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Newton Institute, Cambridge

London Mathematical Society

Workshop on Pure Model Theory
University of East Anglia, Norwich, 4-8 July 2005.

Satellite meeting of the Newton Institute Programme
Model Theory and Applications to Algebra and Analysis

Programme

A programme of the meeting is available as a .pdf file here

Abstracts of Talks

Tutorials


Plenary Talks:


Other Talks

Itay Ben-Yaacov
Model theory in positive and continuous logics


Classically, model theory studies elementary classes using full first order logic. I will discuss two alternatives to this setting which have been studied in recent years, allowing to apply model-theoretic methods to a larger family of classes of structures: I will discuss these two frameworks, the relations between them, and some applications and open problems concerning them.

Ludomir Newelski
Multiplicity; Small profinite structures; type-generated equivalence relations


Talk 1. Multiplicity.
In a stable theory, by a multiplicity of a complete type we mean the number of its stationarizations over the monster model. These stationarizations form a closed set. What distinguishes small superstable theories from the $\omega$-stable ones is the presence of types of infinite multiplicity. So it is reasonable to investigate the ways, in which a type can have infinite multiplicity, and in general, the sets of stationarizations of complete types.

A more general notion here is that of a trace of a type $p(x)$ over a set $A$, being just the set of strong types $r(x)$ over $A$ consistent with $p(x)$. Investigation of traces of types is relevant to the research on Vaught's conjecture for superstable theories, but I believe it is also of independent interest. The main tools for this are meager forking, meager types, m-independence and associated multiplicity rank $\cal{M}$.

I proved that if a superstable theory has few countable models, then the $\cal M$-rank of any type is finite and is not larger than its $U$-rank. Also, I gave a fairly good description of m-independence in this case, proving e.g. that it is m-normal, and proved that in this case the trace of any complete type is the trace of some formula. The arguments rely on some topological properties of forking (meagerness). There are some open problems.

Talk 2. Small profinite structures.
Small profinite structures arose as a combinatorial tool to speak of multiplicities and traces of types in small (super)stable theories. Then they became an object of independent interest. Several results and problems on multiplicities of types have their counterparts in the setting of profinite structures. On the other hand, this setting does not require any knowledge of model theory, so it may have a wider audience.

The main open problems in this subject are restatements of the open questions on multiplical properties of small stable theories, but they are also of independent interest.

One such problem is the question, whether every small profinite group is abelian-by-finite. This question is related to my result saying that every profinite group interpretable in a superstable theory with few countable models has an open abelian subgroup. There are some partial results of Wagner and Krupinski here. Wagner proved the $\cal M$-gap comjecture for small profinite groups.

Another result in this area is the counterpart of Hrushovski's group configuration theorem in the case of m-normal profinite structures. We know no small profinite structure which is not m-normal.

I view profinite structures as a useful tool to speak also of simple theories. They may be viewed as tame hyperimaginaries. On the other hand, m-independence is the strongest refinement of forking independence (in a small stable theory).

Talk 3. Type-generated equivalence relations.
Type-generated equivalence relations are transitive closures of type-definable symmetric relations. The most important example are Lascar strong types. Other examples include the coset relations in a group, with respect to a subgroup generated by a type-definable set. In this talk I will speak of my results in this area using a refined topological analysis of the space of types, particularly with respect to its covering by certain sets. In particular, I will speak of the diameter of the Lascar strong type and of related questions. I will also comment on weak generic types in this context. A common idea in all my three talks is that of using topological properties of the space of types.

Bruno Poizat
AMALGAMES/Amalgams


Copies of the transparencies are available as a .pdf file here

John Baldwin
Perspectives on Expansions


we will discuss recent advances in three approaches to studying whether expansions of models preserve stability properties . The first searches for conditions on a set $A$ such that $(M,A)$ also has the desired property. Recent work by Baizhanov-Baldwin extends the unification of the Baldwin-Benedikt and Poizat lines by Casanovas-Ziegler. This is complimented by Polkowska's work on Bounded Pseudoalgebraically closed structures. Secondly, we report on recent advances by Baldwin-Holland and by Laskowski on "for all there exists" axiomatizability of variant on the Hrushovski construction. And finally we look at work studying the infinitary rather than the first order theory of the expansions.

