A general incidence bound in high dimensions

Thao Do, MIT

Date and time: 6pm, Tuesday, September 25, 2018

Place: Courant Institute, WWH1314

In this talk, I will present a general upper bound for the number of incidences with $k$-dimensional varieties in $\mathbb{R}^d$ such that their incidence graph does not contain $K_{s,t}$ for fixed positive integers $s$, $t$, $k$, $d$ (where $s,t>1$ and $k\lt d$). The leading term of this new bound generalizes previous bounds for the special cases of $k=1$, $k=d-1$, and $k= d/2$. Moreover, we find lower bounds showing that this leading term is tight (up to sub-polynomial factors) in various cases. To prove our incidence bounds, we define $k/d$ as the dimension ratio of an incidence problem. This ratio provides an intuitive approach for deriving incidence bounds and isolating the main difficulties in each proof. If time permits, I will mention other incidence bounds with traversal varieties and hyperplanes in complex spaces.

This is joint work with Adam Sheffer.