Title: Projective geometries arising from Elekes-Szabó problems

Abstract:
This is a continuation of a talk given in December 2017.
I will recall how complex varieties which have asymptotically large 
intersections with grids of unbounded size can be seen, exploiting ideas of 
Hrushovski, to correspond to projective geometries, and how this leads to a 
precise characterisation of such varieties. I will explain moreover a version 
of this which allows the co-ordinates to be points in higher dimensional 
varieties. I will proceed to describe some consequences for generalised 
sum-product estimates and connections to diophantine problems. This is joint 
work with Emmanuel Breuillard.