Mystery of point charges (after C.F. Gauss, J.C. Maxwell and M.Morse)

Abstract：

The following natural question has been considered by several mathematical celebrities in the past. Given N fixed electric charges in R^n, estimated from above the number of points of equilibrium of the electrostatic field. Gauss solved this question for n=2 and Morse applied his famous theory to give the lower bound for this number. Unexpectedly much earlier than Morse J.C.Maxwell has also developed a simple version of Morse theory using certain notions created by the first ever topologist J.B. Listing (who also introduced the word “topology”) and came up with the following guess.

Maxwell’s conjecture. A generic configuration of N charges in R^3 has no more than (N-1)^2 points of equilibrium.

The latter conjecture is still open for N=3. In my talk I will survey recent development in this field.