Combinatorics Seminar

While the topic of geometric incidences existed for several decades as
part of the field of discrete geometry, recently it has been experiencing
a renaissance with new connections to other fields (such as algebraic
geometry and harmonic analysis), as well as new results, techniques, and
applications.

A simple example of an incidences problem: Given a set of n points and set
of n lines, both in the Euclidean plane, what is the maximum number of
point-line pairs such that the point is contained in the corresponding
circle.

In this talk I will present the basics of geometric incidences, describe
the recent technique of polynomial partitioning, and present a new
extension of this technique by relying on Hilbert polynomials. This
extension yields a general bound for incidences between points and
varieties in any dimension.

Joint work with Jacob Fox, Andrew Suk, János Pach, and Joshua Zahl.