The APDE seminar on Monday, 9/12, will be given by Nets Katz (Caltech) in-person in Evans 740, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PST. To participate, email Sung-Jin Oh (sjoh@math.berkeley.edu).

Title: A proto-inverse Szemer\’edi Trotter theorem

Abstract: The symmetric case of the Szemer\’edi-Trotter theorem says that any configuration of $N$ lines and $N$ points in the plane has at most $O(N^{4/3})$ incidences. We describe a recipe involving just $O(N^{1/3})$ parameters which sometimes (that is, for some choices of the parameters) produces a configuration of $N$ point and $N$ lines. (Otherwise, we say the recipe fails.) We show that any near-extremal example for Szemer\’edi Trotter is densely related to a successful instance of the recipe. We discuss the relation of this statement to the inverse Szemer\’edi Trotter problem. (joint work in progress with Olivine Sillier.)