Speaker: Edward B. Saff, Vanderbilt University
Title: Discrete Minimal Energy Problems

Abstract: For a compact set A in Euclidean space we shall
investigate the asymptotic behavior of optimal (and near optimal)
N-point configurations that minimize the Riesz s-energy
(corresponding to the potential 1/r^s for s>0 and log(1/r) for s=0)
over all N-point subsets of A, where r denotes Euclidean distance. If A has
finite and positive d-dimensional Hausdorff measure and s<d, then
the analysis of such points falls under the umbrella of classical
potential theory and is a consequence of the continuous theory. But
what if s>d or s=d ? In such cases, the classical theory does not
apply since A has s-capacity zero and so new techniques are needed
to analyze the behavior of minimal energy configurations. We shall
describe these techniques, which also yield information about
"best-packing points" on A; that is, N points of A for which the
minimal pairwise distance is as large as possible.

References:

0) "Pointillisme", par Laurent Orluc, "Science & Vie, July 2005

1) S.V. Borodachov, D.P. Hardin and E.B. Saff, Asymptotics for
Discrete Weighted Minimal Riesz Energy Problems on Rectifiable
Sets, Trans. Amer. Math. Soc., Vol. 360 (2008), pp. 1559-1580.

2) S.V. Borodachov, D.P. Hardin and E.B. Saff, Asymptotics of Weighted
Best-Packing on Rectifiable Sets, accepted for publication in Math.
Sbornik

3) Henry Cohn and Noam Elkies, New upper bounds on sphere packings. I. 
Ann. of Math. (2)  157  (2003),  no. 2, 689--714

4) D.P. Hardin and E.B. Saff, Discretizing Manifolds via Minimum
Energy Points, Notices of the American Mathematics Society,
November 2004, pp. 1186-1194.

5) D.P. Hardin and E.B. Saff, Minimal Riesz Energy Point
Configurations for Rectifiable d-Dimensional Manifolds, Advances in
Mathematics, Vol. 193, No. 1 (2005), pp. 174-204.

6) N.S. Landkof, Foundations of Modern Potential Theory,
Springer-Verlag, Heidelberg 1972

7) A.L. Levin and E.B. Saff, Potential Theoretic Tools in Polynomial
and Rational Approximation, in Harmonic Analysis and Rational
Approximation, Vol. 327 (Fournier, Grimm, Leblond, Partington,
Eds.), Springer, 2006, pp. 71-94.

8) A. Martínez-Finkelshtein, V. Maymeskul, E.A. Rakhmanov and E.B. Saff,
Asymptotics for Minimal Discrete Riesz Energy on Curves in R^d, Canadian
Journal of Mathematics, Vol. 56 (2004), pp. 529-552.

9) P. Mattila, Geometry of Sets and Measures in Euclidean Space,
Cambridge University Press, 1995

7) E.B. Saff and V.Totik, Logarithmic Potentials with External,
Springer-Verlag, 1997.