Related papers: Topological Crystals
Data analysis often concerns not only the space where data come from, but also various types of maps attached to data. In recent years, several related structures have been used to study maps on data, including Reeb spaces, mappers and…
The discovery of topological insulators has reformed modern materials science, promising to be a platform for tabletop relativistic physics, electronic transport without scattering, and stable quantum computation. Topological invariants are…
Using combinatorics of Young walls, we give a new realization of arbitrary level irreducible highest weight crystals $\mathcal{B}(\lambda)$ for quantum affine algebras of type $A_n^{(1)}$, $B_n^{(1)}$, $C_n^{(1)}$, $A_{2n-1}^{(2)}$,…
Abundant in nature, colloids also find increasingly important applications in science and technology, ranging from direct probing of kinetics in crystals and glasses to fabrication of third-generation quantum-dot solar cells. Because…
Effective homology techniques allow us to compute homology groups of a wide family of topological spaces. By the Whitehead tower method, this can also be used to compute higher homotopy groups. However, some of these techniques (in…
We construct a multiset space $\mathbb{N}[X]$ over a metric space $X$ that simultaneously enjoys desirable topological properties and admits a natural matching metric $d_{\mathbb{N}[X]}$, making it a metrizable abelian topological monoid…
Sormani and Wei proved in 2004 that a compact geodesic space has a categorical universal cover if and only if its covering/critical spectrum is finite. We add to this several equivalent conditions pertaining to the geometry and topology of…
The method of the space dependent basis is applied to study electronic spinors in a crystal. The crystal in the momentum space is described by the Brillouine zone which might contains obstructions or degeneracies for which requires…
Whereas knowledge of a crystalline material's unit cell is fundamental to understanding the material's properties and behavior, there are not obvious analogues to unit cells for disordered materials despite the frequent existence of…
The structure of the densest crystal packings is determined for a variety of concave shapes in 2D constructed by the overlap of two or three disks. The maximum contact number per particle pair is defined and proposed as a useful means of…
We discuss the higher-order topological field theory and response of topological crystalline insulators with no other symmetries. We show how the topology and geometry of the system is organised in terms of the elasticity tetrads which are…
In a dilute two-dimensional electron gas, Coulomb interactions can stabilize the formation of a Wigner crystal. Although Wigner crystals are topologically trivial, it has been predicted that electrons in a partially-filled band can break…
The kth finite subset space of a topological space X is the space exp_k X of non-empty finite subsets of X of size at most k, topologised as a quotient of X^k. The construction is a homotopy functor and may be regarded as a union of…
Given a topological group $G$ that can be embedded as a topological subgroup into some topological vector space (over the field of reals) we say that $G$ has invariant linear span if all linear spans of $G$ under arbitrary embeddings into…
This paper studies the combinatorics of ideals which recently appeared in ergodicity results for analytic equivalence relations. The ideals have the following topological representation. There is a separable metrizable space $X$, a…
Liquid crystals generally support orientational singularities of the director field known as topological defects. These latter modifiy transport properties in their vicinity as if the geometry was non-Euclidean. We present a state of the…
Let $X$ be the prime spectrum of a ring. In [arXiv:0707.1525] the authors define a topology on $X$ by using ultrafilters and they show that this topology is precisely the constructible topology. In this paper we generalize the construction…
The celebrated tenfold-way of Altland-Zirnbauer symmetry classes discern any quantum system by its pattern of non-spatial symmetries. It lays at the core of the periodic table of topological insulators and superconductors which provided a…
A datatset $X$ on $R^2$ is a finite topological space. Current research of a dataset focuses on statistical methods and the algebraic topological method \cite{carlsson}. In \cite{hu}, the concept of typed topological space was introduced…
The fundamental, or first, band gap is of unmatched importance in the study of photonic crystals. Here, we address precisely where this gap can be opened in the band structure of three-dimensional photonic crystals. Although strongly…