SEMINAR: No lattice tiling of Zn by Lee spheres of radius 202-11-2020

Speaker: Yue Zhou

Title: No lattice tiling of Zn by Lee spheres of radius 2

Date/Time: 4 November 2020/ 1:40 - 2:30 pm

Zoom: Meeting ID916 4029 5656

Passcode: Algebra

Abstract: In 1968, Golomb and Welch conjectured that Zn cannot be tiled by Lee spheres with a fixed radius r ≥ 2 for dimension n ≥ 3. Besides its own interest in discrete geometry and coding theory, if we restrict this conjecture to the lattice tiling case it is also equivalent to the nonexistence of abelian Cayley graphs archiving the Moore-like bound for the cardinality of vertices. A question on the existence of abelian Cayley graphs achieving this upper bound except for some trivial examples has been proposed independently in the context of graph theory for many years. In this talk, I will first give a brief survey of known results. Then I will sketch a proof on the nonexistence of lattice tilings of Zn by Lee spheres of radius 2 with n ≥ 3. As a consequence, we will see that the order of any abelian Cayley graph of diameter 2 and degree larger than 5 cannot meet the abelian Cayley Moore-like bound. This talk is based on a recent joint work with Ka Hin Leung.

Bio: Yue Zhou is currently an associate professor at National University of Defense Technology, China. He has obtained the PhD degree at Otto-von-Guerick University Magdeburg, Germany, July 2013. He has worked at University of Naples and Augsburg University.His research interests include various topics in finite geometry, algebraic combinatorics and coding theory. He has published around 40 papers.