Sponsored by the Rutgers University Department of Mathematics and the
Center for Discrete Mathematics and Theoretical Computer Science (DIMACS)
Title: "Magic" numbers in Smale's 7th problem
Speaker: Michael Kiessling, Rutgers University
Date: Thursday, October 17, 2013 5:00pm
Location: Hill Center, Room 705, Rutgers University, Busch Campus, Piscataway, NJ
Abstract:
Smale's 7th problem concerns N-point configurations on the 2-sphere which minimize the logarithmic pair-energy V(r) = - ln r averaged over the pairs in a configuration; here, r is the chordal distance between the points forming a pair. More generally, V(r) may be replaced by the standardized Riesz pair-energy. In a recent paper with Brauchart and Nerattini we inquired into the concavity of the map from the integers >1 into the minimal average standardized Riesz pair-energies of the N-point configurations on the sphere. It is known that this map is strictly increasing and in some Riesz parameter range bounded above, hence ``overall concave.'' It is (easily) proved that for the Riesz parameter 2 it is even locally strictly concave. By analyzing computer-experimental data of putatively minimal average Riesz pair-energies for the Riesz parameters -1,0,1,2,3 and N up to 200 we found that the map in question is locally strictly concave for parameter -1, while not always locally strictly concave for the other parameter values. It is found that the empirical map from the Riesz parameter into the set of convex defect-N is set-theoretically increasing; moreover, the percentage of odd numbers in the range is found to increase with the Riesz parameter. They form a curious sequence of numbers for the logarithmic kernel, reminiscent of the ``magic numbers'' in nuclear physics; it is conjectured that the ``magic numbers'' in Smale's 7th problem are associated with optimally symmetric optimal-energy configurations. The talk emphasizes the role of computer experiments, in particular also of Maple, in our investigation.