TDE Seminar

Online seminar: Topological and Differential Expansions of O-minimal Structures

     

Organizers: Alexander Berenstein (Bogotá)
                      Pantelis Eleftheriou (Leeds)




Zoom ID: 884 0948 8227

The seminar will focus on various expansions of o-minimal structures, such as those with o-minimal open core, d-minimal structures, H-structures, lovely pairs, RCVFs, CODFs, distal and general NIP. We target talks in pure model theory and applications. The intention is to run the seminar once every two weeks. The exact times may slightly vary, so please check below.



Upcoming talks:

Currently none scheduled.


Past talks:


  • December 20, 2021 (Monday) - 15:30 GMT

  • Nigel Pynn-Coates - Ohio State University

    Title: An Ax–Kochen/Ershov theorem for differential-henselian pre-H-fields

    Abstract: Pre-H-fields are a kind of ordered valued differential field introduced by Aschenbrenner and van den Dries as part of their work on the model theory of transseries and Hardy fields. I will describe a class of pre-H-fields, somewhat different from transseries, that admit an Ax–Kochen/Ershov theorem: the theory of each of these pre-H-fields is determined by the theory of its ordered differential residue field. I will also describe a motivating example, which arises by coarsening the valuation of a saturated elementary extension of transseries so that it only distinguishes transexponentially different elements; the residue field of the coarsened valuation is an exponentially bounded model of the theory of transseries. Time permitting, I will mention related results, such as the stable embeddedness of these ordered differential residue fields and an NIP transfer result.

    ➯ RECORDING



  • December 6, 2021 (Monday) - 15:30 GMT --- POSTPONED (due to conflicts):

    December 13, 2021 (Monday) - 15:00 GMT (notice different time)

  • Omar Leon Sanchez - University of Manchester

    Title: Remarks on bi-differential algebras

    Abstract: Abstract: In recent years there have been applications of the model theory of differential fields to the Poisson Dixmier-Moeglin equivalence (related to Poisson spectrums). However, there are some obstacles that we have not been able to overcome, and so we recently proposed a "model-theoretic-type" study of bidifferential algebras (with Poisson algebras as a special case). Turns out that even fundamental results - such as base change - fail dramatically in the bi-differential context. I will talk about some things that work and some that do not. For the latter I will explain how we can "correct" the issues. This joint work with Rahim Moosa.

    ➯ RECORDING



  • November 22, 2021 (Monday) - 15:30 GMT

  • John Goodrick - Universidad de Los Andes

    Title: Ordered abelian groups, dp-minimality, and definable convex subgroups

    Abstract: Dp-minimality is a tameness notion which is a rough analogue of "having weight 1" within the broader class of NIP theories. Dp-minimal theories include all o-minimal theories, P-adically closed fields, weakly minimal stable theories, and more. We will present some new results (obtained jointly with Viktor Verbovskiy) on how dp-minimality relates to definable sets in ordered abelian groups (possibly in an extended language). In particular we will discuss connections with weak o-minimality and the property of having boundedly many definable convex subgroups.

    ➯ RECORDING



  • November 8, 2021 (Monday) - 15:30 GMT

  • Assaf Hasson - Ben-Gurion University

    Title: Zilber's restricted trichotomy

    Abstract: We discuss recent (and somewhat less recent) developments around Zilber's trichotomy conjecture restricted to structures interpretable in various tame settings. We will survey in some detail the proof of Zilber's trichotomy for reducts of algebraic curves and explain why this proof cannot be generalised to almost any other context, and what are the tools needed to generalise certain aspects of the proof to other settings such as the o-minmal and ACVF.

    ➯ RECORDING



  • October 25, 2021 (Monday) - 15:00 GMT

  • Francoise Point - University of Mons

    Title: On exponential topological fields endowed with a generic derivation

    Abstract: We axiomatize a class of existentially closed exponential fields equipped with an E-derivation and endowed with a definable V-topology. We apply our results to the field of real numbers endowed with exp(x) the classical exponential function defined by its power series expansion and to the field of p-adic numbers endowed with the function exp(px) defined on the p-adic integers where p is a prime number strictly bigger than 2 (or with exp(4x) when p=2). Independently, this question has also been considered by A. Fornasiero and E. Kaplan in certain o-minimal expansions of the field of real numbers. This is a joint work with Nathalie Regnault.

