Balázs Keszegh: The number of tangencies between two families of curves

Abstract: 
We prove that the number of tangencies between the members of two families, 
each of which consists of n pairwise disjoint curves, can be as large as 
$\Omega(n^{4/3})$.
From a conjecture about 0-1 matrices it would follow that if the families 
are doubly-grounded then this bound is sharp. 
We also show that if the curves are required to be x-monotone, 
then the maximum number of tangencies is $\Theta(n\log n)$,
which improves a result by Pach, Suk, and Treml. 
Joint work with Dömötör Pálvölgyi.