Short Course
David Evans
An introduction to ampleness in stable theories
Abstract: Hrushovski’s basic counterexample to Zilber’s conjecture on the geometry of strongly minimal structures has a property weaker than one-basedness, known as CM-triviality. This notion was generalised by Pillay into the ampleness hierarchy. In these talks, I will focus on the lower end of the hierarchy in stable theories, as an introduction to recent work on the general hierarchy. The plan of the lectures is:
- The basic Hrushovski construction (the infinite rank version): description of types and forking; weak elimination of imaginaries.
- Complexity of forking in stable theories: one-basedness, CM-triviality and ampleness. Flatness. Triviality. Examples and behaviour under reducts.
- Obtaining ampleness: the free pseudospace and other constructions.
Invited Speakers
Boris Zilber
Geometric dualities and model theory
Abstract: Geometries can be given to us in a naive semantic way, say as a complex or real manifold, or more abstractly, by their co-ordinate algebras and schemes. A duality of this kind becomes highly non-trivial in cases of schemes of arithmetic type and for non-commutative co-ordinate algebras. I will discuss these issues from model-theoretic perspective.
Charlotte Kestner
To be announced
Abstract: To be announced.
Martin Ziegler
Model theory of right-angled buildings
Abstract: Model theory of right-angled countable Tits building B(G) with infinite residues. Using a suitable language, we study the first order theory of B(G). It has a nice axiomatization, is omega-stable, equational and has trivial forking. It is not n-ample, when n is the number of generators of G. (Joint work with A. Baudisch and A. Martin Pizarro).
Talks by Participants
Anja Komatar
Stacking up Ramsey components of shaped partial orders
Abstract: An important step in classification of countable homogeneous coloured (shaped) partial orders is defining an inter densely coloured component. The paper by Torrezão and Truss classifies partial orders with one, two and three components and then shows this in fact yields the whole classification. The talk is about the corresponding process in showing certain classes of shaped partial orders are Ramsey. Assuming that certain classes of shaped partial orders corresponding to one component homogeneous structures are Ramsey, we’ll use them to build classes corresponding to more components.
Benjamin Rigler
Definability of Henselian valuation rings
Abstract: We survey some recent advances in the study of the definability of valuation rings in Henselian valued fields, including results on the existence of defining formulae, uniform definability, potential applications, and some open questions.
Katherina Dupont
Applying Keisler Measures in the context of V-topologies
Abstract: A topology is a V-topology, if the neighbourhoods of zero fullfil six axioms (V 1) to (V 6). It is known that V-topologies are exactly topologies induced by non-trivial absolute values and non-trivial valuations. Under some additional assumptions if a certain topology is a V-topology on a field K, then K admits a non-trivial definable valuation.
In the talk after a general introduction we will concentrate on NIP fields and how Keisler measures can be a applied to show the axiom (V 1).
This work is motivated by conjectures by Hasson and Shelah on the existence of definable valuations on NIP respectively strongly dependent fields.
Levon Haykazyan
Quasiminimality, Regular Types and Homogeneous Pregeometries
Abstract: Quasiminimality, regular types and homogeneous pregeometries all generalise the basic model theory of strongly minimal structures. There are various open questions about them in the spirit of geometric classification theory. In the talk I’ll introduce these concepts and present some connections between them.
Lovkush Agarwal
The Reducts of the Generic Digraph
Abstract: Loosely speaking, a structure N is a reduct of a structure M if N is a less detailed version of M, or, if N is obtained by discarding information from M. The usual set-up is that a structure M is given and one wants to describe the reducts of M. In this talk, I will present work done on determining the reducts of the generic digraph.
Nadav Meir
On reducts of Ramsey structures - survey talk
Abstract: We continue a tradition initiated by D. Bradley-Williams in BPGMT14 of giving a survey talk on a topic closely tied to Model Theory: Counting reducts of ultra- homogeneous structures in a finite language has been a vastly studied subject for decades and is still very active nowadays (the latest result we know of is that of L. Agarwal on the random directed graph). A standing conjecture by Simon Thomas from ’91 states that for every ultrahomogeneous structures in a finite language, there are only finitely many reducts up-to inter-definability. It turns out that this has an equivalent statement in topological dynamic: whether every closed subgroup of S_∞ has finitely many closed subgroups containing it.
An interesting question is what happens when the structure is Ramsey. In 2005 by Kechris, Pestov and Todorcevic found a surprising correspondence to topological dynamics for this case. Following this result, in the past 10 years there has been an extensive work on this in the field of structural Ramsey Theory.
In the talk we will give a survey of some early as well as more recent results and discuss the ramifications this may have towards an answer to the conjecture by Thomas.
Omer Mermelstein
Showing that a reduct of a simple Fraïssé-Hrushovski limit is proper
Abstract: We briefly introduce the notion of a simple Fraïssé-Hrushovski amalgamation class and its unique countable limit. We then present a list of criteria for two simple Fraïssé-Hrushovski classes, assuring that the limit of one is isomorphic to a reduct of the limit of the other. We present Hrushovski’s symmetric non-collapsed construction for a ternary relation and show that it has an infinite descending chain of proper reducts with isomorphic geometries. In the unlikely event that time permits, we will discuss a specific reduct of the structure which has an unusual predimension function.
Ricardo Isaac Bello Aguirre
Generalised stability of pseudofinite residue rings
Abstract: We will present generalised stability properties of ultraproducts of finite residue rings. More specifically we describe the cases for when these ultraproducts are simple, or NIP and non-simple, or NTP2 but not NIP or simple, or TP2.
Samaria Montenegro
Pseudo Real Closed fields and NTP2
Abstract: The notion of PAC fields has been generalized by Basarab and by Prestel to ordered fields. Prestel calls a field M pseudo real closed field(PRC) if M is existentially closed (in the language of rings) in every regular extension L to which all orderings of M extend. Equivalently, if every absolutely irreducible variety defined over M that has a rational point in every real closure of M, has an M-rational point.
In the first part of the talk I will present a short summary of the required preliminaries on pseudo real closed fields. The main theorem is a positive answer to the conjecture by A. Chernikov, I. Kaplan and P. Simon: If M is a PRC field, then Th(M) is NTP2 if and only if M is bounded. In the second part of the talk I will give a sketch of the proof.
Silvain Rideau
Imaginaries in valued differential fields and the independence property
Abstract: In this talk we will consider the theory VDF defined by Scanlon. It is the model completion of valued fields (K,v) with a derivation d such that for all x in K, v(d(x)) is greater of equal to v(x). Since the work of Haskell, Hrushovksi and Macpherson on the imaginaries in algebraically closed valued fields, the question of whether VDF also eliminates imaginaries in the geometric language had remained open. In this talk, I will answer this question positively by relating it to the density of definable types and by showing how the independence property (or rather its absence) can play a role in controlling the canonical basis of definable types.
Vahagn Aslanyan
Ax-Schanuel Type Inequalities in Differentially Closed Fields
Abstract: I consider the problem of existence of Ax-Schanuel type inequalities for given differential equations. This is closely related to definability of a derivation in the corresponding reducts of differentially closed fields. I give some examples and finally conjecture that the definability of a derivation is equivalent to the model-completeness of the reduct.