One of the questions frequently asked nowadays about model theory is whether it is still logic. The reason for asking the question is mainly that more and more of model theoretic research focuses on concrete mathematical fields, uses extensively their tools and attacks their inner problems. Nevertheless the logical roots in the case of model theoretic geometric stability theory are not only clear but also remain very important in all its applications. This line of research started with the notion of a κ-categorical first order theory, which quite soon mutated into the more algebraic and less logical notion of a κ-categorical structure. A structure M in a first order language L is said to be categorical in cardinality κ if there is exactly one, up to isomorphism, structure of cardinality κ satisfying the L-theory of M. In other words, if we add to Th(M) the (non first-order) statement that the cardinality of the domain of the structure is κ, the description becomes categorical. The principal breakthrough, in the mid-sixties, from which stability theory started

The open core of an expansion of a dense linear order is its reduct, in the sense of definability, generated by the collection of all of its open definable sets. In this paper, expansions of dense linear orders that have o-minimal open core are investigated, with emphasis on expansions of densely ordered groups. The first main result establishes conditions under which an expansion of a densely ordered group has an o-minimal open core. Specifically, the following is proved: Let R be an expansion of a densely ordered group (R, <, ∗) that is definably complete and satisfies the uniform finiteness property. Then the open core of R is o-minimal. Two examples of classes of structures that are not o-minimal yet have o-minimal open core are discussed: dense pairs of o-minimal expansions of ordered groups, and expansions of o-minimal structures by generic predicates. In particular, such structures have open core interdefinable with the original o-minimal structure. These examples are differentiated by the existence of definable unary functions whose graphs are dense in the plane, a phenomenon that can occur in dense pairs but not in expansions by generic predicates. The property of having no dense graphs is examined and related to uniform finiteness, definable completeness, and having o-minimal open core.

The four authors present their speculations about the future developments of mathematical logic in the twenty-first century. The areas of recursion theory, proof theory and logic for computer science, model theory, and set theory are discussed independently.

We study forking, Lascar strong types, Keisler measures and definable groups, under an assumption of NIP (not the independence property), continuing aspects of the paper [13]. Among key results are (i) if p = tp(b/A) does not fork over A then the Lascar strong type of b over A coincides with the compact strong type of b over A and any global nonforking extension of p is Borel definable over bdd(A), (ii) analogous statements for Keisler measures and definable groups, including the fact that G 000 = G 00 for G definably amenable, (iii) definitions, characterizations and properties of “generically stable ” types and groups, (iv) uniqueness of invariant (under the group action) Keisler measures on groups with finitely satisfiable generics, (v) a proof of the compact domination conjecture for (definably compact) commutative groups in o-minimal expansions of real closed fields. 1 Introduction and

Abstract. We use a new construction of an o-minimal structure, due to Lipshitz and Robinson, to answer a question of van den Dries regarding the relationship between arbitrary o-minimal expansions of real closed fields and structures over the real numbers. We write a first order sentence which is true in the Lipshitz-Robinson structure but fails in any possible interpretation over the field of real numbers. An o-minimal structure is by definition an expansion M of a linear ordering, such that every definable subset of the linear ordering is a finite union of intervals whose end points are in M ∪ {±∞}. Although o-minimal expansions of discrete linear orderings do exist (e.g. 〈Z, <, z ↦ → z + 1〉), these were recognized early on to have a relatively poor structure and therefore, in the above definition, one often assumes that the linear ordering is dense without endpoints. As was shown in [5], an o-minimal structures is, at least locally,

Introduction In this paper we develop a structure theory for transitive permutation groups definable in o-minimal structures. We fix an o-minimal structure M, a group G definable in M, and a set\Omega and an action of G on\Omega definable in M, and talk of the permutation group (G; \Omega\Gamma/ Often, we are concerned with definably primitive permutation groups (G; \Omega\Gamma/ this means that there is no proper non-trivial definable G-invariant equivalence relation on \Omega\Gamma so it is equivalent to a point stabiliser G ff being a maximal definable subgroup of G. Of course, since any group definable in an o-minimal structure has the descending chain condition on definable subgroups [20] we expect many questions on definable transitive permutation groups to reduce to questi

Abstract. We study the theory of Lovely pairs of þ-rank one theories, in particular O-minimal theories. We show that the class of ℵ0-saturated dense pairs of O-minimal structures studied by van den Dries [6] agrees with the corresponding class of lovely pairs. We also prove that the theory of lovely pairs of O-minimal structures is super-rosy of rank ≤ ω. 1.

The article surveys some topics related to o-minimality, and is based on three lectures. The emphasis is on o-minimality as an analogue of strong minimality, rather than as a setting for the model theory of expansions of the reals. Section 2 gives some basics (the Monotonicity and Cell Decomposition Theorems) together with a discussion of dimension. Section 3 concerns the Peterzil–Starchenko Trichotomy Theorem (an o-minimal analogue of Zil’ber Trichotomy). There follows some material on definable groups, with powerful applications of the Trichotomy Theorem in work by Peterzil, Pillay and Starchenko. The final section introduces weak o-minimality, P-minimality, and C-minimality. These are analogues of o-minimality intended as settings for certain henselian valued fields with extra structure.

Introduction. By and large, definitions of a di#erentiable structure on a set involve two ingredients, topology and algebra. However, in some cases, partial information on one or both of these is su#cient. A very simple example is that of the field R (or any real closed field) where algebra alone determines the ordering and hence the topology of the field: a < b # #x # R (x #= 0 & x 2 = b - a). In the case of the field C, the algebraic structure is insu#cient to determine the Euclidean topology; another topology, Zariski, is associated with the field but this will be too coarse to give a di#erentiable structure. A celebrated example of how partial algebraic and topological data (G a locally euclidean group) determines a di#erentiable structure (G is a Lie group) is Hilbert

were written after the lecture, but have not been carefully polished. The bibliography is not carefully put together. For an old but wonderful introduction to stable group theory, see [13]. For an update, often with results stated in great generality, see [15]. On groups of finite Morley rank, [2] gives the state of play prior to serious intrusion of methods from the classification of finite simple groups, and [1] gives close-to-current information. 1 Motivation for stable group theory Consider an algebraically closed field K, and a linear algebraic group G(K) ≤ GLn(K). This is a subgroup of GLn(K) defined by the vanishing of some polynomials over K. Recall that the case when G(K) is simple is classified – that is, the simple algebraic groups over K are classified, via their root