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Hrushovski’s Fusion
 Festschrift für Ulrich Felgner zum 65. Geburtstag, volume 4 of Studies in Logic
, 2007
"... We present a detailed and simplified exposition of Hrushovki’s fusion of two strongly minimal theories. 1 ..."
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We present a detailed and simplified exposition of Hrushovki’s fusion of two strongly minimal theories. 1
An exposition of Hrushovski’s New Strongly Minimal Set ∗
, 2013
"... In [5] E. Hrushovski proved the following theorem: Theorem 0.1 (Hrushovski’s New Strongly Minimal Set). There is a strongly minimal theory which is not locally modular but does not interpret an infinite group. This refuted a conjecture of B. Zilber that a strongly minimal theory must either be local ..."
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In [5] E. Hrushovski proved the following theorem: Theorem 0.1 (Hrushovski’s New Strongly Minimal Set). There is a strongly minimal theory which is not locally modular but does not interpret an infinite group. This refuted a conjecture of B. Zilber that a strongly minimal theory must either be locally modular or interpret an infinite field (see [7]). Hrushovski’s method was extended and applied to many other questions, for example to the fusion of two strongly minimal theories ([4]) or recently to the construction of a bad field in [3]. There were also attempts to simplify Hrushovski’s original constructions. For the fusion this was the content of [2]. I tried to give a short account of the New Strongly Minimal Set in a tutorial at the Barcelona Logic Colloquium 2011. The present article is a slightly expanded version of that talk. 1 Strongly minimal theories An infinite Lstructure M is minimal if every definable subset of M is either finite or cofinite. A complete Ltheory T is strongly minimal if all its models are minimal. There are three typical examples: • Infinite sets without structure • Infinite vector spaces over a finite field • Algebraically closed fields The algebraic closure acl(A) of a subset A of M is the union of all finite Adefinable subsets. In algebraically closed fields this coincides with the fieldtheoretic algebraic closure. In minimal structures acl has a special property: Lemma 1.1. In a minimal structure acl defines a pregeometry.
Groups of finite dimension in model theory
, 2007
"... The Morley rank is the usual notion of dimension in model theory which encapsulates the more classical notion of dimension of algebraic varieties in algebraic geometry. In this paper we give a survey of results concerning the classification of infinite simple groups of finite Morley rank. We emphasi ..."
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The Morley rank is the usual notion of dimension in model theory which encapsulates the more classical notion of dimension of algebraic varieties in algebraic geometry. In this paper we give a survey of results concerning the classification of infinite simple groups of finite Morley rank. We emphasize both the developments parallel to the Classification of the Finite Simple Groups and the increasing developments of the subject in infinite combinatorics. Since the late sixties, firstorder model theory has become more and more involved with “concrete ” mathematics. The reason is that firstorder properties, despite their weaknesses, mesh well with combinatorics and algebra in general and explain many phenomena at a high level of abstraction. The first theorem bringing model theory to the most modern considerations in algebra and geometry is certainly Morley’s theorem on the categoricity in any uncountable cardinal of firstorder theories categorical in one uncountable cardinal [Mor65]. From this point on, the classification theory by Shelah provided a very clear picture of the firstorder complexity of mathematical structures, with a substansive study of the case of stable theories [She90]. This has had important applications to diophantine problems, and the abstract stability theory also has nowadays generalizations to instable contexts, with developments on simple theories on the one hand and on ominimality and dependent theories on the other. There are however many open questions left in the stable case, notably questions concerning the existence of certain structures and their exact ranking in the stability hierarchy. Such questions arise in particular for structures with an algebraic flavor, especially for groups, even at the very bottom of the stability hierarchy. Here the socalled Morley rank, the usual modeltheoretic notion of dimension which encapsulates the dimension of varieties in algebraic geometry, is finite. The first theorems describing properties of certain algebraic structures satisfying modeltheoretic constraints of that nature were due to Macintyre in the very early seventies. For example, an infinite field of finite Morley rank is algebraically closed [Mac71]. It was also shown by Reineke that a group of Morley rank 1 must be abelianbyfinite [Rei75].