    ➯ RECORDING



  • October 11, 2021 (Monday) - NOTE UNUSUAL TIME: 15:30 GMT

  • Erik Walsberg - University of California at Irvine

    Title: NIP expansions of (ℝ, <, +)

    Abstract: Let M be a first order expansion of (ℝ, <, +). A couple years ago I showed that if M is NIP then every subset of ℝ which is definable in the open core of M either has nonempty interior or an isolated point. This is a weak form of o-minimality that allows us to prove many things about the open core of M. The original proof was pretty short and simple. The version of the proof that appeared in print was much more general and therefore much less accessible. In this talk I will present the original proof for the first time.

    ➯ RECORDING



  • September 27, 2021 (Monday) - NOTE UNUSUAL TIME: 15:30 GMT

  • Mariana Vicaria - University of California at Berkeley

    Title: Towards an imaginary Ax-Kochen Principle

    Abstract: One of the most striking results in the model theory of henselian valued fields is the Ax-Kochen theorem, which roughly states that the first order theory of a henselian valued field of equicharacteristic zero, or of mixed characteristic , unramified and with perfect residue field is determined by the first order theory of the residue field and its value group.

    A model theoretic principle follows from this theorem: any model theoretic question about the valued field can be reduced into a question to its residue field, its value groups and their interaction in the field. A fruitful application of this theorem has been applied to describe the class of definable sets in a valued field, for example Pas proved a relative quantifier elimination statement relative to the residue field once we add an angular component in the equicharacteristic zero case.

    One can therefore ask the following question: Can one obtain an Ax-Kochen style theorem to eliminate imaginaries in a henselian valued field?

    Following the Ax-Kochen principle, it seems natural to look at the problem in two orthogonal directions: one can either make the residue field extremely tame and understand the problems that the value group bring naturally to the picture, or one can assume the value group to be very tame and study the issues that the residue field would contribute to the problem.

    In this talk we will address the first approach. I will present how to eliminate imaginaries in henselian valued fields of equicharacteristic zero with residue field algebraically closed. The results obtained are sensitive to the complexity of the value group. I will start by introducing the problem of imaginaries in order abelian groups according to their combinatorial complexity. Once the picture has been clarified for this setting, we will present how to solve the question for valued fields.

    ➯ RECORDING



  • September 13, 2021 (Monday) - Time: 15:00 GMT

  • Martina Liccardo - Università degli Studi di Napoli Federico II

    Title: Elimination of imaginaries in lexicographic products of ordered abelian groups

    Abstract: We will investigate the property of elimination of imaginaries for some special cases of ordered abelian groups. As main result, we will show that the lexicographically ordered groups Z^n and Z^n \times ​Q eliminate imaginaries once we add finitely many constants to the language of ordered abelian groups.

    ➯ RECORDING



  • August 30, 2021 (Monday) - Time: 15:00 GMT

  • Nadav Meir - University of Wrocław

    Title: Pseudo-finite sets, pseudo-o-minimality

    Abstract: Given a language L, the class of o-minimal L-structures is not elementary, e.g., an ultraproduct of o-minimal structures need not be o-minimal. This fact gives rise to the following notion, introduced by Hans Schoutens: Given a language L, an L-structure is pseudo-o-minimal if it satisfies the common theory of o-minimal L-structures. Of particular importance in pseudo-o-minimal structures are pseudo-finite sets. A definable set in an ordered structure is pseudo-finite if it is closed, bounded, and discrete. Many results from o-minimality translate to pseudo-o-minimality by replacing finite with pseudo-finite. We will review the key role that pseudo-finite sets play in pseudo-o-minimality, as well as other first-order properties of o-minimality such as definable completeness* and local o-minimality**. Finally, we will see how pseudo-finite sets can be used to prove distinctions between generalizations of o-minimality and answer two questions by Schoutens, one of them is whether there is an axiomatization of pseudo-o-minimality by first-order conditions on one-variable formulae only. This also partially answers a conjecture by Antongiulio Fornasiero.

    No knowledge in o-minimality or generalizations thereof will be assumed throughout the talk.

    * An ordered structure is definably complete if every bounded definable subset has a supremum.
    ** An ordered structure is locally o-minimal if, for every definable subset D and every point x, there is an interval with an endpoint in x that is either contained in D or disjoint from D.

    ➯ RECORDING



  • August 16, 2021 (Monday) - Time: 15:00 GMT

  • Neer Bhardwaj - University of Illinois at Urbana-Champaign

    Title: On the Pila-Wilkie theorem

    Abstract: I’ll give an account of the Pila-Wilkie counting theorem and some of its extensions and generalizations. We use semialgebraic cell decomposition to simplify part of the original proof. Very basic knowledge of o-minimality will be assumed; this is joint work with Prof. Lou van den Dries.

