We construct uncountably categorical 3-nilpotent groups of exponent p > 3. They are not one-based and do not allow the interpretation of an infinite field. Therefore they are counterexamples to Zilbers Conjecture. First 2-nilpotent new uncoutably categorical groups were contructed in [3]. Here we use the method of the additive Collapse developed in [5]. Essentially we work with 3-nilpotent graded Lie algebras over the field with p elements.

We study the model theory of countable right-angled buildings with infinite residues. For every Coxeter graph we obtain a complete theory with a natural axiomatisation, which is $\omega$-stable and equational. Furthermore, we provide sharp lower and upper bounds for its degree of ampleness, computed exclusively in terms of the associated Coxeter graph. This generalises and provides an alternative treatment of the free pseudospace.

Since their inception, the Perspectives in Logic and Lecture Notes in Logic series have published seminal works by leading logicians. Many of the original books in the series have been unavailable for years, but they are now in print once again. This volume, the eighth publication in the Perspectives in Logic series, brings together several directions of work in model theory between the late 1950s and early 1980s. It contains expository papers by pre-eminent researchers. Part I provides an introduction to the subject as a whole, as well as to the basic theory and examples. The rest of the book addresses finitary languages with additional quantifiers, infinitary languages, second-order logic, logics of topology and analysis, and advanced topics in abstract model theory. Many chapters can be read independently.

Abstract We show that the class of graded c-nilpotent Lie algebras over a fixed field K is closed under free amalgamation. In [1] this result was applied, but its proof was incorrect. In case of a finite field K we obtain a Fraïssé limit of all finite graded c-nilpotent Lie algebras over K. This gives an example for the following more general considerations. The existence of free amalgamation for the age of a Fraïssé limit implies the universality of its automorphism group for all automorphism groups of substructures of that Fraïssé limit. We use [6] and [5].

Let \({L(n)}\) be the language of group theory with n additional new constant symbols \({c_1,\ldots,c_n}\). In \({L(n)}\) we consider the class \({{\mathbb{K}}(n)}\) of all finite groups G of exponent \({p > 2}\), where \({G'\subseteq\langle c_1^G,\ldots,c_n^G\rangle \subseteq Z(G)}\) and \({c_1^G,\ldots,c_n^G}\) are linearly independent. Using amalgamation we show the existence of Fraïssé limits \({D(n)}\) of \({{\mathbb{K}}(n)}\). \({D(1)}\) is Felgner’s extra special p-group. The elementary theories of the \({D(n)}\) are supersimple of SU-rank 1. They have the independence property.

The ample hierarchy of geometries of stables theories is strict. We generalise the construction of the free pseudospace to higher dimensions and show that the n-dimensional free pseudospace is ω-stable n-ample yet not (n+1)-ample. In particular, the free pseudospace is not 3-ample. A thorough study of forking is conducted and an explicit description of canonical bases is exhibited.

We amalgamate finite 2-nilpotent groups G of exponent p > 2, where G' is contained in a subgroup of the center, generated by n elements. We get Fraisse limits D(n) with superstable elementary theories of SU-rank 1. D(1) is Felgner's extra special p-group.

The general topic of the meeting was "Valued fields and related structures". It included both applications of model theory, as well as so-called "pure" model theory: the classification of first order structures using new techniques extending those developed in stable theories.

We construct a. bad field in characteristic zero. That is, we construct an algebraically closed held which carries a notion of dimension analogous to Zariski-dimension, with an infinite proper multiplicative subgroup of dimension one, and such that the field itself has dimension two. This answers a longstanding open question by Zilber.

Summary. From known examples of theories T obtained by Hrushovski-constructions and of infinite Morley rank, properties are extracted, that allow the collapse to a finite rank substructure. The results are used to give a more model-theoretic proof of the existence of the new uncountably categorical groups in [3].

Abstract We apply Hrushovski-Fraïsseé's amalgamation procedure to obtain a theory of fields of prime characteristic of Morley rank 2 equipped with a definable additive subgroup of rank 1.

We present a detailed and simplified exposition of Hrushovki’s fusion of two strongly minimal theories.

