**Date:** June 21, 2024

**Venue:** Computer Science Building (Friedrich-Hirzebruch-Allee 8)

**Research Area: **C3 Combinatorial optimization, complexity, and chip design

### Friday, June 21

**14:00**

*Coffee and Tea (Room 2.075)*

### Abstracts

#### Lisa Sauermann: Unit distances and distinct distances in typical norms

Given n points in the plane, how many pairs among these points can have distance exactly 1? More formally, what is the maximum possible number of unit distances among a set of n points in the plane? This problem is a very famous and still largely open problem, called the Erdös unit distance problem. One can also study this problem for other norms on R^{2} (or more generally on R^{d} for any dimension d) that are different from the Euclidean norm. This direction has been suggested in the 1980s by Ulam and Erdös and attracted a lot of attention over the years. We give an almost tight answer to this question for almost all norms on R^{d} (for any given d). Furthermore, for almost all norms on R^{d}, we prove an asymptotically tight bound for a related problem, the so-called Erdös distinct distances problem. Joint work with Noga Alon and Matija Bucić.