While the topic of geometric incidences existed for decades as part of the field of discrete geometry, in recent years it has been on the rise due to a large number of new results, techniques, and applications. These involve connections to other fields such as algebraic geometry, theoretical computer science, and harmonic analysis.

This talk will introduce 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 incidence bound for between points and varieties in any dimension.

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