Discrete Potential Theory and Perturbations of Ground State Configurations

本课题主要研究欧几里得空间中紧流形的极小能量点离散化问题，重点研究Riesz能量核。先前关于Riesz - s-能量的离散平衡构型（广义汤姆森问题）的工作表明了研究行为对参数s的依赖的效用。特别感兴趣的是当s等于流形的豪斯多夫维数并且发生从远程相互作用到短程相互作用的转变时发生的临界值。本项目将研究：(i)最小能量的精细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，对于这些维中存在的特殊晶格，用s中的ζ函数确定（或估计）最小能量展开中产生的常数；（iii）自相似集上能量的渐近结果；（iv）“贪心能量点”的行为，特别是在存在外部场的情况下；(v)确定离散的长距离最小能量构型在旋转表面上的极限支撑；（六）流形上均匀分布点快速生成算法的开发与分析。这个研究项目的重点是如何在一个弯曲的表面上的带电粒子安排自己在一个稳定的配置时，通过两粒子排斥相互作用的数学。这项对物质有序的研究将拓宽对膜和薄膜物理的理解，并应用于具有新颖光学和电子特性的新材料的设计。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。这种点生成方法也适用于测试探测设备，如雷达系统。本研究在几个不同的背景下解决了如何最好地从模拟到数字转换的基本问题。

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.