You are here

Discrepancy and Energy on the Sphere

Michelle Mastrianni, University of Minnesota
Thursday, November 17, 2022 - 12:30pm to 1:30pm
PDL C-401

The beautiful and classical problem of how to distribute points on the sphere \$\mathbb{S}^2\$ as uniformly as possible has yielded several notions of "well-distributedness," including the (spherical cap) discrepancy and the Riesz \$s\$-energy. We will first discuss some intriguing connections between the two notions. Then, we will give some examples of deterministic point sets which have good (but not optimal) discrepancy, including a construction called HEALPix first introduced by astrophysicists in 2005 and recently studied by myself, Damir Ferizovic, and Julian Hofstadler. Finally, we will touch on the open problem of finding a deterministic point set which does even better and matches the optimal discrepancy bound given by probabilistic constructions. In particular, numerics suggest that a particular "greedily" chosen sequence appears to do very well in this regard.