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Sum-product inequalities and geometric incidence counting

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If you have a question about this talk, please contact Boris Bukh.

The sum-product theory aims to give as much as possible quantitative development to a paradigm that if all pairs of elements of a finite set A in a ring R generate few distinct sums and products, relative to the size of A, then A must be close to a subring.

A closely related geometric question is to give non-trivial bounds on the number of incidences between a family of straight lines and points in a Desarguesian plane. The first question of this kind, perhaps, is, given a set of points, to provide a lower bound on a number of distinct straight lines determined by all pairs of points.

This talk discusses some reasonably recent results in the Euclidean and prime field settings, along the lines of the interplay of the above two general questions.

This talk is part of the Discrete Analysis Seminar series.

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