Mathematics in the round - a four year journey

Abstract:

I will present an anecdotal foray into my research contributions of the past four years. Focal points will be numerical integration and the minimal energy problem on the sphere. I will give several key results obtained in collaborative effort with the local group and international researchers/visitors from USA and Europe:

• Stolarsky's invariance principle (which relates a geometric property [discrepancy] of a node set with its quality for numerical integration and energy) and its extensions as a central result of the reproducing kernel Hilbert space approach to numerical integration on the sphere (and in space);
• the concept of QMC designs and the strength of a point set sequence (both are connected to the worst-case error of equal weight numerical integration (QMC) methods on the sphere for function in Sobolev spaces);
• the presently sharpest estimate for the so-called spherical cap discrepancy of "constructible'' point sets on the sphere; and applications of minimal energy problems with external fields.