Rothschild Lecture: Thomson's 5 point problem

Thomson's problem, which in a sense goes back to J.J. Thomson's 1904 paper, asks how N points will arrange themselves on the sphere (or the circle, or some other space) so as to minimize their total electrostatic potential.  Mathematicians and physicists have also considered this problem with respect to other potentials, such as power law potentials.  For special values of N, and the sphere of the appropriate dimension, there are spectacular answers which say that the potential minimizers are highly symmetric objects, such as the regular icosahedron or the E8 cell.  In spite of this work, very little has been proved about 5 points on the 2-sphere. In my talk I will explain my computer assisted but rigorous proof that there is a phase transition constant S=15.048... such that the triangular bi-pyramid is the minimizer with respect to a power-law potential if and only if the exponent is less or equal to S.  (This constant was conjectured to exist in 1977 by Melnyk-Knop-Smith.) The talk will have some colorful computer demos.