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A primer of simple theories
 Archive Math. Logic
"... Abstract. We present a selfcontained exposition of the basic aspects of simple theories while developing the fundamentals of forking calculus. We expound also the deeper aspects of S. Shelah’s 1980 paper Simple unstable theories. The concept of weak dividing has been replaced with that of forking. ..."
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Abstract. We present a selfcontained exposition of the basic aspects of simple theories while developing the fundamentals of forking calculus. We expound also the deeper aspects of S. Shelah’s 1980 paper Simple unstable theories. The concept of weak dividing has been replaced with that of forking. The exposition is from a contemporary perspective and takes into account contributions due to S. Buechler, E. Hrushovski, B. Kim, O. Lessmann, S. Shelah and A. Pillay.
Simplicity, And Stability In There
 JOURNAL OF SYMBOLIC LOGIC
, 1999
"... Firstly, in this paper, we prove that the equivalence of simplicity and the symmetry of forking. Secondly, we attempt to recover definability part of stability theory to simplicity theory. In particular, using elimination of hyperimaginaries we prove that for any supersimple T , canonical base of ..."
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Firstly, in this paper, we prove that the equivalence of simplicity and the symmetry of forking. Secondly, we attempt to recover definability part of stability theory to simplicity theory. In particular, using elimination of hyperimaginaries we prove that for any supersimple T , canonical base of an amalgamation class P is the union of names of /definitions of P , / ranging over stationary Lformulas in P . Also, we prove that the same is true with stable formulas for an 1based theory having elimination of hyperimaginaries. For such a theory, the stable forking property holds, too.
Elliptic and Hyperelliptic Curves Over Supersimple Fields
"... We prove that if F is an infinite field with characteristic di#erent from 2, whose theory is supersimple, and C is an elliptic or hyperelliptic curve over F with generic moduli then C has a generic F rational point. The notion of generity here is in the sense of the supersimple field F . ..."
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Cited by 4 (1 self)
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We prove that if F is an infinite field with characteristic di#erent from 2, whose theory is supersimple, and C is an elliptic or hyperelliptic curve over F with generic moduli then C has a generic F rational point. The notion of generity here is in the sense of the supersimple field F .
Dimension and measure in finite first order structures
, 2005
"... The candidate confirms that the work submitted is his own and that appropriate credit has been given where reference has been made to the works of others. This copy has been supplied on the understanding that it is copyright material and that no quotation from the thesis may be published without pro ..."
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Cited by 2 (2 self)
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The candidate confirms that the work submitted is his own and that appropriate credit has been given where reference has been made to the works of others. This copy has been supplied on the understanding that it is copyright material and that no quotation from the thesis may be published without proper acknowledgement. 2Acknowledgments The research undertaken in Chapter 6 was performed jointly by myself and Mark Ryten. My contribution to the work was throughout: in the formation of the initial strategy, in the development of the methods, in the exposition of the proofs, and in their correction and finetuning. Many thanks to my PhDsupervisor Dugald Macpherson for his guidance, assistance, and support. Thanks also to the University of Leeds and especially the staff of the School of Mathematics for providing me with the facilities and support to undertake this research. Thanks to Agatha WalczakTypke for her help with LaTeX.
MAIN GAP FOR LOCALLY SATURATED ELEMENTARY SUBMODELS OF A HOMOGENEOUS STRUCTURE
"... We prove a main gap theorem for locally saturated submodels of a homogeneous structure. We also study the number of locally saturated models, which are not elementarily embeddable into each other. 676 revision:20000413 modified:20000905 Hard experience has indicated that before we speak on this ..."
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We prove a main gap theorem for locally saturated submodels of a homogeneous structure. We also study the number of locally saturated models, which are not elementarily embeddable into each other. 676 revision:20000413 modified:20000905 Hard experience has indicated that before we speak on this particular paper, we should say something on classification theory for nonelementary classes and of the specific context chosen here. Classification theory for first order theories is so established now that many tend to forget that there are other possibilities. There are some good reasons to consider these other possibilities: first, it is better to understand a more general context, we like to classify more; second, concerning applications many classes arising in ’nature ’ are not first order; third, understanding more general contexts may shed light on the first order one. Of course, we may suspect that applying to a wider context will leave us with less content, but only trying will teach us if there are enough interesting things to discover.
Simplicity simplified
 Rev. Colombiana Mat., 41(Numero Especial):263–277
, 2007
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Supersimple omegaCategorical Groups And Theories
"... An #categorical supersimple group is finitebyabelianbyfinite, and has finite SUrank. Every definable subgroup is commensurable with an acl(#) definable subgroup. Every finitely based regular type in a CMtrivial #categorical simple theory is nonorthogonal to a type of SUrank 1. In particu ..."
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An #categorical supersimple group is finitebyabelianbyfinite, and has finite SUrank. Every definable subgroup is commensurable with an acl(#) definable subgroup. Every finitely based regular type in a CMtrivial #categorical simple theory is nonorthogonal to a type of SUrank 1. In particular, a supersimple #categorical CMtrivial theory has finite SUrank.
PERSISTENCE AND NIP IN THE CHARACTERISTIC SEQUENCE
, 908
"... Abstract. For a firstorder formula ϕ(x; y) we introduce and study the characteristic sequence 〈Pn: n < ω 〉 of hypergraphs defined by Pn(y1,...,yn): = (∃x) ∧ i≤n ϕ(x; yi). We show that combinatorial and classification theoretic properties of the characteristic sequence reflect classification theo ..."
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Abstract. For a firstorder formula ϕ(x; y) we introduce and study the characteristic sequence 〈Pn: n < ω 〉 of hypergraphs defined by Pn(y1,...,yn): = (∃x) ∧ i≤n ϕ(x; yi). We show that combinatorial and classification theoretic properties of the characteristic sequence reflect classification theoretic properties of ϕ and vice versa. Specifically, we show that some tree properties are detected by the presence of certain combinatorial configurations in the characteristic sequence while other properties such as instability and the independence property manifest themselves in the persistence of complicated configurations under localization. 1.
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"... The starting point is a question about the structure of Keisler’s order, a preorder on theories which compares the difficulty of producing saturated regular ultrapowers. In Chapter 1 we show that Keisler’s order reduces to the analysis of types in a finite language, i.e. that the combinatorial barr ..."
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The starting point is a question about the structure of Keisler’s order, a preorder on theories which compares the difficulty of producing saturated regular ultrapowers. In Chapter 1 we show that Keisler’s order reduces to the analysis of types in a finite language, i.e. that the combinatorial barriers to saturation are contained in the parameter spaces of the formulas of T. In Chapter 2 we define the characteristic sequence of hypergraphs 〈Pn: n < ω 〉 associated to a formula which describe the relevant incidence relations, and develop a general framework for analyzing the complexity of a formula in terms of the complexity of its characteristic sequence. Specifically, we are interested in analyzing consistent partial types, which correspond to sets A such that An ⊂ Pn for all n. The key issues studied in Chapter 2 are localization and persistence, which describe the difficulty of separating some 2fixed complex configuration from a complete graph under analysis by progressive restrictions of the base set. We characterize stability and simplicity of ϕ in terms of persistence in the characteristic sequence. Chapter 3 restricts attention to the behavior of the graph P2 in the character