Department of Computer Science | Institute of Theoretical Computer Science | CADMO

Theory of Combinatorial Algorithms

Prof. Emo Welzl and Prof. Bernd Gärtner

Mittagsseminar (in cooperation with A. Steger, D. Steurer and B. Sudakov)

Mittagsseminar Talk Information

Date and Time: Tuesday, May 10, 2016, 12:15 pm

Duration: 45 minutes

Location: OAT S15/S16/S17

Speaker: Marko Kabic

Random runners are very lonely

The general form of the Lonely runners conjecture still remains an open problem, after almost 50 years since it first appeared in Jörg Wills's paper (1967). It states that if k runners are running on a circular track of unit length, each running with constant speed and starting at the same point, then, sooner or later, each runner will become "lonely", in the sense that all the other runners will be at distance of at least 1/k from him. Some special cases of the conjecture have already been proven - e.g. for the specific number of runners or for some specific choices of speeds, but the proof for the general case has not yet been found. However, if initial speeds of runners are chosen uniformly at random, then much stronger separation result was proven! In the talk, we will see what the idea behind this proof is and what the non-trivial consequences of such a result are. It elegantly uses Fourier analytic method on finite Abelian groups. For the sake of completeness, I will try to intuitively give a short introduction to this method and to show how it was used to prove this beautiful conjecture in the random setting.

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