# 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.