(Special Colloquium) On d and M problems for Newtonian potentials

During the Covid pandemic I decided to have another look at a conjecture which I made some progress on in my early mathematical career. The conjecture concerns Newtonian potentials which have all their mass on the unit sphere about the origin in Euclidean n space and are normalized to be one at the origin. The conjecture essentially divides these potentials into subclasses whose criteria for membership is that a given member have its maximum on the closed unit ball at most M and its minimum at least d. It then lists the potentials in each subclass which solve certain extremal problems. During my talk I will discuss some special cases of this conjecture, together with reminiscences of past work with coauthors. Also time permitting, I will outline recent work on the conjecture showing existence of the proposed extremal potentials, as well as, proof of an integral inequality on spheres about the origin, involving so called extremal potentials in two different d and M subclasses.