    ➯ RECORDING



  • August 2, 2021 (Monday) - Time: 15:00 GMT

  • Gareth Boxall - Stellenbosch University

    Title: Some finiteness results concerning points on a curve with a power on a curve

    Abstract: Let X and Y be two geometrically irreducible closed algebraic curves in the algebraic torus of dimension 3. Suppose a generic point of X satisfies no non-trivial multiplicative relations. We denote by [n]X the set of all n-th powers of points in X. It is a conjecture that at most finitely many x in X will have the property that there is a positive integer n such that x^n is in Y and [n]X is not contained in Y. I shall discuss proofs of several cases of this conjecture. Work of Bays, Kirby and Wilkie, which established an analogue of Schanuel's conjecture for the operation of raising to an exponentially transcendental power, plays an important role.

    ➯ RECORDING



  • July 19, 2021 (Monday) - Time: 15:00 GMT

  • Andrew Harrison-Migochi - Stellenbosch University

    Title: Pila-Wilkie in Tame Expansions of a P-adic Structure

    Abstract: We adapt a paper of Eleftheriou on an analogue for the Pila-Wilkie theorem in expansions of the real analytic structure to a p-adic context. We prove a structure theorem for definable sets in an expansion o the p-adic subanalytic structure by a dense independent set along the way.

    ➯ RECORDING



  • July 5, 2021 (Monday) - Time: 15:00 GMT

  • Antongiulio Fornasiero - University of Florence

    Title: O-minimal structures with many dense substructures.

    Abstract: We study expansion of o-minimal structures with two or more predicates dense "independent" substructures.

    ➯ RECORDING



  • June 21, 2021 (Monday) - Time: 15:00 GMT

  • Tsinjo Rakotonarivo - Stellenbosch University

    Title: The definable (p,q)-theorem for dense pairs of distal geometric structures where acl=dcl.

    Abstract: The definable (p,q)-conjecture is a model-theoretic version of the combinatorics (p,q)-theorem. In this talk, I will discuss how we proved that the definable (p,q)-conjecture holds for dense pairs of distal geometric structures where the algebraic closure is equal to the definable closure.

    This is a joint work with Gareth Boxall.

    ➯ RECORDING



  • June 7, 2021 (Monday) - Time: 15:00 GMT

  • Philipp Hieronymi - University of Illinois at Urbana-Champaign

    Title: A strong version of Cobham’s theorem and other decidability results in expansions of Presburger arithmetic

    Abstract: Let k,l>1 be two multiplicatively independent integers. Cobham’s famous theorem states that a subset of the natural numbers is both k-recognizable and l-recognizable if and only if it is Presburger-definable. We show the following strengthening. Let X be k-recognizable, let Y be l-recognizable such that both X and Y are not Presburger-definable. Then the theory of (N,<,+,X,Y) is undecidable. This is joint work with Christian Schulz.

    ➯ RECORDING



  • May 24, 2021 (Monday) - Time: 15:00 GMT

  • Ayhan Günaydın - Boğaziçi University

    Title: Expanding the Additive Group of Integers by Beatty Sequences

    Abstract: The Beatty Sequence generated by an irrational r>1 is ([nr] : n>0), where [a] denotes the integer part of a real number a. After presenting some properties of such a sequence, we will investigate the expansion of (Z, +) by the unary subset B consisting of the terms of a Beatty sequence. We will mention a quantifier elimination result and an axiomatization for the theory of such an expansion. If time allows, we will also show that there are no intermediate structures between (Z, +) and (Z, +, B). (This is a joint work with Melissa Özsahakyan.)

    ➯ RECORDING



  • May 10, 2021 (Monday) - Time: 16:30 GMT

  • Deirdre Haskell - McMaster University

    Title: Tameness properties of theories of valued fields with analytic functions

    Abstract: An important motif in model-theoretic algebra over the last thirty years has been the concept of tameness and the impact it has for understanding the definable sets of a structure. In this talk, I will describe some of the ways this motif occurs in the case of valued fields, especially ordered convexly valued fields, when equipped with additional function symbols which, on the standard model, are interpreted by functions defined by convergent power series. All of these notions will be defined in the course of the talk.

    ➯ RECORDING



  • April 26, 2021 (Monday) - Time: 16:30 GMT

  • Kobi Peterzil - University of Haifa

    Title: Interpretable fields in various valued fields

    Abstract: Difficulties in analyzing interpretable objects arise when we lack (a simple) elimination of imaginaries. It turns out that in several dp-minimal settings it is possible to circumvent this difficulty by focusing on one-dimensional subsets and by reducing these to several relatively understandable sorts.

