Recognizing Groups in Erdős Geometry and Model Theory
Erdős-style geometry is concerned with combinatorial questions about simple geometric objects, such as counting incidences between finite sets of points, lines, etc. These questions can be typically viewed as asking for the possible number of intersections of a given (semi-)algebraic variety with large finite grids of points. An influential theorem of Elekes and Szabó indicates that such intersections have maximal size only for varieties that are closely connected to algebraic groups. Techniques from model theory - Hrushovski's group configuration and its variants - are very useful in recognizing these groups, and allow to obtain higher arity and dimension generalizations of the Elekes-Szabó theorem. In fact, all of this is not just about polynomials and works in the larger setting of definable sets in o-minimal structures.
Joint work with Kobi Peterzil and Sergei Starchenko.