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Model theory of special subvarieties and Schanuel-type conjectures
We use the language and tools available in model theory to redefine and
clarify the rather involved notion of a {\em special subvariety} known from the
theory of Shimura varieties (mixed and pure)
Projective geometries arising from Elekes-Szab\'o problems
We generalise the Elekes-Szab\'o theorem to arbitrary arity and dimension and
characterise the complex algebraic varieties without power saving. The
characterisation involves certain algebraic subgroups of commutative algebraic
groups endowed with an extra structure arising from a skew field of
endomorphisms. We also extend the Erd\H{o}s-Szemer\'edi sum-product phenomenon
to elliptic curves. Our approach is based on Hrushovski's framework of
pseudo-finite dimensions and the abelian group configuration theorem.Comment: 48 pages. Minor improvements in presentation. To appear in ASEN
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