Energy Minimizing Periodic Point Sets

对于给定的对势函数，最小化能量的点构型出现在不同的数学背景及其应用中。近年来，普遍最优点构型的研究揭示了一些引人注目的新现象。在这个项目中，我们将在周期点构型的研究中开辟新的领域，即晶格平移的有限并。创建和使用改进的数值工具进行计算实验，我们期望揭示新的现象，并收集普遍最优周期点集存在或不存在的证据。我们将开发和应用周期点构型空间的局部最优性的新准则。本文将对最近观测到的形式对偶周期点集现象进行研究，并尽可能对相应的点集进行分类。项目期间开发的软件将会发表，为其他研究人员今后的研究提供一个有用的工具。

Point configurations minimizing energy for a given pair potential function occur in diverse contexts of mathematics and its applications. In recent years the study of universally optimal point configurations has revealed some striking new phenomena. In this project we will break new grounds in the study of periodic point configurations, that is, for finite unions of translates of a lattice.Creating and using improved numerical tools for computational experiments, we expect to reveal new phenomena and collect evidence for the existence or non-existence of universally optimal periodic point sets. We will develop and apply new criteria for local optimality in the space of periodic point configurations. The recently observed phenomenon of formally-dual periodic point sets will be studied and, as far as possible, corresponding sets will be classified. The software developed during the project will be published, giving other researchers a useful tool for future studies.