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Ample dividing
 J. Symbolic Logic
"... We construct a stable onebased, trivial theory with a reduct which is not trivial. This answers a question of John B. Goode. Using this, we construct a stable theory which is nample for all natural numbers n, and does not interpret an infinite group. ..."
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We construct a stable onebased, trivial theory with a reduct which is not trivial. This answers a question of John B. Goode. Using this, we construct a stable theory which is nample for all natural numbers n, and does not interpret an infinite group.
The geometries of the Hrushovski constructions
, 2009
"... In 1984 Zilber conjectured that any strongly minimal structure is geometrically equivalent to one of the following types of strongly minimal structures, in the appropriate language: Pure sets, Vector Spaces over a fixed Division Ring and Algebraically Closed Fields. In 1993, in his article ‘A new st ..."
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Cited by 4 (2 self)
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In 1984 Zilber conjectured that any strongly minimal structure is geometrically equivalent to one of the following types of strongly minimal structures, in the appropriate language: Pure sets, Vector Spaces over a fixed Division Ring and Algebraically Closed Fields. In 1993, in his article ‘A new strongly minimal set’ Hrushovski produced a family of counterexamples to Zilber’s conjecture. His method consists in two steps. Firstly he builds a ‘limit’ structure from a suitable class of finite structures in a language consisting only of a ternary relational symbol. Secondly, in a step called the collapse, he defines a continuum of subclasses such that the corresponding ‘limit’ structures are new strongly minimal structures. These new strongly minimal structures are non isomorphic but Hrushovski then asks if they are geometrically equivalent. We first analyze the pregeometries arising from different variations of the construction before the collapse. In particular we prove that if we repeat the construction starting with an nary relational symbol instead of a 3ary relational symbol, then the pregeometries associated to the corresponding ‘limit’ structures are not locally isomorphic when we vary the arity. Second we prove that these new strongly minimal structures are geometrically equivalent. In fact we prove that their geometries are isomorphic to the geometry of the ‘limit’ structure obtained before the collapse.
Matroid theory and Hrushovski’s predimension construction
, 2011
"... We give an exposition of some results from matroid theory which characterise the finite pregeometries arising from Hrushovski’s predimension construction. As a corollary, we observe that a finite pregeometry which satisfies Hrushovski’s flatness condition arises from a predimension. ..."
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We give an exposition of some results from matroid theory which characterise the finite pregeometries arising from Hrushovski’s predimension construction. As a corollary, we observe that a finite pregeometry which satisfies Hrushovski’s flatness condition arises from a predimension.
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"... PU Public PP Restricted to other programme participants (including the Commission Services) RE Restricted to a group specified by the consortium (including the Commission Services) CO Confidential, only for members of the consortium (including the Commission Services) 1 2Report on Workpackage MI: Pu ..."
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PU Public PP Restricted to other programme participants (including the Commission Services) RE Restricted to a group specified by the consortium (including the Commission Services) CO Confidential, only for members of the consortium (including the Commission Services) 1 2Report on Workpackage MI: Pure Model Theory In the following, members of Modnet teams are identified by an asterisk (*) when first mentioned; Modnet fellows are indicated by a double asterisk (**); external experts and collaborators who were identified as having a close involvement with the project in the original proposal are identified by a triple asterisk (***). Results Task I.1: Theoretical stability and simplicity Result of Task I.1.e: Find new unstable structures with many stable, stably embedded sets and develop model the
Report on EPSRC Grants GR/R32048/01 and GR/R32123/01
"... Model theory of generic structures and simple theories As intended in the original proposal, the grants employed Dr Massoud Pourmahdian as a research assistant at UEA for a year, starting in October 2001. Pourmahdian was then employed by Edinburgh for a year, starting in October 2002. The original g ..."
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Model theory of generic structures and simple theories As intended in the original proposal, the grants employed Dr Massoud Pourmahdian as a research assistant at UEA for a year, starting in October 2001. Pourmahdian was then employed by Edinburgh for a year, starting in October 2002. The original grant applications were submitted as being related and this report covers activities supported by both of the grants. The start dates were later than in the original proposals due to Pourmahdian’s commitments in Iran. Pourmahdian spent much of the second year based in Oxford, where Macintyre was visiting on sabbatical leave from Edinburgh. A. Description of Reseach Undertaken The objectives were to study open problems in model theory by investigating and extending Hrushovski’s constructions of generic structures. The work done focused around the following themes: (1) Hrushovski’s unstable, non uniformly locally finite generic. (2) Stable forking and Lascar strong types in generics. (3) Construction of nonCMtrivial structures.