Erdős-style geometry is concerned with difficult questions about simple geometric objects, such as counting incidences between finite sets of points, lines, etc. These questions can be viewed as asking for the possible number of intersections of a given 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 - variants of Hrushovski’s group configuration and of Zilber’s trichotomy principle - are very useful in recognizing these groups, and led to far reaching generalizations of Elekes-Szabó in the last decade. I will overview some of the recent developments in this area, in particular explaining how all of this is not just about polynomials and works for definable sets in o-minimal structures.
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