independent if any finite number is considered at a time; but as a whole they are tied together by the product formula. In an appropriate normalization, this formula asserts that the product of the norms of any nonzero number is identically equal to 1.

This relation opens the door to an entirely new geometry, beyond the sum of the local geometries (the theory of heights is a basic example.) The product formula can be expressed in real-valued continuous logic; a joint project with Itai Ben Yaacov aims to study it model-theoretically. I will discuss some aspects of this theory, focusing on a geometric interpretation of quantifier-free types and quantifier-free stability.

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Itay Kaplan, Hebrew University of Jerusalem

On Kim-independence in NSOP₁ theories

NSOP₁ is the “lowest” class of theories in the SOP hierarchy, and it turns out that there are natural examples of such non-simple theories, both combinatorial and algebraic.

I will describe and discuss the properties of Kim-independence, a generalization of the usual non-forking independence, that works very well in NSOP₁ theories and coincides with non-forking-independence in simple theories.

This talk will be based on joint works with Nick Ramsey and Saharon Shelah.

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Franz-Viktor Kuhlmann, Uniwersytet Śląski

Pushing back the barrier of imperfection

The word “imperfection” in our title not only refers to fields that are not perfect, but also to the defect of valued field extensions. The latter is not necessarily directly connected with imperfect fields but may always appear when at least the residue field of a valued field has positive characteristic. For important open problems in algebraic geometry in positive characteristic, such as resolution of singularities and its local form, local uniformization, both forms of imperfection are a severe hurdle. The same is true for the model theory of valued fields, in particular the open question whether the elementary theory of the imperfect Laurent Series Field F_p((t)) is decidable. This problem is also open for the perfect hull of F_p((t)), which admits extensions with nontrivial defect. Another only partially answered question is under which conditions the existence of a rational place for a function field F|K implies that K is existentially closed in F. Here one can nicely study the interplay of imperfection, the notion of large field, model theory and resolution of singularities.

Beating or avoiding imperfection, as is done in the theory of tame valued fields, leads to partial answers to the above open problems. One possible way of generalizing these answers is to push back the barrier of imperfection, that is, to consider notions of potentially imperfect valued fields with otherwise strong properties, such as the extremal valued fields, or of perfect valued fields which allow only defects of a sort that we can still handle, such as deeply ramified fields (in the sense of Gabber-Ramero) or separably tame fields.

We will give a survey on what is known and what (hopefully) could be done next. The results I will mention come from various projects in which the following coauthors were involved: Sylvy Anscombe, Salih Azgin, Anna Blaszczok, Hagen Knaf, Koushik Pal, Florian Pop.

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Dugald Macpherson, University of Leeds

Cardinalities of definable sets in finite structures

I will discuss recent work with Anscombe, Steinhorn and Wolf, related work with Harrison-Shermoen, and also the PhD thesis of Bello Aguirre, building on the concept of `asymptotic class’ of finite structures and ultimately on results of Chatzidakis, van den Dries and Macintyre on definable sets in finite fields. We consider classes C of finite structures over a given language, such that for any formula φ(x,y) (with x and y tuples) the cardinalities of definable sets φ(x,a) are tightly constrained in structures M in C as a varies through M, either asymptotically (multidimensional asymptotic classes) or exactly (multidimensional exact classes).

The emphasis will be on the latter: for example, which homogeneous structures over a finite relational language are elementarily equivalent to an ultraproduct of a multidimensional exact class?

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Jana Maříková, Western Illinois University

Quantifier elimination for a certain class of convexly valued o-minimal
fields

Let (R,V) be an o-minimal field with a convex subring V such that the
corresponding residue field is o-minimal. Then (R,V) eliminates quantifiers
relative to R after enriching the language by constants for some elementary
substructure of (R,V).

