Development and applications of discrete geometric analysis

该项目的主要目的是提供对现代晶体学的数学见解，这是一门典型的实用科学，起源于对观察到的晶体形状进行分类。我们所采用的工具是从代数拓扑学，在纯数学领域在上个世纪上半叶培育。更具体地说，覆盖空间和同调的基本理论有效地用于与晶体相关的3D网络的研究。此外，我们制定了一个最小原则的离散几何分析，这为我们提供了标准实现的概念，一个规范的方式来放置一个给定的晶体结构在空间中，以产生最对称的微观形状的框架中的晶体。尽管它的纯数学性质，这一概念结合同源性理论证明适合系统的设计和枚举晶体结构，一个领域的相当大的科学兴趣多年。同时，标准实现出现在拓扑晶体上随机游动的渐近行为，晶体结构的抽象，并与代数几何中Abel-Jacobi映射的离散模拟密切相关。

The primarily purpose of this project was to provide a mathematical insight into the modern crystallography, a typical practical science that originated in the classification of the observed shapes of crystals. The tools we employed are adopted from algebraic topology, a field in pure mathematics cultivated during the first half of the last century. More specifically the elementary theory of covering spaces and homology is effectively used in the study of 3D networks associated with crystals. Further we formulate a minimum principle for crystals in the framework of discrete geometric analysis, which provides us with the concept of standard realizations, a canonical way to place a given crystal structure in space so as to produce the most symmetric microscopic shape. In spite of its pure-mathematical nature, this concept combined with homology theory turns out to fit with a systematic design and enumeration of crystal structures, an area of considerable scientific interest for many years. Meanwhile, standard realizations show up in asymptotic behaviors of random walks on topological crystals, the abstraction of crystal structures, and are closely related to a discrete analogue of Abel-Jacobi maps in algebraic geometry.