Polynomial partitioning and incidence problems in higher dimensions

Abstract

Incidence geometry is the study of the intersection patterns of simple geometric objects. One of the breakthroughs in this field is the polynomial partitioning technique introduced by Guth and Katz. In this thesis, I will present two results on incidence problems with high-dimensional objects: an almost tight bound on the number of joints formed by varieties in Rn and a tight bound on the number of flags in Rn. The proofs are based on the polynomial partitioning technique and its variations..