NSFCBMS Conference on Additive Combinatorics from a Geometric Viewpoint
Orit Raz
University of British Columbia http://www.math.ubc.ca/~oritraz/ 

Abstract 
Polynomials vanishing on Cartesian products
Let F(x,y,z) be a real trivariate polynomial of constant degree, and let A,B,C be three sets of real numbers, each of size n. How many roots can F have on A x B x C? This question has been studied by Elekes and Rónyai and then by Elekes and Szabó about 15 years ago. One of their results is that either F vanishes at o(n^2) number of points of A x B x C, or else the surface {F=0} must have a certain special form. In the talk I will discuss several recent results, in which the analysis is greatly simplified, and the bounds become sharp: If {F=0} does not have the special form, the number of roots is at most O(n^{11/6}). This setup arises in various Erdöstype problems in extremal combinatorial geometry, and the result mentioned above provides a unified tool for their analysis. If time allows, I will discuss some applications of this kind. 