Self organization by local interaction

A surprising number of problems in various fields of science and engineering -- ranging from biology over chemistry and physics to computer science and logistics -- can be formulated as a discrete energy problem. A classic example from physics is Thomson`s problem of determining the minimal potential energy distribution of N electrons confined to the unit sphere that repel each other with a force given by Coulomb`s law. When tuning the underlying potential and thus the force, one encounters multielectron bubbles in superfluid helium, equal-weight numerical integration formulas, virus morphology, crystals, error-correcting codes, multibeam laser implosion devices, and the optimal placement of mobile phone antennas, communication satellites, and storage facilities. Indeed, ``How to efficiently stack oranges in a crate`` has mesmerized grocers, armorers, and chemists alike for centuries. Only recently, the three- dimensional case known as Kepler`s Conjecture was solved by Hales. For higher dimensions, such arrangement questions are still mysterious and persistently resurface in the analysis of the large N behavior of minimal N-point energy considered in this project. In the mathematical abstraction, such a diversity in self organization of point arrangements is observed when points interact according to an inverse power law (with exponent s) and assume minimal energy positions. This is the setup for the discrete minimal energy problem studied in the project. The strength of this approach becomes apparent when a simple change of the exponent s enables one to deal with topics as diverse as (I) the worst-case behavior of numerical integration using the average value of the integrated function at well-chosen nodes (Quasi-Monte Carlo rules), (II) the position of electrons in the most-stable equilibrium and arrangements of protein subunits which form viral capsids, (III) discretization of manifolds, and (IV) best-packing arrangements. A substantial part of the project is devoted to fundamental mathematical questions regarding the asymptotic behavior of an appropriate notion of the energy of infinite periodic and quasi-periodic point sets and the connection to the minimal energy problem in the compact setting. For theoretical computations novel combinations of number-theoretic, algebraic, combinatoric, and graph-theoretical methods will be needed. Other specific questions concern (a) the discrete minimal energy problem on curves and torus for general exponent s; (b) the extension of the framework for minimal energy problem in the external field setting to cover all exponents s in the potential-theoretic regime; and (c) the analysis of explicitly constructible point configurations for Quasi-Monte Carlo rules. This project will advance the existing mathematical knowledge for presently unresolved potential- theoretic questions.

The Meitner Project M2030-N32 "Self organization by local interaction: minimal energy, external fields, and numerical integration" studied questions of point distributions using methods of potential theory, harmonic analysis, and approximation theory. A central concept was the energy of point sets based on pair interaction via a Riesz potential induced by an application. Of special interest were configurations with minimal or small energy. On the other hand, estimates of the energy of point sets provided bounds for the L2-discrepancy (a measure of uniformity) and the worst-case error of numerical integration for Quasi-Monte Carlo methods. Uniformity and Hyperuniformity: The new concept of hyperuniformity of a point set sequence on compact sets, specifically, on a sphere in (d+1)-dimensional Euclidean space was introduced and studied. The fluctuations of the number of points in a spherical cap moving on the sphere was quantified. A sequence is hyperuniform if this fluctuation does not change with the area of the cap. Three regimes were identified. Certain nonrandom sequences (t-designs with optimal order of points, points that maximize the sum of all mutual pairwise distances, and QMC design sequences) were shown to be hyperuniform in all three regimes. Further work concerned random processes on the sphere. One key aspect of the project was the explicit construction of point sets on the sphere with small L2-discrepancy. The project made further progress in this fundamental unresolved question of discrepancy theory. Limit Distributions and External Fields: The project studied the limit distribution of points interacting according to a logarithmic or Riesz potential on finite cylinders, circular tori, and more general sets of revolution in 3-dimensional space. Numeric results clarified the behavior on the boundary of the support of the equilibrium and methods to determine the density function were developed. Somewhat surprising was the result that for logarithmic interaction a weak external field generated by points on the sphere led to a uniform limit distribution outside of spherical caps free of mass about the field sources. External fields were also applied to obtain asymptotically uniform distribution of points in the unit square (d-dimensional cube) by means of maximizing the sum of all mutual pairwise distances. (Without external field such points would lie on the boundary of the cube). Function approximation: Further work of the project concerned the complete asymptotic expansion of the logarithmic potential energy of points on the interval, the approximation on the sphere by means of so-called needlets, and the introduction of trial functions of specified Sobolev space smoothness to test the quality of numerical integration methods. The proofs used classical orthogonal polynomials and special functions. Several of the project's papers arose from networked research with projects of the SFB project "Quasi-Monte Carlo Methods: Theory and Applications" of the FWF.