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45
On groups and rings definable in ominimal expansions of real closed fields
 Bull. London Math. Soc
, 1996
"... Let </?, <, +,> be a real closed field, and let M be an ominimal expansion of R. We prove here several results regarding rings and groups which are definable in Jt. We show that every ^definable ring without zero divisors is definably isomorphic to R, R(V(—l)) or the ring of quaternions o ..."
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Cited by 20 (6 self)
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Let </?, <, +,> be a real closed field, and let M be an ominimal expansion of R. We prove here several results regarding rings and groups which are definable in Jt. We show that every ^definable ring without zero divisors is definably isomorphic to R, R(V(—l)) or the ring of quaternions over R. One corollary is that no model of 7^xp is interpretable in a model of 7^,. 1.
The theory of exponential differential equations
 Ph. D. thesis
, 2006
"... To the memory of my mother, who always encouraged my interest in mathematics but did not quite live to read these words, and to my father, who has always been wonderfully supportive. Acknowledgements My supervisor, Boris Zilber, has been a marvellous source of ideas and inspiration. It is a great pl ..."
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Cited by 16 (4 self)
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To the memory of my mother, who always encouraged my interest in mathematics but did not quite live to read these words, and to my father, who has always been wonderfully supportive. Acknowledgements My supervisor, Boris Zilber, has been a marvellous source of ideas and inspiration. It is a great pleasure for me to be able to confirm and build on his predictions in this thesis. Over the course of the DPhil I have had helpful conversations with many
Die böse Farbe
, 2007
"... We construct a bad field in characteristic zero. That is, we construct an algebraically closed field which carries a notion of dimension analogous to Zariskidimension, with an infinite proper multiplicative subgroup of dimension one, and such that the field itself has dimension two. This answers a ..."
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Cited by 13 (6 self)
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We construct a bad field in characteristic zero. That is, we construct an algebraically closed field which carries a notion of dimension analogous to Zariskidimension, 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.
Constructing ωstable Structures: Rank 2 fields
, 2001
"... We provide a general framework for studying the expansion of strongly minimal sets by adding additional relations in the style of Hrushovski. We introduce a notion of separation of quantifiers which is a condition on the class of expansions of finitely generated models for the expanded theory to ha ..."
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Cited by 12 (4 self)
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We provide a general framework for studying the expansion of strongly minimal sets by adding additional relations in the style of Hrushovski. We introduce a notion of separation of quantifiers which is a condition on the class of expansions of finitely generated models for the expanded theory to have a countable ωsaturated model. We apply these results to construct for each sufficiently fast growing finitetoone function µ from `primitive extensions ' to the natural numbers a theory T^µ of an expansion of an algebraically closed field which has Morley rank 2. Finally, we show that if µ is not finitetoone the theory may not be ωstable.
The Definable Multiplicity Property and Generic Automorphisms
 Annals of Pure and Applied Logic
, 2000
"... Let T be a strongly minimal theory with quantifier elimination. We show that the class of existentially closed models of T # {"# is an automorphism"} is an elementary class if and only if T has the definable multiplicity property, as long as T is a finite cover of a strongly minimal ..."
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Cited by 9 (3 self)
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Let T be a strongly minimal theory with quantifier elimination. We show that the class of existentially closed models of T # {"# is an automorphism"} is an elementary class if and only if T has the definable multiplicity property, as long as T is a finite cover of a strongly minimal theory which does have the definable multiplicity property. We obtain cleaner results working with several automorphisms, and prove: the class of existentially closed models of T # {"# i is an automorphism " : i = 1, 2} is an elementary class if and only if T has the definable multiplicity property. 1 Introduction Given a complete theory T with quantifier elimination in a language L, we consider the (incomplete) theory T # = T # {"# is an automorphism"} in the language L # {#}. For M a model of T , and # # Aut(M) we call # a # Supported by a grant from Tokai University + Supported by an NSF grant 1 generic automorphism of M if (M, #) is an existentially closed model of T # . A general...
Fusion over a vector space
, 2006
"... 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. ..."
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Cited by 8 (4 self)
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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.
Model Theory of Finite Difference Fields and Simple Groups
, 2007
"... The candidate confirms that the work submitted is his own and that appropriate credit has been given where reference has been made to the work 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 prop ..."
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Cited by 6 (0 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 work 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. 2 Asymptotic classes are classes of finite structures which have uniformly definable estimates for the cardinalities of their firstorder definable sets akin to those in finite fields given by the LangWeil estimates. The goal of the thesis is to prove that the finite simple groups of a fixed Lie type and Lie rank form asymptotic classes. This requires the following: 1. The introduction describes the background. 2. Chapter 4 shows a general method of generating one asymptotic class of structures from another through the notion of biinterpretability. Specifically, the notions