Wilfrid Hodges (Thursday 5.00-6.00)
The history of model theory
The beginning of model theory proper can be dated to the 1950 International Congress of Mathematicians, where Alfred Tarski and Abraham Robinson presented the theory. We will draw together some of the threads from mathematics and philosophy that led to the creation of model theory. Then we will discuss what happened around 1950, and how the aims of the theory developed after that.
Anand Pillay (Friday 2.30-3.30)
Separably closed fields, type-definable groups, and Mordell-Lang revisited
John K Truss (Short Course, handout)
Homogeneous structures and descendant-homogeneous digraphs I shall introduce and compare the three notions 'homogeneous', 'ultrahomogeneous', and 'descendant-homogeneous'. Corresponding to the first of these, I give some classical results of Keisler, Morley, and others, on the number of homogeneous models of a complete theory in various cardinalities. In the uncountable case, the GCH is usually assumed. I shall give several examples, some of which are due to Kudaibergenov, who answered a number of questions raised by Keisler and Morley in their 1967 paper. As a contrast, I then briefly describe ultrahomogeneity and recall Fraïssé's theorem about this situation. In the last lecture I shall introduce a related notion, called 'descendant-homogeneity', which applies to directed graphs, present an analogue of Fraïssé's theorem in this context, and give some examples and theorems (joint work with Daniela Amato and David Evans).
Paul Baginski (Friday 11.00-11.30)
Baer-Suzuki Theorems for Infinite Groups
The Baer-Suzuki Theorem is a classic result in finite group theory that characterizes nilpotence on a whole group (or orbit of the group) in terms of nilpotence on two-generated subgroups. We will discuss the classic theorem and its importance and then present extensions of this theorem to classes of infinite groups using model theory. Some of these results have been known for many years, while others are quite recent. We will also discuss avenues for extending these results to new classes of groups and Baer-Suzuki-style results.
Elias Baro (Wednesday 5.00-5.30)
On the homeomorphic type of definably compact abelian definable groups in o-minimal structures
In this talk we show that every definably connected, definably compact abelian definable group in an o-minimal expansion of a real closed field of dimension not 4 is definably homeomorphic to a torus of the same dimension. Moreover, in the semialgebraic case the result holds for all dimensions.
Katharina Dupont (Friday 5.00-5.30) (slides)
V-topologies, t-henselian fields and definable valuations
V-topologies are field topologies induced by absolute values or valuations. The class of these topologies can be defined in second order logic by a set of local sentences which is equivalent to being first order definable.
We will first give a short introduction to V-topologies and local sentences. Afterwards we will define when we call a field with a V-topology t-henselian first and give two equivalences. We will then talk about the relation of V-topologies and definable valuations. We will discuss a theorem about definable valuations on t-henselian fields and give an overview over current and planned research on definable valuations.
Jochen Koenigsmann , Definable Valuations, Prèpublication de l'Equipe de Logique No. 54, 1995
Alexander Prestel, Martin Ziegler , Modeltheoretic methods in the theory of topological fields, J. reine angew. Math. 299 /300 (1978), page 318 – 341
Bernhard Elsner (Thursday 11.00-11.30)
Preanalytic Topological Strucktures
We consider Topological Structures with Good Dimension in sense of B. Zilber and ask for sufficient conditions to extend the (AF) and (FC) axioms to general projective sets.
Antongiulio Fornasiero (Wednesday 4.30-5.00) (slides)
Lovely pairs for independence relations
In the literature there are two different notions of lovely pairs of a theory T, according to whether T is simple or geometric. We introduce a notion of lovely pairs for an independence relation, which generalizes both the simple and the geometric case, and show how the main theorems for those two cases extend to our general notion.
Joint work with G. Boxall.
Zaniar Ghadernezhad (Wednesday 5.30-6.00) (slides)
Topological simplicity of automorphism group of some generic ab-initio structures
In the early 90's, Lascar proved that in a countable almost strongly minimal structure, the bounded automorphisms form a normal subgroup of the group of strong automorphisms and their quotient group is simple. The topological structure of the automorphism group of a countable structure is also known. There are new results by Macpherson and Tent, proving the topological simplicity of the automorphism groups of certain generalized n-gons. In the generic ab-initio case, looking at the group-structure of the automorphism group first leads us to investigate the bounded automorphisms. By giving an answer to the existence of non-trivial bounded automorphisms in the some cases of uncollapsed generic ab-initio structures and following the same method by Macpherson and Tent, we will prove the topological simplicity of the uncollapsed generic ab-initio structure.
Gabriel Giabicani (Thursday 3:45-4:30)
The Cavalieri-Weil ring of a difference field
In the theory of existentially closed difference fields ACFA, a definable set of finite dimension should be considered, for reasons I will detail, as an "object of finite nature" even if it has actually infinitely many points. A natural question is to determine when two such sets have "the same number of points".
I will describe the construction of the Cavalieri ring of a given theory T. This assigns, to a definable set X of finite nature, a "motivic cardinality" that plays the role of cardinality for finite sets.
I will then explain why, for a given valued difference field K with residue field k, the Cavalieri rings of K and k are equal. This will lead to a notion of "rational equivalence" for definable sets in ACFA.
