The ACO Seminar (2012-2013)

The polynomial partitioning theorem of Guth and Katz has led to many new bounds on the number of incidences between points and various types of geometric objects such as lines, planes, spheres, etc. The general paradigm is as follows. Simple geometric considerations give crude bounds on the number of incidences between points and various types of geometric objects. These bounds can then be strengthened by partitioning Euclidean space into disjoint "cells," applying the crude bound within each cell, and then summing the resulting contributions. I will discuss these techniques, some recent results, and some open problems in this field.