ZusammenfassungWir konstruieren einen schlechten Körper der Charakteristik Null. Mit anderen Worten, wir konstruieren einen algebraisch abgeschlossenen Körper mit einem Dimensionsbegriff analog der Zariski-Dimension, zusammen mit einer unendlichen echten multiplikativen Untergruppe der Dimension Eins, so daβ der Körper selbst Dimension Zwei hat. Dies beantwortet eine alte Frage von Zilber.

The workshop Model Theory and Groups , organised by Andreas Baudisch (Berlin), David Marker (Chicago), Katrin Tent (Bielefeld) and Frank Wagner (Lyon) was held January 14th–20th, 2007. This meeting focused on interactions between classical model theoretic investigations of groups and their applications to geometric group theory and vice versa. It was well attended with 55 scientists, both model theorists as well as geometric group theorists, including 11 women and a relatively large number of young researchers and students. Needless to say that participants came from a broad geographical background. For many years groups have played a central role in model theory, both in applied model theory where one is focused on understanding algebraic structures and, more surprisingly, in pure model theory where one is studying structures from an abstract viewpoint. At first, only the most basic tools from the general theory were needed in applications, but, over the last ten years, some of the most sophisticated ideas from pure model theory have played an important role in applications, most notably Hrushovski's proof of the Mordell–Lang Conjecture for function fields. The investigation of variations of Mordell–Lang like theorems in different situations played an important role in a number of talks. Geometric group theory and model theory have started interacting in the context of free groups and surface groups as well as in the study of the asymptotic behaviour of geometric properties on groups. This was a second main topic of the conference which particularly profited from the fact that researchers from different areas attended the meeting and presented their results. At the core of model theoretic investigations of groups were the reports on groups of finite Morley rank around the Cherlin–Zilber Conjecture which states that every simple group of finite Morley rank is an algebraic group over an algebraically closed field. While originally attempts at proving this conjecture have followed the lines for the classification of algebraic groups, more recent advances have been made by adapting and generalising ideas from the classification of finite simple groups, in particular the study of the 2-Sylow subgroup, which has allowed a distinction into three cases: even characteristic, odd characteristic (including 0 ) and degenerate (no involutions). The even case is solved, and important progress has been made in the other cases. The recent construction of so-called bad fields, i.e. fields of finite Morley rank with a distinguished multiplicative subgroup also added new impetus to the search for new proofs not involving assumptions on the non-existence of such fields. The organisers asked Dugald Macpherson and Charles Steinhorn before the conference to give a three-lecture tutorial on asymptotic classes and measurable structures. This is a new development in model theory generalising results on finite and pseudofinite fields. In addition, 27 participants were invited to report on their research (18 long and 9 short talks). Altogether it was a very successful workshop which inspired a number of new cooperations and further projects. The reader may find here extended abstracts of all talks (in the order in which the talks were given).

We exhibit a simplified version of the construction of a field of Morley rank p with a predicate of rank p − 1, extracting the main ideas for our construction from previous papers and refining the arguments. Moreover, an explicit axiomatization is given and ranks are computed.

We construct a bad field in characteristic zero. That is, we construct an algebraically closed field which carries a notion of dimension analogous to Zariski-dimension, with an infinite proper multiplicative subgroup of dimension one, and such that the field itself has dimension two. This answers a longstanding open question by Zilber.

We exhibit a simplified version of the construction of a field of Morley rank p with a predicate of rank p-1, extracting the main ideas for the construction from previous papers and refining the arguments. Moreover, an explicit axiomatization is given, and ranks are computed.

Let T1 and T2 be two countable strongly minimal theories with the DMP whose common theory is the theory of vector spaces over a fixed finite field. We show that T1 ∪ T2 has a strongly minimal completion.

