Model Theory, Combinatorics and valued fields
Institut Henri Poincaré
Winter 2018

Monday and Wednesday, from 11am to 12:30pm
Check the calendar below for the room

Organization: Alf Onshuus and Silvain Rideau

Calendar of the trimester

Next week's talks

April 3rd, 2-3:30pm: Asymptotic and multidimensional asymptotic classes of finite structures, Charles Steinhorn.
Asymptotic classes of finite structures and measurable structures were introduced by Macpherson and me in an effort to develop a model theory for classes of finite structures that reflects contemporary infinite model theoretic themes. Some of this work developed over several years by us and others is reviewed in this talk. Current research that generalizes these concepts to what are called multidimensional asymptotic classes and generalized measurable structures then will be discussed. This most recent work is joint with Macpherson, S. Anscombe, and D. Wolf.

April 4th: On a problem of Gaifman concerning invariant measures, Rehana Patel.
In his seminal 1964 paper Concerning measures in first order calculi Gaifman asked for a characterisation of “those theories which possess a measure-model with strict equality, satisfying also the symmetry condition.” In modern terminology, symmetric measure-models with strict equality are essentially probability measures on the collection of structures with a fixed underlying set (in some fixed language) that are invariant under permutations of that set. In this talk I will give some context for Gaifman’s problem and provide a solution, in the countable case, due to Ackerman, Freer and myself.

Past talks

March 21st: Equationality of differentially and separably closed fields, Amador Martin-Pizarro.
A complete first-order theory is equational if every definable set is a Boolean combination of instances of equations, that is, of formulae such that the family of finite intersections of instances has the descending chain condition. Equationality, as introduced by Srour and later studied together with Pillay, is a strengthening of stability. Typical examples of equational theories are the theory of an equivalence relation with infinite many infinite classes, completions of the theory of modules over a fixed ring, algebraically closed fields of some fixed characteristic, as well as differentially closed fields of characteristic 0 and separably closed fields of finite imperfection degree. So far, the only known "natural" example of a stable non-equational theory is the free non-abelian finitely generated group, as recently shown by Sela. Proving that a particular stable theory is equational is nontheless far from obvious, in general.
The theory of differentially closed fields of positive characteristic, which has been thoroughly studied by Wood, was shown to be equational by Srour. I willsent in this talk int work with Martin Ziegler on how to deduce from Srour's result the equationality of the theory of separably closed fields of infinite imperfection degree.

March 19th: Large groups are o-minimal, Pantelis Eleftheriou.
We work in an expansion (M, P) of an o-minimal structure M by a dense set P, such that three tameness conditions hold. Examples include dense pairs, expansions of M by an independent set, and expansions by a multiplicative group with the Mann property. We prove that a definable group in (M, P) is definable in M if and only if it its dimension coincides with the dimension of its topological closure. As an application, we obtain that if P is independent, then every definable group in (M, P) is already definable in M.

March 14th: Between model theory and physics, Boris Zilber.
There are several important issues in physics which model theory have potential to help with. First of all, there is the issue of adequate language and formalism, and closely related to this there is a more specific problem of giving rigorous meanings to limits and integrals used by physicists. I will present a variation of 'positive model theory' which addresses these issues and discuss some progress in defining and calculating oscillating integrals of importance in quantum physics.

March 12th: Difference Picard group and difference cohomology, Piotr Kowalski.
This is joint work with Marcin Chałupnik. More than 10 years ago, Anand Pillay and I proved that any finite dimensional difference module over an algebraically closed Frobenius difference field (or over a model of ACFA) is isotrivial. This result can be generalized to provide a description of the difference Picard group (to be defined) of any difference field. In this talk, I will discuss the difference Picard groups of difference rings/schemes and connect these groups to the notion of difference cohomology which has been developed by Chałupnik and myself.

February 28th: Differential valued fi elds of kappa-bounded generalized power series, Salma Kuhlmann.
A divisible ordered abelian group is an exponential group if its rank as an ordered set is isomorphic to its negative cone. Exponential groups appear as the value groups of ordered exponential fields, and were studied in [3]. In [6] we gave an explicit construction of exponential groups as Hahn groups of series with support bounded in cardinality by an uncountable regular cardinal kappa. These kappa-bounded Hahn groups are used in turn for the construction of the kappa-bounded exponential logarithmic series fields, which are models of real exponentiation with restricted analytic functions. These models are particularly interesting, since they are naturally similar to Conway-Gonshor's exp-log field of Surreal Numbers No [1], and can therefore be exploited to investigate its properties. In this talk I will discuss aspects of these models of particular interest to No [5]: their differential structure [4], their T_exp - convexity rank, their trans-exponential functions, their exponential integer parts with their co-fi nal sequences of primes [2]. This is ongoing joint work with A. Berarducci, V. Mantova and M. Matusinski.

[1] A. Berarducci and V. Mantova. Surreal numbers, derivations and transseries. To appear in: Journal of the European Mathematical Society, 1 - 47 (2015).
[2] D. Biljakovic, M. Kotchetov, S. Kuhlmann.Primes and Irreducibles in Trun- cation Integer Parts of Real Closed Fields. In: Logic, Algebra and Arithmetic, Lecture Notes in Logic, Volume 26, Association for Symbolic Logic, AK Peters, 42-65 (2006).
[3] S. Kuhlmann. Ordered exponential fi elds. The Fields Institute Monograph Series, vol 12. Amer. Math. Soc. (2000).
[4] S. Kuhlmann and M. Matusinski. Hardy type derivations on fi elds of expo- nential logarithmic series. Journal of Algebra, 345, 171-189 (2011).
[5] S. Kuhlmann and M. Matusinski. The exponential-logarithmic equivalence classes of surreal numbers. ORDER - A Journal on the Theory of Ordered Sets and its Applications, Volume 32, Issue 1, 53-68 (2014).
[6] S. Kuhlmann and S. Shelah. -bounded Exponential-Logarithmic power series fields. Annals Pure and Applied Logic, 136, 284 - 296 (2005).