Byunghan Kim
Generalized type-amalgamation and n-simplicity I


It is recently revealed by de Piro, Kim and Young that the key property of getting the canonical hyperdefinable group from the group configuration is Hrushovski's P^-(4)-amalgamation(= 4-amalgamation). Then we (Kim, Kolesnikov and Tsuboi) study the notion of higher amalgamation. Indeed, Kolesnikov's ideas in his thesis go thru in the n-amalgamation context with the corresponding changes of definitions. Hence it is conjectured, analogously to the one in the thesis, that K(n)-simplicity (defined via infinite indiscernible sequences) implies (n+2)-amalgamation. Case n=1 is the Independence Theorem, and case n=2 is done by Kolesnikov. (He also proves the conjecture under an assumption L_{n+1}.) However for each n> 2, we build a counterexample, a generic model of a certain class of finite graphs. Accordingly the notion is modified to `n-simplicity' (via finite indiscernible sequences), which is shown to be equivalent to (n+2)-CA(= having k-amalgamation for all k\leq n+2). We also obtain some of positive results under K(n)-simplicity by extending Lascar-Pillay's notion of heir base.

Chris Laskowski
Interpreting groups in strictly stable theories


In joint work with Saharon Shelah we prove a `trichotomy theorem' for strictly stable (i.e., stable, unsuperstable) theories in a countable language. Any such theory either (1) has DOP or (2) has NDOP and is deep or (3) interprets a strictly stable group. This allows us to prove `nonstructure' results for all strictly stable theories by considering each of the three cases separately. As an application of this, we prove that the Karp complexity of the class of models of any strictly stable theory is uncontrolled. Finally, we discuss the connection between these results and an attempt at proving the fabled `Main Gap for aleph_1-saturated models' which is still open.

Kobi Peterzil
Pillay's group conjecture and its solution


Pillay's group conjectures states: Let G be a definably compact group in an o-minimal structure. Then there exists a minimal type-definable subgroup G^00 of G of bounded index, such that the group G/G^00, when endowed with a Logic topology, is isomorphic to a Lie group whose dimension equals the o-minimal dimension of G.

I will discuss the recent solution of this conjecture, for an arbitrary o-minimal expansion of a real closed field, in a joint work with Hrushovski and Pillay. The last component of the proof makes use of connections between theories without the independence property and existence of measures on definable sets, a conncetion which was pointed out by Keisler in his paper "Measure and Forking".

I will also discuss earlier works, leading to that solution, by Berarducci, Otero, Pillay and myself.

Mike Prest
Functorial methods in model theory

In the context of additive definable categories of structures there is a model-theoretic / functor-category-theoretic "dictionary". I will describe this and indicate how it has been used in the, now numerous, applications of model-theoretic methods in module theory.

This all started in the context of modules but has been extended to rather general additive contexts. I will describe some of that and then discuss extension of the link between the category of imaginaries and functor categories to non-additive contexts and consequent links with ideas from accessible categories and topos theory.

Hans Adler
The lattice of algebraically closed sets

I will relate thorn-forking with some basic notions from lattice theory.

Alfred Dolich
On Model Theoretic Properties of Structures Expanding the Real Closed Field

Given a theory T expanding that of real closed fields we consider the influence of various model theoretic properties on the "tameness" of models of T. In particular we consider how close such a theory T having properties such as the exchange property for algebraic closure or various forms of not having the independence property is to being o-minimal.

Marko Djordjevic
The finite submodel property and omega-categorical structures.

I will survey a few partly interconnected results concerning the finite submodel property for omega-categorical structures; a structure M has the finite submodel property if every sentence which is true in M is true in a finite substructure of M.

Assaf Hasson
On the Theory of Envelopes and Homogeneous Approximations.

We consider the following question: given a quasi-finite structure M intepreted in a structure N, find conditions to obtain a family of approximasions N(k) to N such that M(k) is finite (of prescribed size). This question and variants thereof have been central in a number of important results in model theory, such as (to mention just a few) Zilber's proof of the non-finite axiomatizability of totally categorical theories, the Cherlin-Harrington-Lachlan analysis of totally transcendental countably categorical theories and Hrushovski's construction of new uncaountably categorical structrues. In the talk we will present a general framework in which this question can be partially answered.

