A Short Lecture on Topological Crystallography and a Discrete Surface Theory

Abstract

In this note, we discuss topological crystallography, which is a mathematical theory of crystal structures. The most symmetric structure among all placements of the graph is obtained by a variational principle in topological crystallography. We also discuss a theory of trivalent discrete surfaces in 3-dimensional Euclidean space, which are mathematical models of crystal/molecular structures. IntroductionGeometric analysis is a field of analysis on geometric objects such as manifolds, surfaces, and metric spaces. Discrete geometric analysis is an analysis on discrete geometric objects, for example, graphs, and contains the spectral theory and the probability theory of graphs. Topological crystallography and a discrete surface theory based on crystal/molecular structures are also parts of discrete geometric analysis.Topological crystallography is a mathematical theory of crystal structures, which is founded by M. Kotani and T. Sunada [20,21,22,39]. In physics, chemistry, and mathematics, crystal structures are described by space groups, which denote symmetry of placements of atoms. The usual description of crystals contains bonds between atoms in crystals. However, space groups do not consider such atomic bonds. Graphs are also natural notions to describe crystal structure, since vertices and edges of graphs correspond to atoms and atomic bonds in crystals, respectively. On the other hand, one of the important notions of physical phenomena is the principle of the least action, which corresponds to the variational principle in mathematics. That is to say, to describe physical phenomena, first we define an energy functional, which is called a Lagrangian in physics, and then we may find such phenomena as minimizers of the energy functional. There is no direct relationship between descriptions of crystal structures by using space groups and the principle of the least action.Topological crystallography gives us a direct relationship between symmetry of crystal structures and the variational principle. Precisely, for a given graph structure which describes a crystal, we define the energy of realizations of the graph (placements of vertices of the graph in suitable dimensional Euclidean space), and obtain a "good" structure as a minimizer of the energy. Moreover, such structures give us most symmetric among all placements of the graph, which is proved by using the theory of random walk on graphs.Molecular structures can be also described by using graph theory. The Hückel molecular orbital method, which is an important theory in physical chemistry, and the tight binding approximation for studying electronic states of crystals can be regarded as spectral theories of graphs from mathematical viewpoints. In this way, discrete geometric analysis can be applied to physics, chemistry, and related technologies.In the first few sections, we discuss topological crystallography including graph theory and geometry. The most important bibliography of this part is T. Sunada's lecture note [38]. The author discusses an introduction to t...