Discrete Potential Theory and Perturbations of Ground State Configurations

本研究项目主要研究欧氏空间中紧致流形通过极小能量点的离散化，特别强调Riesz能量核。以前的工作的离散平衡配置的Riesz s-能量（广义汤姆森问题）表明调查的依赖性的行为的参数s的效用。 特别感兴趣的是临界值时，发生的S等于豪斯多夫维数的流形和过渡发生从远程到短程相互作用。 该项目将调查：（i）最小能量的更精细的渐近性及其与流形的曲率和光滑性的联系;（ii）对于2，8和24维，确定（或估计）的常数出现在最小的能量扩展的zeta函数在s中的特殊格存在于这些方面;（iii）渐近结果的能量自相似集;（iv）“贪婪能量点”的行为，特别是在外场存在的情况下;（v）旋转曲面上离散长程最小能量组态的极限支撑的确定;和（六）发展和分析算法的快速生成均匀分布的点的流形上。这个研究项目的重点是数学的带电粒子如何在一个当通过两粒子排斥相互作用相互作用时，弯曲表面以稳定构型排列它们自己。 对物质有序性的研究将拓宽对膜和薄膜物理学的理解，并应用于设计具有新颖光学和电子特性的新材料。 该项目的一个相关方面是在曲面（如地球）上快速生成数据采样点，可用于测量各种物理特性。 用于点生成的这种方法也可用于测试诸如雷达系统的检测设备。 这项研究在几个不同的背景下解决了如何最好地从模拟转换为数字的基本问题。

This research project focuses on the study of discretizations of compact manifolds in Euclidean space via minimal energy points, with special emphasis on Riesz energy kernels. Previous work on the discrete equilibrium configurations for the Riesz s-energy (generalized Thomson problem) showed the utility of investigating the dependence of behavior on the parameter s. Of particular interest is the critical value that occurs when s equals the Hausdorff dimension of the manifold and a transition occurs from long range to short range interactions. This project will investigate: (i) finer asymptotics for the minimal energy and its connection with the curvature and smoothness properties of the manifold; (ii) for dimensions 2, 8, and 24, the determination (or estimation) of constants arising in the minimal energy expansion in terms of the zeta functions in s for special lattices existing in these dimensions; (iii) asymptotic results for energy on self-similar sets; (iv) the behavior of "greedy energy points," especially in the presence of an external field; (v) the determination of the limiting support of discrete long range minimal energy configurations on surfaces of revolution; and (vi) development and analysis of algorithms for the fast generation of uniformly distributed points on manifolds.This research project focuses on the mathematics of how charged particles on a curved surface arrange themselves in a stable configuration when interacting through two-particle repulsive interactions. This study of the ordering of matter will broaden the understanding of the physics of membranes and films and has applications to the design of new materials with novel optical and electronic properties. A related aspect of the project is the rapid generation of data sampling points on curved surfaces (such as the earth) which can be used to measure a variety of physical properties. Such methods for point generation are also useful for testing detection devices such as radar systems. This research addresses in several different contexts the fundamental problem of how best to convert from analog to digital.