February 26th: Additive reducts of valued difference fields, Gonenç Onay.
Let (K < L, v) be a valued field extension and f be a continuous ring endomorphism of L. We are interested in the K-vector space structure of L equipped with f and v. This situation can be studied more generally in the language of the "valued modules" that we will introduce during the talk. We will then classify the C-minimal valued modules (C-minimality can be thought as a generalization of the o-minimality for ultrametric structures; for instance, C-minimal fields are exactly the algebraically closed valued fields).

February 21st: Definable topologies and definable compactness in o-minimal structures, Margaret Thomas.
A definable set together with a definable family which forms a basis for a topology is a definable topological space. We consider various properties of topological spaces definable in o-minimal structures, and related properties of definable `directed sets'. From our main result we derive the equivalence of different notions of definable compactness for such spaces (at least if the underlying structure expands a group), identify an appropriate notion of `definable first countability', and, in turn, obtain implications for the classification of definable topological spaces. This is joint work with Pablo Andujar Guererro and Erik Walsberg.

February 19th: Valuation independence and vs-defectless extensions of valued fields, Pablo Cubides Kovacsics.
Valuation independence is a natural relation which strengthens linear independence in the framework of valued fields and valued vector spaces. A valued field extension (L|K,v) is called vs-defectless if every finitely generated K-vector subspace of L admits a K-valuation independent basis. Somewhat surprisingly, model theoretic methods are useful to deal with this notion. In this talk we will revisit some results of Françoise Delon about vs-defectless extensions and solve two questions left open about them. It is a joint work with Anna Blaszczok and Franz-Viktor Kuhlmann.

February 14th: Effective elimination for difference equations, Tom Scanlon.
Roughly speaking, the elimination problem for difference equations asks for a procedure to decide for any system of difference equations $f_1(x_1,\ldots,x_n;y_1, \ldots,y_m) = \cdots = f_m(x_1,\ldots,x_n;y_1,\ldots,y_m) = 0$ for which parameters $b = (b_1,\ldots,b_m)$ is the system $f_1(x;b) = \cdots = f_m(x;b) = 0$ consistent. If by \emph{consistent} we mean that the system has a solution in some difference field, then this problem may be solved using the quantifier simplification theorem for ACFA. However, if we ask for a solution in a sequence ring, that is, a difference ring of the form $K^{\mathbb{N}}$ where $K$ is an algebraically closed field and the distinguished endomorphism acts as a shift operator, then theorems of Hrushovski and Point about the undecidability of the theories of these and all related difference rings suggest that this problem is hopelessly difficult. Nevertheless, we find a feasible procedure to answer the consistency problem when $m = 0$ (so, an effective difference Nullstellensatz) and another feasible procedure for general $m$ to compute a low order consequence in the $y$ variables of the consistency of this system (so, an effective difference elimination theorem in the algebraic sense).
The central theorems are proved through a thorough reworking of the projection-prolongation-elimination method taking into account the failure of the usual geometric axioms. In passing from elimination techniques for rings of the form $K^\mathbb{N}$ with $K$ uncountable to general sequence rings, we make essential use of the limit theory of the Frobenius automorphism.
(This is a report on joint work with Alexei Ovchinnikov and Gleb Pogudin.)

February 12th: Gromov-Hausdorff limits of curves with flat metrics and non-Archimedean geometry, Dmitry Sustretov.
Let X be a smooth family of complex curves of genus >=1 over a punctured disc, and let \Omega be a relative holomorphic 1-form. The (1,1)-form (\Omega \wedge \bar\Omega)_s defines a flat Kahler metric with conical singularities on a fibre X_s. Assuming a certain degeneracy condition on X, we will describe the Gromov-Hausdorff limit of X_s with the metric normalized so that the diameter is 1, as s tends to zero. The limit is a metric graph which can be described as a quotient of a set X_t definable in the theory of algebraically closed valued fields by a definable equivalence relation, where t is a parameter infinitesimally close to zero.

February 7th: Interpretability of tensor products and the regularity lemma, Tomás Ibarlucía
I will discuss a simple remark (observed with Itaï Ben Yaacov) that connects the Szemerédi regularity lemma to the problem of interpreting tensor powers of certain metric structures in the original structures.

February 5th: Remarks on regularity and group regularity theorems, and applications, Anand Pillay.

January 24th: INP-minimal coubtably categorical groups, Frank Wagner.
Inp-minimality is the analogue of strong minimality for NTP_2 theories. A conjecture by Krupinski, in analogy to the strongly minimal case, states that an inp-minimal countably categorical group should be virtually finite-by-abelian. I shall present a proof of the conjecture. This is joint work with Jan Dobrowolski.

January 22nd: Mutually algebraic structures and theories, Chris Laskowski.
Mutual algebraicity is a strong combinatorial property (much stronger than stability) with some remarkable closure relations. Any substructure of a mutually algebraic structure remains so, as well as any expansion by any number of unary predicates and/or injective unary functions.
I will define and develop mutual algebraicity, state many equivalents, including some aimed at identifying mutually algebraic structures in the wild, and survey how the concept has been used in various contexts.

January 17th: Some results in distal theories , Charlotte Kestner.
Distal theories were conceived as being those NIP theories 'furthest' away from being stable. I will give an introduction to distal theories. I will then go on to discuss some results in distal theories. In particular I will discuss the definable (p,q)-theorem for distal theories, and the more recent result that T is distal provided it has a model M such that the theory of the Shelah expansion of M is distal. This is joint work with G. Boxall.