Title: Lonely Runner Conjecture.
Speaker: Hrishikesh Venkatraman
Date and Venue: 18 April, 2023, Seminar Room
Abstract: Imagine that you are running on a circular track. For company, you have a few friends with you. No two of you run with the same speed, and all of you have a constant, non-zero speed. If you all run in the same direction, will there ever be a moment when you are at least a fixed minimum distance from the friend nearest you? Does the same thing happen with each of your friends?
Lonely Runner Conjecture, proposed by Jörg M. Wills and so nomenclatured by Luis Goddyn, has been an object of interest since it was first conceived in 1967 : Given positive integers k and n1, n2,...,nk there exists a positive real number t such that the distance of t⋅nj to the nearest integer is at least 1⁄k+1, for all 1 ≤ j ≤ k. In a recent article Beck, Hosten and Schymura described the Lonely Runner polyhedron and provided a polyhedral approach to identifying families of lonely runner instances. In this talk, we revisit the Lonely Runner Conjecture through the lens of polyhedral theory and highlight some new families of instances satisfying the conjecture.