Dimensions and homogeneity in mathematical structures (Q2758398)

This expository paper provides a very nice and readable description of some model-theoretic matters related to strong minimality. The major emphasis is naturally laid on the Trichotomy Conjecture raised by the author in 1983, and proved jointly with Ehud Hrushovski in 1996 in the restricted framework of strongly minimal Zariski structures. Some noteworthy applications of the last result are discussed, in particular it is observed that Trichotomy is satisfied by minimal infinite analytic sets in compact complex manifolds, and by strongly minimal structures in differentially closed fields of characteristic 0 (a theorem of Hrushoski and Sokolovic playing a key role towards Hrushovski's model-theoretic solution of the Mordell-Lang Conjecture). Of course, the author deals also with Hrushovski's counterexamples to the general Trichotomy Conjecture, and underlines their positive relevance. In fact, he notes that these counterexamples, and in particular the construction of a strongly minimal set carrying two field structures with no definable isomorphism between them, ``rather than being mathematical pathologies show that model theoretical stability ideology might be applicable in understanding classical analytic studies'', for instance in approaching the model theory of the complex exponentiation \(({\mathbb C}, +,\cdot,\exp)\) and its reducts.NEWLINENEWLINEFor the entire collection see [Zbl 0971.00010].