Generalizations of the Szemerédi-Trotter Theorem

In this paper, we generalize the Szemer\'edi-Trotter theorem, a fundamental result of incidence geometry in the plane, to flags in higher dimensions. In particular, we employ a stronger version of the polynomial cell decomposition technique, which has recently shown to be a powerful tool, to generalize the Szemer\'edi-Trotter Theorem to an upper bound for the number of incidences of complete flags in $\mathbb R^n$ (i.e. amongst sets of points, lines, planes, etc.). We also consider variants of this problem in three dimensions, such as the incidences of points and light-like lines, as well as the incidences of points, lines, and planes, where the number of points and planes on each line is restricted. Finally, we explore a group theoretic interpretation of flags, which leads us to new incidence problems.