Martin Hils
Generalised Fusion

We study the possibility of generalising Hrushovski's fusion of strongly minimal theories to the relative context, where the two strongly minimal theories in question share a common reduct which is assumed to be totally categorial. Under mild assumptions - including the case where the common subtheory is an infinite vector space over some finite field - the "free fusion", i.e. the amargamation construction without collapse is omega-stable of rank omega. If the assumptions are weakend we still obtain a supersimple theory of SU-rank omega. In the special case when the initial theories are all locally modular, the collapse onto a strongly minimal theory can be done (this is joint work with Assaf Hasson).

Alexei Kolesnikov
Generalized type-amalgamation and n-simplicity II

The study of generalized type amalgamation properties in the context of simple theories was started in the speaker's Ph.D. thesis, inspired by the results of Hrushovski on ACFA and Shelah's work on excellent classes. In general, some simple theories fail to have generalized amalgamation, and this leads to a hierarchy of "simplicities". It became clear soon that there are several ways in which one can define the amalgamation properties, and this resulted in several different higher simplicity notions. This is a continuation of the talk by Byunghan Kim. I will say more about the connection between the properties L_n; the generalized amalgamation properties; and the extension of the their base notion.

Alf Onshuus
th-forking

In this talk we will talk about a notion of independence (th-independence). This notion was defined by Thomas Scanlon and has been studied together with Clifton Ealy and Thomas Scanlon during the last few years. In this talk we will state some of the known facts about th-independence, concentrating particularly in giving an intuition for the definition, together with results that prove that th-independence is ``universal'' in some sense (which will become clear from the talk).

Nick Peatfield
Generic functions and analytic Zariski structures

Work of Wilkie and Koiran has shown that the complete theory of a Liouville function provides a genuine complex analytic example of a Hrushovski-type theory, supporting Zilber's idea that some of these theories may come from analysis. Wilkie's functions have the same theory as that of an abstract generic function (without derivatives) as introduced by Zilber. But the models of this theory do not have all the properties required of an "analytic Zariski structure" - there are not enough closed sets for the notion of irreducibility to make sense. There may be a somewhat trivial way to enrich the language in order to get these properties, following earlier work by the speaker and Zilber, but the basic irreducible closed sets produced by this method do not agree with the complex analytic propotype. In this talk we will introduce a new language for the theory of a generic function with with derivatives with more closed (positive definable) sets, which supports a "realistic" analytic Zariski structure, and explain in what way it is more realistic. We will show that the topology we obtain from this language is only a slight refinement (by zero dimensional sets) of the language introduced by Zilber to describe how the derivatives of the generic functions are related to one another. Details can be found here.

Ziv Shami
Unidimensionality and the forking topology

Some properties of the forking topology in unidimensional simple theories and more general contexts will be discussed.

Alex Usvyatsov
Dependent theories

I will discuss recents developments in the study of dependent theories (i.e. theories without the independence property), mostly due to Shelah. I intend to focus on the theorem saying that given a dependent theory T and its model M, one can expand the language by predicates for externally definable sets in M (e.g. making certain types over M definable) and obtain again a dependent theory T' (and if T has QE, so does T'). The relevant reference for this result is [Sh783].

Alex Usvyatsov
Local stability in continuous logic

Developing local (formula-by-formula) stability for metric structures was one of the motivations for introducing continuous logic. Local stability provides an alternative approach to studying stable continuous theories and stable Hausdorff cats, it generalizes Krivine and Maurey's work on stable functions on Banach spaces and Iovino's study of stable Banach structures. This is a joint work with Itay Ben-Yaacov.

Evgueni Vassiliev
Weak nfcp and related conditions in simple theories

The weak non-finite cover property (wnfcp) is a "simple analogue" of the non-finite cover property, and is equivalent to "axiomatizability" of lovely pairs of models of a simple theory. We will talk about the question of preservation of the wnfcp when passing to the lovely pairs expansion, and its connection with n-tuples of models of a simple theory. We will also discuss the condition of weak lowness, introduced in a joint work with Anand Pillay.