MASCOS - Analysis Seminar


Tuesday, 8th March 2005


Speaker: Mr Johann S Brauchart
University of Technology Graz, Austria

Title: Points on the sphere and some Proof Techniques

Date: Wednesday March 9th
Time: 2:00 pm
Place: Room 3084, Rec Centre


In dealing with points on an unit sphere in an Euclidean
space $\mathbb{R}^{d+1}$, $d\geq2$, which are 'good' in some sense,
one faces certain technical difficulties to answer questions like

  • What are the explicit coordinates of such points?
  • How good do such points approximate the uniform measure on the sphere ---> 'equidistribution' ?
  • How can this be quantified ---> 'discrepancy(-bounds)' ?
  • What is the distance of two closest points--> well-separation' ?
  • What is their 'Riesz-$s$-energy' when these points are thought to interact through a Riesz-potential $1/r^s$.
  • If these points are used as nodes for a numerical integration rule what can be said about the error of integration ---> 'worst-case error bounds' ?

We want to consider some of these questions and look mostly at the techniques used to prove results.