Related papers: Introduction to Periodic Geometry and Topology

Mathematical crystal chemistry views crystal structures as the optimal solutions of mathematical optimization problem formalizing inorganic structural chemistry. This paper introduces the minimum and maximum atomic radii depending on the…

Topological materials are quantum materials with nontrivial ground-state entanglement that are irremovable so long as certain rules, like invariance under symmetries and the existence of an energy gap, are respected. They showcase…

We try to convince the reader that the categorical version of differential geometry, called Synthetic Differential Geometry (SDG), offers valuable tools which can be applied to work with some unsolved problems of general relativity. We do…

Doubly periodic tangles, or DP tangles, are embeddings of curves in the thickened plane that are periodically repeated in two directions. They are defined as universal covers of their generating cells, the flat motifs, which represent knots…

Statistical Topology emerged since topological aspects continue to gain importance in many areas of physics. It is most desirable to study topological invariants and their statistics in schematic models that facilitate the identification of…

A periodic structure sandwiched between two homogeneous media can support bound states in the continuum (BICs) that are valuable for many applications. It is known that generic BICs in periodic structures with an up-down mirror symmetry and…

Time-periodic form or expression is a ubiquitous natural and man-made phenomenon observable in all the scientific and engineering disciplines. In this article, we propose a theory of periodic sequence (TPS), which can be formulated as a…

The tenfold classification provides a powerful framework for organizing topological phases of matter based on symmetry and spatial dimension. However, it does not offer a systematic method for transitioning between classes or engineering…

Topological insulators are materials that conduct on the surface and insulate in their interior due to non-trivial topological order. The edge states on the interface between topological (non-trivial) and conventional (trivial) insulators…

Disordered hyperuniform systems are exotic states of matter that completely suppress large-scale density fluctuations like crystals, and yet possess no Bragg peaks similar to liquids or glasses. Such systems have been discovered in a…

Crystals are paradigms of ordered structures. While order was once seen as synonymous with lattice periodic arrangements, the discoveries of incommensurate crystals and quasicrystals led to a more general perception of crystalline order,…

This work introduces a novel framework for quantifying the presence and strength of recurrent dynamics in video data. Specifically, we provide continuous measures of periodicity (perfect repetition) and quasiperiodicity (superposition of…

Topological acoustic and elastic waves have recently emerged as an exciting interdisciplinary field which is still mainly focused on low-dimensional structures and model systems. Here we demonstrate numerically an elastic-wave analogue of…

The topological mechanics is a perfect tool that can bridge the gap between the quantum and Newtonian physics and mechanics of materials. It requires discrete models of the material with analogies with the topological characteristics of…

Coarse geometry studies metric spaces on the large scale. The recently introduced notion of coarse entropy is a tool to study dynamics from the coarse point of view. We prove that all isometries of a given metric space have the same coarse…

This article proposes a topological method that extracts hierarchical structures of various amorphous solids. The method is based on the persistence diagram (PD), a mathematical tool for capturing shapes of multiscale data. The input to the…

The paper is devoted to a categorical study of the category of probabilistic metric spaces. The study is based on an isomorphic description of the category of probabilistic metric spaces. The isomorphic description was obtained in [3] and…

Topological semimetals have emerged as an important class of quantum materials with novel electronic responses and unconventional transport phenomena. Among them, nodal-line semimetals are distinguished by band crossings that extend along…

Metric graphs are ubiquitous in science and engineering. For example, many data are drawn from hidden spaces that are graph-like, such as the cosmic web. A metric graph offers one of the simplest yet still meaningful ways to represent the…

Crystal structures are characterised by repeating atomic patterns within unit cells across three-dimensional space, posing unique challenges for graph-based representation learning. Current methods often overlook essential periodic boundary…