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Monday, October 29, 2007
3:00 PM - 4:00 PM
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Discrete Minimal Energy Problems

Edward B. Saff
Vanderbilt University

For a closed and bounded surface A in 3-space, such as a sphere or torus, we analyze the behavior (for large N) of N-point equilibrium onfigurations on A for the potential (1/r)^s, where s>0 is a parameter and r denotes Euclidean distance between points. (The case s=1 corresponds to the familiar Coulomb potential, while large s corresponds (in the limit) to best-packing.) If d=dim(A) and s greater than d (the case of long range interactions), 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 is greater than d or s equals d ? In such cases, the classical theory does not apply and 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. The research has relevance to the study of self-assembling materials and has extensions to higher dimensions. References: 1)D.P. Hardin and E.B. Saff, Discretizing Manifolds via Minimum Energy Points, Notices of the American Mathematics Society,November 2004, pp.1186-1194. 2)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. 3)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. (2008) 4)D. Hardin, E.B. Saff and H. Stahl, The Support of the Logarithmic Equilibrium Measure on Sets of Revolution in R3, J. Math. Physics, Vol. 48, No. 2 (2007), 022901, 14 pp. 5) S. Borodachov, D.P. Hardin, and E.B. Saff, Asymptotics of Best-Packing on Rectifiable Sets, Proc. Amer. Math. Soc., Vol. 135 (2007), pp. 2369-2380 --------------------------------------- Edward B. Saff Professor of Mathematics and Director, Center for Constructive Approximation Phone:(615) 322-2014 Fax:(615) 343-0215 Vanderbilt University Department of Mathematics Nashville, TN 37240 Constr. Approx. (615) 343-4107 Homepage

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