Oberseminar Geometrie Fribourg
Speaker: Hugo Parlier (Fribourg)
Date: 27th February 2013 (planned talk was postponed)
Title: Kissing numbers for surfaces
Abstract: The classical kissing number problem for sphere packings is the search for an optimal upper bound on the number of n-dimensional euclidean unit spheres, pairwise disjoint in their interior, that can be tangent to a fixed unit sphere. In the case of lattice sphere packings, one asks that the centers of the spheres be points lying on a lattice. Another classical problem for lattices is the study of Hermite constants which is the problem of finding optimal bounds on the length of a shortest non-trivial vector of a unit volume lattice of dimension n.
Schmutz Schaller introduced a nice parallel between these problems and problems related to systoles on closed hyperbolic surface of genus g. A systole of a surface is a homotopically non-trivial curve of shortest length and the parallel problems focus on the maximum number of systoles and their maximum length in function of genus. In this talk I'll explain this parallel and why a hyperbolic surface of genus g cannot have more than roughly $g^2$ systoles.