In this article, we discuss applications of topology to materials science. We explain how topology is used to express the structure and shape of materials. The structure of a multicyclic polymer is described by applying graph theory and the 3-dimensional shape is described by applying knot theory. Polycontinuous patterns of microphase separation of a block copolymer melt can be analyzed by applying the 3-dimensional topology. We also discuss the topological structures of interlocking molecules composed of metal-peptide rings using the concept of polyhedral links.

Geometric structures of carbon networks are key in designing their material properties. In particular, optimization of curved structures through the introduction of topological defects and doping of heteroatoms in the lattice is crucial to the design of carbon-based, non-noble-metal-free catalysts. A simple and practical mathematical model based on discrete geometric analysis is proposed to describe the geometric features of carbon networks and their relationships to their material properties. This model can pre-screen candidates for novel material design, and these candidates can be further examined by the density functional theory (DFT). Inspired by observations regarding the preferential doping of heteroatoms at local curved sites, the important characteristics of the candidate material were experimentally realized, and its enhanced catalytic activity facilitated by chemical dopants was confirmed in the designed carbon network.

Polycrystalline materials are ubiquitous, and their macroscopic properties strongly depend on the defects in crystalline structures. Grain boundaries (GBs) are the interfaces between crystal grains, which admit unique atomic arrangements different from those of crystals. It is essential to reveal the polyhedral arrangement in GB atomic structures to identify the origin of structure-property relationships in polycrystalline materials. We found that the GB region can only be packed by the bulk or GB-type polyhedral units with minor variations. Moreover, the GB hierarchy directly follows the distribution of rational numbers that is represented by the Farey diagram. The singular GB is defined to be a GB packed by at the most two types of polyhedral units in a period, and any GB can be packed by the polyhedral units that form the singular GBs. This approach allows us to predict the polyhedral arrangement in any tilt GBs including random GBs.

We review our recent study on the development of a quantitative data analysis technique for angle-resolved photoemission spectroscopy. The new method enables us to precisely disentangle one-particle band dispersion and many-body effects from ARPES data based on Bayesian inference.

Molecules can be regarded as a kind of graph with atoms as vertices and bonds as edges. Therefore, it is possible to discuss the physical properties and reactivity of various molecules by applying the ideas of graph theory and network theory. Such research has flourished in the field of chemical graph theory. In this manuscript, it is shown that the lowest occupied molecular orbital (LOMO) can be interpreted graph theoretically as eigenvector centrality. This is one measure of centrality that characterizes the importance or influence of a node on a network. As such, LOMO coefficient can be regarded as a manifestation of centrality in an aggregate of atoms, indicating which atom plays the most important role in that aggregate or has the greatest influence on the atom network. Using such properties of LOMO, an analysis of the network of metal atoms in metal clusters is performed. The predictability of the binding energies of the constituent atoms of the metal clusters using LOMO coefficients is discussed.

Polymers are raw materials for products in our daily life such as plastics, resin, rubber, and organic glasses. The long molecular chains in polymers generate versatile and complex structure in mesoscopic scale, which affects the material property. Therefore, it is hard to deal with the mesostructure-property relationships by simple experiment or theory. Recently, the author and coworkers have developed the mesostructured-property relationships in polymer science based on mathematical and informatics methods. First, description of rubber elasticity of elastomers by complex network analysis are represented. Then, process-mesostructure-property relationships in crystalline polymers by machine learning techniques are mentioned. Furthermore, reverse analysis of biodegradable polymers with both toughness and degradability are included.

We study non-local effects on inhomogeneous flows of soft athermal particles near the jamming transition. We employ molecular dynamics simulations to demonstrate Kolmogorov flows, where a sinusoidal flow profile with fixed wave number is externally imposed, resulting in a spatially inhomogeneous shear rate. We find that the rheology of soft athermal particles is strongly wave number-dependent and particle migration is not sufficient to explain the resulting stress profiles within conventional local constitutive relations. We show that stress profiles can be described by non-local constitutive relations that account for gradients to fourth order.

We introduce our state-of-the art of “vacuum electrochemistry” to investigation of the interfaces between oxides and ionic liquid (IL). Pulsed laser deposition (PLD) has been one of the powerful and sophisticated techniques to realize nanoscale preparation of high-quality oxide thin films. On the other hand, electrochemistry is a simple, very sensitive, and non-destructive analysis technique for solid-liquid interfaces. To ensure the reproducibility in experiment of the interfaces of such epitaxial oxide films, as well as bulk oxide single-crystals, with IL, we employ a home-built PLD-electrochemical (EC) system with IL as an electrolyte. The system allows one to perform all-in-vacuum experiments from preparation of well-defined oxide electrode surfaces to their electrochemical analyses. The topics include electrochemical evaluations of the oxide's own properties, such as carrier density and dielectric constant, and the interfacial properties of oxides in contact with IL, such as flat band potential and electric double layer (EDL) capacitance.