To participate in this seminar, please register here: https://www.msri.org/seminars/25206

Abstract:

Erdõs and Szemerédi observed the following "sum-product" phenomenon: there is some c>0 such that for any finite set A of reals, max{|A+A|, |A*A|} > |A|^{1+c}.

Elekes and Rónyai generalized this by showing that for any polynomial f(x,y) we must have |f(AxA)|>|A|^{1+c}, unless f is either additive or multiplicative (i.e. of the form g(h(x) + i(y)) or g(h(x) * i(y)) for some univariate polynomials g,h,i respectively). A remarkable theorem of Elekes and Szabó provides a conceptual generalization, showing that for any polynomial F(x,y,z) such that its set of solutions has dimension 2, if F has a maximal possible number of zeroes n^2 on finite n-by-n-by-n grids, then F is the graph of multiplication of an algebraic group, up to a finite correspondence (in the special case above, either the additive or the multiplicative group of the field). I will overview recent related work by a number of people  and connections to model-theoretic methods for recognizing groups from generic data.

I will present a generalization of this theorem to hypergraphs of any arity and dimension definable in arbitrary o-minimal structures, as well as in a large class of stable structures  (including differentially closed fields). 

Joint work with Kobi Peterzil and Sergei Starchenko.

Recognizing Groups In Model Theory And Erdõs Geometry