# Matthias Beck: Lonely Runner Polyhedra

**Time: **
Wed 2019-12-04 10.15 - 11.00

**Location: **
KTH, 3418

**Participating: **
Matthias Beck

### Abstract

We study the *Lonely Runner Conjecture*, conceived by Jörg M. Wills in the 1960's: Given positive integers \(n_1, n_2, \ldots n_k\), there exists a positive real number t such that for all \(1 ≤ j ≤ k\) the distance of \(tn_j\) to the nearest integer is at least \(1/(k+1)\). Continuing a view-obstruction approach by Cusick and recent work by Henze and Malikiosis, our goal is to promote a polyhedral ansatz to the Lonely Runner Conjecture. Our results include geometric proofs of some folklore results that are only implicit in the existing literature, a new family of affirmative instances defined by the parities of the speeds, and geometrically motivated conjectures whose resolution would shed further light on the Lonely Runner Conjecture.

This is joint work with Serkan Hosten (SF State) and Matthias Schymura (Lausanne).