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# Lonely Runners and My Favorite Polyhedron (Matthias Beck, San Francisco State University)

## November 8, 2023 @ 4:15 pm - 5:30 pm

**Title: **Lonely Runners and My Favorite Polyhedron

**Speaker:** Matthias Beck, Department of Mathematics, San Francisco State University

**Abstract: **We study the *Lonely Runner Conjecture*, conceived by Wills in the 1960’s, and originally phrased in terms of Diophantine approximation: Given positive integers n_1, n_2, …, n_k, there exists a positive real number t such that for all 1 ≤ j ≤ k the distance of t n_j to the nearest integer is at least 1/(k+1). This conjecture can be recast in lay terms: if k 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/k to the others. This (in)famous conjecture in combinatorial number theory is open for k ≥ 7. We will give a brief history of the Lonely Runner Conjecture and some of its variants, emphasizing 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 affirmative instances that become (quite literally) visible through polyhedral geometry.

Based on joint work with Serkan Hosten (SF State) and Matthias Schymura (Rostock).

Matthias Beck is a professor of mathematics at San Francisco State University and has had visiting positions at SUNY Binghamton, the Mathematical Sciences Research Institute in Berkeley, the Max-Planck-Institute in Bonn, Cornell University, Vassar College, and the Freie Universität in Berlin. Matt’s research is in combinatorics and number theory, in particular, counting integer points in polyhedra and the application of these enumeration functions to various mathematical topics and problems. He (co-)authored four books and numerous research papers, many of which feature student coauthors. Matt was honored with the Mathematical Association of America’s Haimo Award for Distinguished College or University Teaching of Mathematics in 2013.x`