Modularity of extremal algebraic configurations

Martin Bays (M√ľnster)

Frank Adams 1,

Large finite subsets of characteristic zero fields which form combinatorially extreme configurations with respect to solutions to polynomial equations often appear "linear". One example of this is the Szemeredi-Trotter phenomenon, where exceeding a certain bound on  the number of incidences in an algebraic incidence system forces the configuration to be somewhat degenerate. Another is the Elekes-Szabo theorem which shows that an (abelian) algebraic group must explain extremal properly ternary relations. Elaborating on work of Hrushovski, we discuss a formalism in which these phenomena become instances of modularity of an associated geometry; this is a precise notion of "linearity" with strong consequences, yielding in particular higher arity versions of Elekes-Szabo. This is part of a project with Emmanuel Breuillard.

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