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Samaria Montenegro, Universidad de los Andes

Definable groups in PRC fields

The class of PRC fields was introduced by Prestel and Basarav as a generalization of real closed fields and pseudo algebraically closed fields. We know that the complete theory of a bounded PRC field is NTP₂ and we have a good description of forking. In this talk we will study the class of pseudo real closed fields (PRC-fields) from a model theoretical point of view. We will focus in the description of the groups with f-generic types definable in bounded PRC fields.

(Joint work with Alf Onshuus and Pierre Simon.)

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Ludomir Newelski, Uniwersytet Wrocławski

Topological dynamics of stable groups

The setting from the title provides new tools to measure G-types, alternative to forking. Namely, every G-type may be identified with an endomorhpism d_p of the G-algebra of definable subsets of G. The size of p is directly and inversely correlated with the size of the kernel and image of d_p, respectively. We will discuss possible usage of these tools in the case of the 2-step theorem on generating type-definable subgroups of G. We will discuss possible generalizations of the methods to unstable settings.

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Alf Onshuus, Universidad de los Andes

Hrushovski's stabilizer theorem and groups in NTP₂ theories

In this talk, I will sketch a proof of a variation of Hrushovski's Stabilizer Theorem, and use it to study amenable groups definable in NTP₂ theories. This is joint work with Samaria Montenegro and Pierre Simon.

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Daniel Palacín, Hebrew University of Jerusalem

A property of pseudofinite groups

In this talk, I aim to prove that every pseudo-finite group contains an infinite abelian subgroup. In addition, I shall discuss the existence of a finite-by-abelian finite index subgroup in any pseudofinite group satisfying a certain condition on centralizers. This is joint work with Nadja Hempel.

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Silvain Rideau, University of California, Berkeley

Imaginaries in pseudo-p-adically closed fields

A field is said to be pseudo-p-adically closed (ppc) if it is existentially closed in every regular extension to which each of its p-adic valuation extends. Recent work of Montenegro led to a much better understanding of the model theory of bounded ppc fields (for example, we now know that they are NTP₂). But one natural question was left open: elimination of imaginaries.

The goal of this talk will be to show how the lack of interaction between the p-adic valuations of a bounded ppc field can be used to classify its imaginaries and show that they can all be described in terms of the imaginaries induced by each valuation. I will also describe a general criterion for elimination of imaginaries, inspired by the proofs of that result in various simple theories, combining quantifier free invariant extensions of types and amalgamation.

(Joint with Samaria Montenegro.)

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Pierre Simon, University of California, Berkeley

Finitely generated dense subgroups of Aut(M)

We address the following question: given an omega-categorical structure M, does Aut(M) admit a finitely generated dense subgroup? We show that if M admits some form of canonical amalgamation satisfying a transitivity axiom, then it has a 2-generated dense subgroup. Furthermore, if M has an expansion with this property (plus an extra condition on acl(0)), then it has a 4-generated dense subgroup. This raises the interesting question of whether all omega-categorical structures have such an expansion, a question related to dynamical properties of the automorphism group.

(Joint work with Itay Kaplan.)

Rizos Sklinos, Université Claude Bernard – Lyon 1

Some model theory of the free group

We will give a concise overview of what is known about the model theory of nonabelian free groups. In particular, we will give emphasis to results that radically changed the picture of the model theory of groups.

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Sergei Starchenko, University of Notre Dame

Model Theory and Combinatorial Geometry, I

Let G be a bipartite graph definable in a structure M, and G be the family of all finite induced subgraphs of G.

In this talk we consider combinatorial properties of this family under various model theoretic assumptions on M or on the edge relation of G.

We introduce the notion of a distal relation and prove that if the edge relation of G is distal then the Strong Erdos-Hajnal Property holds for the family G.

We will also discuss so-called Cutting Lemma that plays essential role in our proofs of Strong Erdos-Hajanal Property.

This is a joint work with A. Chernikov and D. Galvin.