Lorna Gregory (Thursday 3.00-3.45)
Decidability of the theory of modules of a valuation domain via the Ziegler Spectrum
The Ziegler spectrum ZgR of a ring R is a topological space attached to the category of R-modules which encodes large amounts of information about the theory of modules TR. In this talk I will describe how to use the Ziegler spectrum as a tool to prove decidability results for the theory of modules TR. Specifically applying this method to valuation domains, we will indicate how to prove the following theorem: Let V be an effectively given valuation domain. The following are equivalent: 1) The theory of V-modules, TV, is decidable. 2) There exists an algorithm which, given a, b ∈ V, answers whether a ∈ rad(bV).
This theorem was conjectured for valuation domains with dense value group by Puninski, Puninskaya and Toffolori in "Decidability of the theory of modules over commutative valuation domains" (2007).
Simon Iosti (Friday 10.30-11.00)
The Tannakian formalism from the model-theoretic point of view
The aim of this lecture is to present (and comment) a version of the following theorem : "Any algebraic group defined over a field K can be reconstructed from the category of its finite-dimensional linear representations over K, together with the functor 'forgetting of the representation' with values into the category of finite dimensional K-vector spaces." I will present a quite general approach, developed by Yves André, consisting in replacing K by a (generalized) differential ring, and permitting to prove this theorem in particular for differential fields. A model-theoretic proof due to Moshe Kamensky, based on a result of elimination of imaginaries and on the model-theoretic notion of internality (and its links with the binding groups and groupoids, abstract version of Galois groups and groupoids), will be presented.
This theorem permits in particular to study some interdefinability questions between a field (a ring?) and definable groups in this field.
Noa Lavi (Thursday 10.00-10.30) (slides)
Some positivstellensatze in real closed valued fields
The purpose of this talk is to give a generalization of Hilbert's seventeenth problem in real closed valued fields, that is, to give an algebraic characterization, for a definable set, of the set of polynomials which get only non-negative values on it. We give a general characterization of the positive semi-definite polynomials for any definable set with a Ganzstellensatz, and we also give a representation of those polynomials in the sense of Hilbert 17th problem (that is, in terms of sums of squares) for definable sets from a certain kind.
Vincenco Mantova (Friday 4.30-5.00)
About Zilber's fields
Zilber's fields are imaginative algebraic structures which mimic the structure of the complex field equipped with the exponential function, but whose model-theoretic properties are well-known and rather well-behaved. Indeed, provided that we use a suitable infinitary language, their theory is axiomatizable, uncountably categorical and quasi-minimal. It is an open question if analogous properties hold for the classical complex exponential field, and if the complex field itself is just an example of Zilber's field. I will review some basic motivations for the use of Schanuel's Conjecture and existential closure in the definition of Zilber's fields, and I will show an easy way to explicitly build pseudoexponential functions which satisfy the axioms of Zilber's fields.
Daniel Palacin (Thursday, 10.30-11.00) (slides)
A new result on elimination of hyperimaginaries
A simple CM-trivial theory eliminates hyperimaginaries whenever it eliminates finitary ones. In particular, a small simple CM-trivial theory eliminates hyperimaginaries. Also, we will make some remarks on different levels of elimination of hyperimaginaries.
Tristram de Piro (Friday 4.00-4.30)
On a conjecture of Severi's
Using uniformities in Newton's theorem, we show how to correct the erroneous step in Severi's proof of his conjecture that the variety parametrising curves with a given degree, and a given number of nodes, is irreducible.
Pierre Simon (Wednesday 3.00-3.45)
Distal NIP theories
NIP theories are often thought of as being built, in some vague sense, of stable and ordered pieces. One side of the picture, the stable part, is well understood. We are interested in the other extreme. I will present a notion of pure instability called "distality" and give equivalent definitions. The class of distal theories includes for example o-minimal theories and the p-adics. I will also briefly explain how in non-distal theories one can pick out the stable part of types.
Existential definability in t-henselian fields.
Prestel and Ziegler proved a field is t-henselian if and only if it satisfies the implicit function theorem. We use this characterisation to give a local picture of existentially definable sets in t-henselian fields K. Let X be an existentially definable subset of K. If the parameters come from the perfect core Kp∞ then X contains the Frobenius image of a ball; but it may not, otherwise. We draw various conclusions relating to existential definability in t-henselian fields. Using Pop's theorem that large fields are existentially closed in a certain henselian field, we extend our results to large fields. Throughout, the difficulties are mainly due to imperfection.
The ℵ0-Categorical Trees
This poster outlines the description of the ℵ0-categorical linear orders given by Rosenstein and shows how this result can be used to classify the aleph-null categorical semilinear orders.
Jordan groups and homogeneous structures
Real Closed Rings
The Absolute Galois group of a Function Field
The absolute Galois group of a field K contains a lot of information about K, but it can never determine the field up to isomorphism. Here we consider a richer structure, namely the isomorphism type of the canonical projection pr: GK(t) → GK. We outline that this sequence encodes the field K up to isomorphism for a large class of fields. The conjecture is that one can extend this result and replace K(t) by any function field in one variable over K.
Generic automophisms of abelian groups
Andres Aranda López
Homogeneous structures with simple theory
Type-definable Subgroups of the Multiplicative Group in SCF
Completeness of Projective Varieties via Positive Quantifier Elimination (pdf)