Geometry Seminar
Lattice zonotopes and the Lonely Runner Conjecture.
Speaker: Francisco “Paco" Santos, University of Cantabria
Location: Online
Date: Tuesday, November 24, 2026, 2 p.m.
Synopsis:
The lonely runner conjecture (LRC) is the following statement formulated by Jörg Wills in 1968:
If \(n+1\) “runners” move along a circle of length one, all starting at the same point, each with its own (constant) velocity then for each individual runner \(i\) there is a time \(t_i\) at which she is at distance at least \(1/(n+1)\) from all the others.
In its shifted version (sLRC) the runners are allowed to start each at a different position and the velocities are assumed distinct, since otherwise easy counter-examples exist. The conjectures have been approached from different perspectives, and as of 2025 they were proved until \(n=6\) in the original version (Barajas and Serra, 2008) and \(n=3\) in the shifted version (Cslovjecsek, Malikiosis, Naszódi, and Schymura, 2022).
In 2017 Malikiosis and Schymura developed a connection of the conjectures to the convex geometry of certain zonotopes, a connection that was later (2025) used by them together with myself to prove that for each number \(n\) of runners, if the conjecture holds for integer velocities adding up to at most \(\binom{n+1}{2}^{n-1}\) then it holds for arbitrary velocities. A similar finiteness result, with a worse and much less explicit bound, had been proved by Tao in 2018.
The new bound has then been used to prove the conjectures for additional values of \(n\):
- Rosenfeld (2026) devised a computational method based on analysing prime factors of candidate velocities. Using it he proved \(n=7,8\) and Sungkawichai and Trakulthongchai proved \(n=9,10,11,12\) (2026+).
- With Alcántara and Criado (2026) I have computationally proved the shifted version for \(n=4\), and with Blanco and Criado (2026+) I have found counter-examples to the shifted conjecture starting at \(n=5\).
In this talk I will summarise the zonotopal approach to LRC and this recent progress. The talk will be self-contained and assume only some basic familiarity with polytope theory.
Notes:
Only on Zoom. Please contact Boris Aronov to be put on the email announcement list and obtain Zoom details.