Ben Lund, Incidences between points and n-flats in PG(n+d,q)

Let $P$ be a set of points in $PG(n+d,q)$, and let $L$ be a set of $n$-flats. Here, $n$-flat is a short name for $n$-dimensional projective subspaces. A classical bound of Haemers, rediscovered in an influential paper of Vinh, gives an upper bound on the difference between the number of incidences between $P$ and $L$ and the expected number of incidences for random sets of points and flats with the same cardinalities as $P$ and $L$. Haemers’ bound is tight as a function of $|P|$ times $|L|$. Recent work of Kong and Tamo improves the bound under the assumption that $|L|$ is not too large. I will discuss recent work, joint with Tao Zhang, that improves the bound of Kong and Tamo. The proof depends on an independently interesting upper bound on the number of pairs $(l_1,l_2)$ of flats in $L$ such that $\dim(l_1 \cap l_2)=j$, for $0 \leq j \leq n$.