    We consider an interpretable field F in either a real closed valued field K or T-convex expansions of K. In this case one can reduce the analysis to the four sorts K, k, the value group, and K/O (for O the valuation ring), then eliminate the last two sorts, and conclude that F is either definable in the field K or in k. As a result, F is definably isomorphic to K,K(i), k or k(i).

    Similar analysis can be carried out in certain P-minimal structures (in particular, in p-adically closed fields), and probably more.

    (part of a joint work with Y. Halevi and A. Hasson)

    ➯ RECORDING



  • April 12, 2021 (Monday) - Time: 16:30 GMT

  • Alexi Block Gorman - University of Illinois at Urbana-Champaign

    Title: Fractal Dimensions and Definability from Büchi Automata

    Abstract: Büchi automata are the natural extension of finite automata, also called finite-state machines, to a "machine" that accepts infinite-length inputs. We say a subset X of the reals is r-regular if there is a Büchi automaton that accepts (one of) the base-r expansions of every element in X, and rejects the base-r expansions of each element in its complement. We can analogously define r-regular subsets of higher arities of the reals, and these sets often exhibit fractal-like behavior--e.g., the Cantor set is 3-regular. There are several known--and remarkable--connections in logic to Büchi automata, including the fact that the expansion of the real additive group by every r-regular subset of [0,1] (for some fixed positive integer r) interprets the monadic second-order theory of the natural numbers with successor. In this talk, I will focus on some of the geometric behavior of closed r-regular sets in terms of fractal dimensions, and discuss how closed r-regular subsets of [0,1] with and without integer Hausdorff dimension form a dichotomy in terms of first order definability in expansions of the real additive group by a predicate for a specific r-regular set.

    ➯ RECORDING



  • March 29, 2021 (Monday) - Time: 16:30 GMT

  • Vincenzo Mantova - University of Leeds

    Title: A survey on exponential-algebraic closure

    Abstract: Zilber conjectured that complex exponentiation is quasiminimal in 1997 (if not before) and produced different quasiminimal structures. He later formulated the exponential-algebraic closedness conjecture (EAC), which would imply quasiminimality of complex exponentiation.

    I will summarise what has been proved so far around EAC, including extensions to abelian exponentials, modular functions, the special case of raising to powers, and an odd spin-off with o-minimal open core.

    ➯ RECORDING



  • March 15, 2021 (Monday) - Time: 16:30 GMT

  • Rosario Mennuni - Universität Münster

    Title: The domination monoid in o-minimal theories

    Abstract: The product of invariant types and domination-equivalence are not always compatible but, when they are, they allow to define the "domination monoid" associated to a monster model U of a first-order theory. In the superstable case, this object parameterises "finitely generated saturated extensions of U" and how they can be amalgamated independently. After recalling the basic definitions and facts, I will talk about some results from my thesis, concerning the study of this monoid in a different context, that of o-minimal theories. This includes a reduction of the problem to showing generation by classes of 1-types, and a proof that this holds in RCF. I will then discuss the open problem of showing generation by 1-types in general, and some possible lines of attack.

    ➯ RECORDING



  • March 1, 2021 (Monday) - Time: 16:30 GMT

  • Vahagn Aslanyan - University of East Anglia

    Title: Blurrings of the j-function

    Abstract: I will define blurred variants of the j-function and its derivatives, where blurring is given by the action of a subgroup of GL2(ℂ). For a dense subgroup (in the complex topology) an Existential Closedness theorem holds which states that all systems of equations in terms of the corresponding blurred j with derivatives have complex solutions, except where there is a functional transcendence reason why they should not. The proof is based on the Ax-Schanuel theorem and Remmert's open mapping theorem from complex geometry. For the j-function without derivatives we prove a stronger theorem, namely, Existential Closedness for j blurred by the action of a subgroup which is dense in GL2+(ℝ), but not necessarily in GL2(ℂ). In this case apart from the Ax-Schanuel theorem and some basic complex geometry, o-minimality is also used in the proof (I will present the proof in this case). If time permits, I will also discuss some model theoretic properties of the blurred j-function such as stability and quasiminimality. This is a joint work with Jonathan Kirby.

    ➯ RECORDING



  • February 15, 2021 (Monday) - Time: 16:30 GMT

  • Elliot Kaplan - University of Illinois Urbana-Champaign

    Title: Generic derivations on o-minimal structures

    Abstract: Let T be a model complete o-minimal theory which extends the theory of real closed ordered fields (RCF). We introduce T-derivations: derivations on models of T which cooperate with T-definable functions. The theory of models of T expanded by a T-derivation has a model completion. If T = RCF, then this model completion is the theory of closed ordered differential fields (CODF) as introduced by Singer. We can recover many of the known facts about CODF (open core, distality) in our setting. This is joint work with Antongiulio Fornasiero.

