Math Encounters: The Lonely Runner: an unsolved mystery of mathematics - M. Beck on April 13, 2022

Math Encounters: "The Lonely Runner: an unsolved mystery of mathematics" with Matthias Beck on April 13, 2022
The Lonely Runner Conjecture is an (in)famous open problem in combinatorial number theory: if n runners with different (constant) speeds move around a circular track of length 1, then for each runner there will be a time when they have distance at least 1/n to the others. How did this problem come about, how is it related to the approximation of real numbers by rationals, and how can we visualize this problem using two- or three-dimensional shapes? Join mathematician Matthias Beck as we explore this fascinating mathematical mystery.
Special introduction by Nathan Kaplan, Associate Professor of Mathematics at University of California, Irvine.
Math Encounters is a public presentation series celebrating the spectacular world of mathematics, presented by the Simons Foundation and the National Museum of Mathematics.
For further information, call the National Museum of Mathematics at
212-542-0566 or e-mail [e-mail].

[@DavidVonR]

Great talk. I worked on this problem for 5-6 years and got some good results, but didn't prove the whole thing. The case of three runners is pretty easy and is equivalent to a simple Diophantine approximation problem.
For 4 or more runners, the lonely runner conjecture is equivalent to a very complex Diophantine approximation problem. The best thing I could prove is that given any set of speeds s1 < s2 < ... < sk, and given epsilon > 0, there exists speeds v1 < v2 < ... < vk such that |sk - vk| < epsilon for which the conjecture is true.

[@xyh6552]

Your statement is easy to gain by diophantine approximation, one of the key points of the conjecture is that one of the obstacles comes from number theory.

[@Bryce386]

super high and I just solved this but I’m not telling anyone

[@sajjadakbar6649]

what other methods one can look at this problem other than Diophantine approximation?

[@daveolifent4477]

Possibly Fourier transforms. If you take each dynamic position relative to a runner as a number, then their time and position can be represented as a sine wave. If you take the summation of all other runners' absolute value, then you can calculate when this runner is lonely as the absolute sum of all numbers would be larger than 1/n.

[@SaveSoilSaveSoil]

Excellent talk! I learned a lot from him. I just hope that he stops adding "oh yeah?" at the end of a sentence when it's not needed.

[@jblangcua2726]

*yeah not oh yeah