Ujue Etayo: Thomson Problem Revisited - Distributing Points on a Sphere
Time: Wed 2019-11-20 13.15 - 14.14
Participating: Ujue Etayo, Technische Universität Graz
What is a configuration of well-distributed points on the sphere? The question, trivial on the sphere of dimension 1, gets clearly complicated when moving to higher dimensions. The first obstacle that we are dealing with is that we are missing a unique definition of well separated points. During this talk we will present different definitions of "good-distribution", their internal relationships and their connection with some other mathematical problems as finding a sequence of polynomials that are well-conditioned.
In the article "Minimal discrete energy on the sphere", Saff, Rakhmanov and Zhou presented a sequence of configurations of points in the sphere of dimension 2 that are somehow well distributed and through this configurations of points they were able to prove some properties concerning Riesz potentials. An extensive literature verses about the numerical results of this family of points, although there are hardly none analytic results. In recent work, together with C. Beltrán we introduced a new family of points on the sphere called the Diamond ensemble that can be read as a continuation of the work of Saff, Rakhmanov and Zhou and for which we are able to compute analytically the logarithmic energy. Further studies of this family allows us to also compute the spherical cap discrepancy.