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Predrag Tanović, Matematički Institut SANU

Colored orders with equivalence relations

We continue the work of Rubin, Poizat and Simon on colored linear orders. We find a close 'geometric' description of subsets definable in a colored order: they are boolean combinations of intervals, unary 0-definable sets and classes of convex 0-definable equivalence relations., which follows from the characteriziation of saturated colored orders with convex equivalences.

Theorem: Let  (M,<,...) be a saturated linearly ordered structure satisfying: for each convex C ⊆ M and f ∈ Aut(M) fixing C the map g agreeing with f on C and with identity outside C is in Aut(M). Then the original structure is definitionally equivalent to the structure (M,<,P_i,E_j)_i∈I,j∈J with all the 0-definable unary predicates and convex equivalences named. Moreover, if we name all the successor relations S_j(x,y), saying that x<y are in consecutive E_j-classes, we have elimination of quantifiers.

We also characterize colored orders with almost convex equivalence
relations (E is almost convex if each class of the minimal convex
equivalence containing E consists of finitely many E-classes).

Katrin Tent, Universität Münster

The complexity of topological group isomorphism (joint with A. Kechris and A. Nies)

Countable first order structures can be studied via their automorphism groups. This motivates our study of the complexity of the isomorphism relation for various classes of closed subgroups of S_∞. We use reducibility between equivalence relations on Polish spaces.

For profinite, locally compact, and Roelcke precompact groups, we show that the complexity is the same as the one of countable graph isomorphism.

Todor Tsankov, Institut de Mathématiques de Jussieu – Paris 7

Model theory of measure-preserving actions

I will discuss a continuous logic approach to the study of measure-preserving actions of countable groups. For actions of Z (or, more generally, an amenable group G), the theory is well-understood: all free actions of G are elementarily equivalent and the theory is stable and eliminates quantifiers. (This is a result of Ben Yaacov, Berenstein, Henson, and Usvyatsov.) For non-amenable groups, the situation is more complicated and no naturally axiomatizable complete theories are known. Nonetheless, many dynamical notions such as weak containment of actions are naturally expressed in model-theoretic language and this leads to a new approach to some known rigidity results as well as few new ones. This is joint work with Tomás Ibarlucía and François Le Maître.

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Lou van den Dries, University of Illinois, Urbana-Champaign

Hardy fields and transseries: a progress report on ongoing joint work with Matthias Aschenbrenner and Joris van der Hoeven.

Conjecture (1): all maximal Hardy fields are elementarily equivalent to the ordered differential field of transseries. Conjecture (2): all maximal Hardy fields are η₁. Conjecture (1) amounts to showing that every Hardy field can be extended to one that is omega-free and newtonian; this has been done for the “omega-free” part. Much of Conjecture (2) has been established modulo Conjecture (1). Conjecture (1) and (2) together imply, under CH, that all maximal Hardy fields are isomorphic as ordered differential fields. I will also discuss the connection to the Intermediate Value Property for differential polynomials.

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Ningyuan Yao, Fudan University

Groups with definable f-generic types in NIP structures

The concepts of compact Lie group and torsion free Lie group are mutually “orthogonal” in RCF. Taking a model theoretic view, the orthogonality of these two classes of groups is strongly connected with two model theoretic invariants suggested by definable topological dynamics: definable f-generic (abbreviated as dfg) and finitely satisfiable  f-generic (abbreviated as fsg), since fsg groups and dfg groups coincide with compact Lie groups and torsion free Lie groups respectively in RCF context. The observation is also witnessed by the distality of RCF: definable types and finitely satisfiable type are weakly orthogonal. So we believe  this ‘definable f-generic’ might be a dual concept to fsg,  and probably a useful tool to describe the analogs of torsion free o-minimal groups in p-adic (or even arbitrary distal NIP) context.

In this talk, will briefly introduce the new invariants suggested by definable typological dynamics, including almost periodic types, minimal flows, Ellis groups and various generics. and then give our new results on dfg to explain why we are interested in dfg. At the end of this talk, we will post our conjectures and problems on dfg.

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