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"... Abstract. With any connected affine algebraic group G over an algebraically closed field K of characteristic zero, we associate another connected affine algebraic group D over K and a finite central subgroup F of D such that, up to isomorphism of algebraic groups, affine algebraic groups over K abst ..."
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Abstract. With any connected affine algebraic group G over an algebraically closed field K of characteristic zero, we associate another connected affine algebraic group D over K and a finite central subgroup F of D such that, up to isomorphism of algebraic groups, affine algebraic groups over K abstractly automorphism of D and s is an integer satisfying s = 0 when the derived subgroup of G contains the identity component of the center of G. It follows from the latter that any two abstractly isomorphic connected algebraic groups over K have a common algebraic central extension. The construction of D lies heavily on model theory and groups of finite Morley rank. In particular, it needs to prove that, for any two connected algebraic groups over K, the elementary equivalence of the pure groups implies
Homogeneous structures, ωcategoricity and amalgamation constructions ∗
"... The main purpose of these lectures is to give an exposition of some basic material on homogeneous structures, ωcategorical structures and their automorphism groups. There is nothing new in the the talks: most of what is in the first two sessions will be familiar to anyone who has done any work in ..."
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The main purpose of these lectures is to give an exposition of some basic material on homogeneous structures, ωcategorical structures and their automorphism groups. There is nothing new in the the talks: most of what is in the first two sessions will be familiar to anyone who has done any work in the area; the third session on Hrushovski’s predimension construction is a bit more specialised. The plan of the lectures is: 1. Homogeneous structures, Fräıssé’s theorem and examples; ωcategoricity, the RyllNardzewski Theorem, more examples. 2. Automorphism groups as topological groups; imaginaries and biinterpretability for ωcategorical structures. 3. Generalizations of the Fräısse ́ construction. Hrushovski’s predimension construction and amalgamation. Using the Hrushovski construction to produce ωcategorical structures. These notes are rather casual about attribution of results: the references indicated below provide more background information and detail. In many places, there are strong similarities to the notes of Macpherson [23]. General background on model theory can be found in standard texts such as [25], [13] or [27]. A short appendix (Section 4, essentially reproduced from [11]) covers some of the basics. Introductory material on ωcategoricity can be found in the introduction to [19] (and many other places), and the book [6] focuses on the connections with permutation groups. The paper [10] gives a survey of constructions of ωcategorical structures, including the examples described here. Macpherson’s MALOA lectures [23], and the paper [24], give an extensive survey of work on homogeneous structures and their automorphism groups. The introduction to [7] surveys work on classification of homogeneous structures.
SPLITTING IN SOLVABLE GROUPS OF FINITE MORLEY RANK
"... Abstract. We exhibit counterexamples to a Conjecture of Nesin, since we build a connected solvable group with finite center and of finite Morley rank in which no normal nilpotent subgroup has a nilpotent complement. The main result says that each centerless connected solvable group G of finite Morle ..."
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Abstract. We exhibit counterexamples to a Conjecture of Nesin, since we build a connected solvable group with finite center and of finite Morley rank in which no normal nilpotent subgroup has a nilpotent complement. The main result says that each centerless connected solvable group G of finite Morley has a normal nilpotent subgroup U and an abelian subgroup T such that G = U ⋊ T, if and only if, for any field K of finite Morley rank, the connected definable subgroups of K ∗ are pseudotori. Also we build a centerless connected solvable group G of finite Morley rank with no definable representation over a direct sum of interpretable fields. 1.
RELATIVE GEOMETRIC CONFIGURATIONS
, 2009
"... This is a survey of a recent work done by the three authors, in which an analysis of geometric properties of a structure relative to a reduct is initiated. In particular, definable groups and fields in this context are considered. In a relatively 1based theory every group is definably isogenous to ..."
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This is a survey of a recent work done by the three authors, in which an analysis of geometric properties of a structure relative to a reduct is initiated. In particular, definable groups and fields in this context are considered. In a relatively 1based theory every group is definably isogenous to a subgroup of a group definable in the reduct. For relatively CMtrivial theories (which encompass certain Hrushovski’s amalgams, such as the fusion of two strongly minimal theories or coloured fields), we prove that every group can be mapped by a homomorphism with central kernel to a group definable in the reduct.