Richard Schwartz

Divide and Conquer:  5 Point Energy Minimization

Thomson’s classic problem asks, for any N, which configurations of N points on the sphere minimize the electrostatic potential. A more general version asks which which points minimize the potential of the configuration with respect to a power law potential. I will describe the progress I have made to date on this problem. For one thing, I solved Thomson’s problem in the case N=5. More generally, I rigorously proved the existence of phase transition, first noticed by Melnyk-Knop-Smith in the 1970s.

Their original discovery is that the triangular bi-pyramid is the minimum potential configuration with respect to all power laws up to about 15.04 and then the answer changes to a pyramid with square base. My proof is a mixture of computer algebra, interval arithmetic, and symmetrization. I will explain the main ideas and also give some neat computer demos showing the proof in action.

CV