FWF - Self WW - Self organization by local interaction: minimal energy, external fields, and numerical integration

Project: Research project

Description

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.
StatusFinished
Effective start/end date1/10/1628/02/19