## UBC Discrete Math Seminar: Olivine Silier

- Date: 03/28/2023
- Time: 16:00

University of British Columbia

A Proto Inverse Szemerédi-Trotter Theorem

A point-line incidence is a point-line pair such that the point is on the line. The Szemer\'edi-Trotter theorem says the number of point-line incidences for n (distinct) points and lines in R^2 is tightly upperbounded by O(n^{4/3}). We advance the inverse problem: we geometrically characterize `sharp' examples which saturate the bound by proving the existence of a nice cell decomposition we call the \textit{two bush cell decomposition}. The proof crucially relies on the crossing number inequality from graph theory and has a traditional analysis flavor.

Our two bush cell decomposition also holds in the analogous point-unit circle incidence problem. This constitutes an important step towards obtaining an ϵ improvement in the unit-distance problem. (Ongoing work with Nets Katz)

No background required, all welcome!

**Location**: ESB 4127

**Time**: 4pm Pacific