Lonely Runner spectra

Dirichlet's Theorem says that for any real number t, there is some v in {1,2,...,n} such that tv is within 1/(n+1) of an integer. The Lonely Runner Conjecture of Wills and Cusick asserts that the constant1/(n+1)in this theorem cannot be improved by replacing {1,2,...,n} with a different set of n nonzero real numbers. The conjecture, although now more than 50 years old, remains wide open for n larger than 6. In this talk I will describe the "Lonely Runner spectra" that arise when one considers the "inverse problem" for the Lonely Runner Conjecture, and I will explain the (a priori surprising) "hierarchical" relations among these spectra. Based on joint work with Vikram Giri.