    ➯ RECORDING



  • February 1, 2021 (Monday) - Time: 16:30 GMT

  • Itay Kaplan - Hebrew University of Jerusalem

    Title: Compressible types in NIP theories

    Abstract: I will present some work in progress joint with Martin Bays and Pierre Simon. I will discuss compressible types and relate them to uniform definability of types over finite sets (UDTFS), and to uniformity of honest definitions. All notions will be defined during the talk.

    ➯ RECORDING



  • January 18, 2021 (Monday) - Time: 16:30 GMT

  • Tingxiang Zou - Universität Münster

    Title: Geometric random graphs

    Abstract: Geometric random graphs are graphs on a countable dense set of some underlying metric space such that locally in any ball of radius one, it is a random graph. The geometric random graphs on R^n and on circles have been studied by probabilists and graph theorists. In this talk we will present some model theoretic views. In particular, we will show that under some mild assumptions, the geometric random graphs based on a fixed metric space will have the same theory. We will also talk about some geometric properties of the underlying metric space that can be recovered from the graphs. This is a work in progress joint with Omer Ben-Neria and Itay Kaplan.

    ➯ RECORDING



  • January 4, 2021 (Monday) - Time: 16:30 GMT

  • Chris Miller - Ohio State University

    Title: Connectedness in structures on the real numbers

    Abstract: We consider structures on the set of real numbers having the property that connected components of definable sets are definable. Our main analytic-geometric result is that any such expansion of the real additive group by boolean combinations of closed sets (of any arities) is either o-minimal (with respect the usual order) or undecidable, and if the set of integers is definable, then so is integer multiplication. It is known that all o-minimal structures on the real line have the property, as do all expansions of the real field that define the integers (easy modulo some basic descriptive set theory). We show that fusions of the real ordered additive group with expansions of the ring of integers are also examples. All results hold with "connected component" replaced by "path component" or "quasicomponent". (Joint with A. Dolich, A. Savatovsky and A. Thamrongthanyalak. Preprint available on MODNET and arXiv.)

    NO RECORDING



  • December 21, 2020 (Monday) - Time: 16:30 GMT

  • Christian d'Elbée - Hebrew University of Jerusalem

    Title: Generic expansions by a reduct

    Abstract: Consider the expansion TS of a theory T by a predicate for a submodel of a reduct T0 of T. This generalizes the generic predicate construction and some theories of lovely pairs. We present a setup in which this expansion admits a model companion TS. We show that the nice features of the theory T transfer to TS. In particular, by studying independence relations, we find conditions for which this expansion preserves the NSOP1, simplicity or stability of the starting theory T. We will also give concrete examples of new model-companion obtained by this process, among them new NSOP1 theories such as the expansion of an algebraically closed field of positive characteristic by an additive subgroup (ACFG) and the expansion of an algebraically closed field of any characteristic by a generic multiplicative subgroup. This construction also gives some very wild expansions of fields, such as the expansion of an algebraically closed valued field of positive characteristic by a generic additive subgroup, which has TP1 and TP2.

    ➯ RECORDING



  • December 7, 2020 (Monday) - Time: 16:30 GMT

  • Artem Chernikov - University of California, Los Angeles

    Title: Distality in valued fields and related structures

    Abstract: In this talk we discuss distality, a model-theoretic notion of tameness generalizing o-minimality, in valued fields and related structures. In particular, we characterize distality in certain ordered abelian groups, provide an Ax-Kochen-Ershov style characterization for henselian valued fields, and demonstrate that certain expansions of fields, e.g. the valued differential field of logarithmic-exponential transseries, are distal. This relies in particular on a general quantifier elimination result for pure short exact sequences of abelian groups. Joint work with Matthias Aschenbrenner, Allen Gehret and Martin Ziegler.

    ➯ RECORDING



  • November 23, 2020 (Monday) - Time: 16:30 GMT

  • Alexander Berenstein - Universidad de Los Andes

    Title: A review of expansions by predicates and some preservation theorems

    Abstract: We say that a theory is geometric if the algebraic closure satisfies the exchange property and eliminates the quantifier exists infinitely. Examples include dense o-minimal theories, strongly minimal theories and SU-rank one theories. In this talk we will introduce geometric theories, review some of its expansions by predicates and the structural properties (stability, simplicity, NIP, NTP2) that these expansions preserve.

    ➯ RECORDING