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35
The MordellLang conjecture for function fields
 J. Amer. Math. Soc
, 1996
"... In [La65], Lang formulated a hypothesis including as special cases the Mordell conjecture concerning rational points on curves, and the ManinMumford conjecture on torsion points of Abelian varieties. Sometimes generalized to semiAbelian varieties, and to positive characteristic, this has been call ..."
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Cited by 63 (2 self)
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In [La65], Lang formulated a hypothesis including as special cases the Mordell conjecture concerning rational points on curves, and the ManinMumford conjecture on torsion points of Abelian varieties. Sometimes generalized to semiAbelian varieties, and to positive characteristic, this has been called the MordellLang conjecture;
SIMPLE HOMOGENEOUS MODELS
, 2002
"... Geometrical stability theory is a powerful set of modeltheoretic tools that can lead to structural results on models of a simple firstorder theory. Typical results offer a characterization of the groups definable in a model of the theory. The work is carried out in a universal domain of the theor ..."
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Cited by 16 (2 self)
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Geometrical stability theory is a powerful set of modeltheoretic tools that can lead to structural results on models of a simple firstorder theory. Typical results offer a characterization of the groups definable in a model of the theory. The work is carried out in a universal domain of the theory (a saturated model) in which the Stone space topology on ultrafilters of definable relations is compact. Here we operate in the more general setting of homogeneous models, which typically have noncompact Stone topologies. A structure M equipped with a class of finitary relations R is strongly λ−homogeneous if orbits under automorphisms of (M, R) have finite character in the following sense: Given α an ordinal < λ ≤ M  and sequences ā = { ai: i < α}, ¯ b = { bi: i < α} from M, if (ai1,..., ain) and (bi1,..., bin) have the same orbit, for all n and i1 < · · · < in < α, then f(ā) = ¯ b for some automorphism f of (M, R). In this paper strongly λ−homogeneous models (M, R) in which the elements of R induce a symmetric and transitive notion of independence with bounded character are studied. This notion of independence, defined using a combinatorial condition called “dividing”, agrees with forking independence when (M, R) is saturated. The concept central to the development of geometrical stability theory for saturated structures, namely the canonical base, is also shown to exist in this setting. These results broaden the scope of the
Analytic and pseudoanalytic structures
 Proc. Logic Colloquium
, 2000
"... 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. Neverthe ..."
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Cited by 9 (0 self)
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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 κcategorical first order theory, which quite soon mutated into the more algebraic and less logical notion of a κcategorical 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 Ltheory of M. In other words, if we add to Th(M) the (non firstorder) statement that the cardinality of the domain of the structure is κ, the description becomes categorical. The principal breakthrough, in the midsixties, from which stability theory started
Differential arcs and regular types in differential fields
 J. REINE ANGEW. MATH
, 2007
"... We introduce differential arc spaces in analogy to the algebraic arc spaces and show that a differential variety in characteristic zero is determined by its arcs at a point. Using differential arcs, we show that if (K, +, ×, δ1,..., δn) is a differentially closed field of characteristic zero with n ..."
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Cited by 8 (3 self)
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We introduce differential arc spaces in analogy to the algebraic arc spaces and show that a differential variety in characteristic zero is determined by its arcs at a point. Using differential arcs, we show that if (K, +, ×, δ1,..., δn) is a differentially closed field of characteristic zero with n commuting derivations and p ∈ S(K) is a regular type over K, then either p is locally modular or there is a definable subgroup G ≤ (K, +) of the additive group having a regular generic type that is nonorthogonal to p.
Prospects for mathematical logic in the twentyfirst century
 BULLETIN OF SYMBOLIC LOGIC
, 2002
"... The four authors present their speculations about the future developments of mathematical logic in the twentyfirst century. The areas of recursion theory, proof theory and logic for computer science, model theory, and set theory are discussed independently. ..."
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Cited by 8 (0 self)
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The four authors present their speculations about the future developments of mathematical logic in the twentyfirst century. The areas of recursion theory, proof theory and logic for computer science, model theory, and set theory are discussed independently.
Meromorphic groups
 Transactions of the American Mathematical Society
"... We introduce the notion of a meromorphic group, weakening somewhat Fujiki’s definition ([4]). We prove that a meromorphic group is meromorphically an extension of a complex torus by a linear algebraic group, generalizing results in [4]. A special case of this result, as well as one of the ingredient ..."
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Cited by 7 (4 self)
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We introduce the notion of a meromorphic group, weakening somewhat Fujiki’s definition ([4]). We prove that a meromorphic group is meromorphically an extension of a complex torus by a linear algebraic group, generalizing results in [4]. A special case of this result, as well as one of the ingredients in the proof, is that a strongly minimal “modular” meromorphic group is a complex torus, answering a question of Hrushovski. As a consequence, we show that a simple compact complex manifold has algebraic and Kummer dimension zero if and only if its generic type is trivial. 1
Compact complex manifolds with the DOP and other properties.
