Distributing points on spheres and other manifolds

Abstract: 

Distributing points on spheres or other sets is a very classical problem. Its modern formulation in terms of energy-minimising configurations is due to the discoverer of the electron J. J. Thomson who, in 1904, posed the question: in which position within some set, such as a ball or a sphere, would N electrons lie in order to minimise their electrostatic potential? There are many different approaches to the definition of what a sensibly distributed collection of spherical points is. Apart from the aforementioned minimisation of energy, other definitions include minimising the separation distance, having a small discrepancy, providing exact integral formulas for low degree polynomials, etc. In this talk we will review some of these problems and address the problem of defining minimal-energy points in an arbitrary compact manifold from an intrinsic viewpoint.