Boris Shapiro: Mystery of points charges after C.F. Gauss, J.C. Maxwell and M. Morse

Abstract

 Any configuration of \(K\) point charges in \(\mathbb{R}^n\) creates an electrostatic field and we are interested in finding/estimating the number of its points of equilibrium. This question has been considered by Gauss, Maxwell and Morse (among others). In particular, Maxwell conjectured and provided an incomplete proof of the following surprising claim. Maxwell’s conjecture. For any \(K\) points charges in \(\mathbb{R}^3\), the number of its points of equilibrium (assumed finite) does not exceed \((K-1)^2\). Strikingly, this conjecture is still open already for \(K = 3\). In my talk I will tell what is currently known in this direction. Certain asymptotic arguments/guesses resembling that of tropical geometry will be presented.