Ben Lund, Point-plane incidence bounds

In the early 1980s, Beck proved that, if P is a set of n points in the real plane, and no more than g points of P lie on any single line, then there are $\mathrm{\Omega }(n(n-g))$ lines that each contain at least 2 points of P. In 2016, I found a generalization of this theorem, giving a similar lower bound on the number of planes spanned by a set of points in real space. I will discuss this result, along with a number of applications and related open problems.