 J. Symbolic Logic
, 2001
"... We point out that a certain complex compact manifold constructed by Lieberman has the dimensional order property, and has Urank di#erent from Morley rank. We also give a su#cient condition for a Kahler manifold to be totally degenerate (that is, to be an indiscernible set, in its canonical lang ..."
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Cited by 6 (5 self)
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We point out that a certain complex compact manifold constructed by Lieberman has the dimensional order property, and has Urank di#erent from Morley rank. We also give a su#cient condition for a Kahler manifold to be totally degenerate (that is, to be an indiscernible set, in its canonical language) and point out that there are K3 surfaces which satisfy these conditions. 1
Diophantine geometry from model theory
 Bull. Symbolic Logic
, 2001
"... $1. Introduction. With Hrushovski's proof of the function field MordellLang conjecture [16] the relevance of geometric stability theory to diophantine geometry first came to light. A gulf between logicians and number theorists allowed for contradictory reactions. It has been asserted that Hrushovski ..."
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Cited by 6 (2 self)
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$1. Introduction. With Hrushovski's proof of the function field MordellLang conjecture [16] the relevance of geometric stability theory to diophantine geometry first came to light. A gulf between logicians and number theorists allowed for contradictory reactions. It has been asserted that Hrushovski's proof was simply an algebraic argument masked in the language of model theory. Another camp held that this theorem was merely a clever oneoff. Still others regarded the argument as magical and asked whether such sorcery could unlock the secrets of a wide coterie of number theoretic problems. In the intervening years each of these prejudices has been revealed as false though such attitudes are still common. The methods pioneered in [16] have been extended and applied to a number of other problems. At their best, these methods have been integrated into the general methods for solving diophantine problems. Moreover, the newer work suggests limits to the application of model theory to diophantine geometry. For example, all such known applications are connected with commutative algebraic groups. This need not be an intrinsic restriction, but its removal requires serious advances in the model theory of fields.
Constructing the hyperdefinable group from the group configuration
 J. Math. Logic
"... Abstract. Under P(4) −amalgamation, we obtain the canonical hyperdefinable group from the group configuration. The group configuration theorem for stable theories given by Hrushovski [5], which extends Zilber’s result for ωcategorical theories [17], plays a central role in producing deep results ..."
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Cited by 5 (0 self)
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Abstract. Under P(4) −amalgamation, we obtain the canonical hyperdefinable group from the group configuration. The group configuration theorem for stable theories given by Hrushovski [5], which extends Zilber’s result for ωcategorical theories [17], plays a central role in producing deep results in geometric stability theory (For a complete exposition of it, see [14]). For example, it is pivotal in the proof of the dichotomy theorem for Zariski ’ structures (See [9]). It is fair to say the group configuration theorem is one of the foundational theorems in geometric stability theory and its applications to algebraic geometry. The theorem roughly says that one can get the canonical nontrivial typedefinable group from the group configuration, a certain geometrical configuration, in stable theories. The complete generalization of the theorem into the context of simple theories seemed unreachable. In their topical paper [1], BenYaacov, Tomasic and Wagner generalize the group configuration theorem by obtaining an invariant group from the group configuration in simple theories. However the group they produce does not completely fit into the firstorder context.
Problems on `pathological structures
 Proceedings of 10th Easter Conference in Model Theory, Wendisches Rietz, April 1219
, 1993
"... The following three major results of Hrushovski create a new area for model theoretic studies. 0.1 Theorem. There exist nontrivial strongly minimal sets which do not interpret a group. 0.2 Theorem. Any two reasonable strongly minimal sets can be amalgamated to form a new strongly minimal set. 0.3 Th ..."
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Cited by 3 (1 self)
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The following three major results of Hrushovski create a new area for model theoretic studies. 0.1 Theorem. There exist nontrivial strongly minimal sets which do not interpret a group. 0.2 Theorem. Any two reasonable strongly minimal sets can be amalgamated to form a new strongly minimal set. 0.3 Theorem. There exist countable stable and @0categorical structures which are not!stable. The first and second examples strongly refuted Zil'ber's conjecture that the `known ' examples of strongly minimal sets exhausted the class. The third answered a similar question of Lachlan limiting the variety of stable theories. One direction of research, pursued with great success by Zil'ber and Hrushovski [14], [19], is to further refine the notion of strongly minimal set to see which strongly minimal sets are closely connected to such classical fields of study as algebraic geometry and complex manifolds. Another is to explore the realm of the more paradoxical examples. A general method of construction underlies all three of these results. We will describe this method