The workshop consisted of 2 tutorials of four 1-hours talks each, 10 1-hours talks, and 5 half-hour talks. The tutorials were given by Ya'acov Peterzil and Sergei Strachenko on Complex analytic geometry, an o-minimal viewpoint , and by Boris Zil'ber and Alex Wilkie on Pseudoanalytic structures and Hrushovski's construction . For many years there were two main lines of research in model theory: At first, only the most basic tools from the general theory were needed in applications, but, over the last ten years, some of the most sophisticated ideas from stability theory have played an important role in applications, most notably Hrushovski's proof of the Mordell–Lang Conjecture for function fields. At the same time, these applications have given us new examples of stable structures which have led to new insights in the general theory. We shall briefly describe some of the recent work. Compact Complex Spaces. Zil'ber showed that a compact complex space equipp\-ed with all analytic relations is an \omega -stable structure with quantifier elimination. He and Hrushovski showed that any strongly minimal set definable in these structures is either locally modular or closely related to the field of complex numbers. This type of dichotomy is the fundamental insight in many of the modern applications of model theory. Pillay began the systematic model theoretic study of these structures and was able to show that many interesting model theoretic phenomena arise naturally in this context. For example, simple non-algebraic tori are exactly the locally modular groups. In addition to giving us new examples of locally modular strongly minimal sets, this result led Pillay to a model theoretic methods to extend Falting's theorem to a proof of Mordell–Lang Conjecture for complex tori. Pillay, in collaboration with Scanlon and Kowalski, have carried on a detailed model theoretic analysis of the groups definable in compact complex spaces, their results extend and generalize Fujiki's work on meromorphic groups. A highlight of this work is Pillay and Scanlon's proof that any meromorphic group is an extension of a complex torus by a linear algebraic group, a generalization of Chevalley's theorem for algebraic groups. Recently Pillay was able to show how results of Campana and Fujiki on cycle spaces leads to a relatively easy proof of the dichotomy theorem for strongly minimal sets. With this as a model he and Ziegler were able to find new proofs of the dichotomy theorem in several other important settings (differential fields, difference fields of characteristic 0) that greatly simplify and offer new insights to some applications of model theory to diophantine geometry. In model theory one often needs to not only understand the structures we are studying but also their nonstandard extensions. While these extensions have no classical analogs, problems about nonstandard extensions often give rise to interesting classical problems about uniformity. An important recent result in this direction is Moosa's proof of the nonstandard Riemann Existence Theorem. Quasi-analytic structures. Zil'ber originally conjectured that the dichotomy property was true for all strongly minimal sets. Hrushovski refuted this by giving a very combinatorial construction of a counterexample. Zil'ber's current research program is designed to show that the type of examples constructed by Hrushovski actually arise naturally. The first major success of this program was recently completed by Koiran, building on work of Wilkie, who showed that one could construct analytic functions f such that the structure (\mathbb C,+,\cdot,f) is isomorphic to an expansion built by a Hrushovski construction. The most intriguing part of this program is Zil'ber's work on pseudoexponentiation. Zil'ber has shown that a Hrushovski style construction can be used to expand the complex field by adding an homomorphism from the additive to multiplicative group with very good model theoretic properties. The proof uses a wide array of ingredients including some diophantine geometry of intersections of varieties in algebraic tori developed by Zil'ber and Shelah's very abstract work on the classification theory of excellent classes. The most remarkable part of this program is Zil'ber's conjecture that the structure he has built is actually the complex field with the usual exponential function. An outright proof of this is unlikely, as it would require strong forms of Schanuel's conjecture, but, if true, this would give us a much better understanding of the model theory of this structure. For example, one could prove that any subset of \mathbb C definable using exponentiation is either countable or co-countable, and show that there are many automorphism. These are two difficult open problems. A related question is whether one can obtain model theoretically interesting new structures on the complex numbers by adding sets defined in tame expansions of the real field. Marker showed that it was impossible to add any new real algebraic structure and Peterzil and Starchenko recently generalized this to show that one cannot add any o-minimal structure. It is still interesting to ask of o-minimal structures have any interesting \omega -stable reducts. In a slightly different direction, Miller and Speissegger have shown that the logarithmic spiral is d-minimal and Zil'ber believes this can be used to obtain some natural models of the theory of bi-colored field first built by Poizat using a Hrushovski construction.

Abstract Let T be a model-complete theory that eliminates the quantifier ∃∞x For T we construct a theory T+ such that any element in a model of T+ determines a model of T. We show that T+ has a model companion T1. We can iterate the construction. The produced theories are investigated.