Related papers: A Short Lecture on Topological Crystallography and…
Crystals which have a uniform distribution of defects are endowed with a Lie group description which allows one to construct an associated discrete structure. These structures are in fact the discrete subgroups of the ambient Lie group. The…
The geometric shape, symmetry, and topology of colloidal particles often allow for controlling colloidal phase behavior and physical properties of these soft matter systems. In liquid crystalline dispersions, colloidal particles with low…
Topological surface states, a new kind of electronic state of matter, have recently been observed on the cleaved surfaces of crystals of a handful of small band gap semiconductors. The underlying chemical factors that enable these states…
Topological defects -- locations of local mismatch of order -- are a universal concept playing important roles in diverse systems studied in physics and beyond, including the universe, various condensed matter systems, and recently, even…
Physical systems are frequently modeled as sets of points in space, each representing the position of an atom, molecule, or mesoscale particle. As many properties of such systems depend on the underlying ordering of their constituent…
Sunada's work on crystallography emphasizes the role of the "maximal abelian cover" of a graph $X$. This is a covering space of $X$ for which the group of deck transformations is the first homology group $H_1(X,\mathbb{Z})$. An embedding of…
Multi-connected universe models with space identification scales smaller than the size of the observable universe produce topological images of cosmic sources. We generalise to locally hyperbolic spaces the crystallographic method, aimed to…
Crystalline assemblages of identical sub-units packed together and elastically bent in the form of a torus have been found in the past ten years in a variety of systems of surprisingly different nature, such as viral capsids, self-assembled…
In this paper we develop some combinatorial models for continuous spaces. In this spirit we study the approximations of continuous spaces by graphs, molecular spaces and coordinate matrices. We define the dimension on a discrete space by…
We use molecular dynamics to study the ordering of a nematic liquid crystal around a spherical particle or droplet. Homeotropic boundary conditions and strong anchoring create a hedgehog director configuration on the particle surface and in…
These are notes of a series of lectures on mirror symmetry and topological string theory given at the Mathematical Sciences Center at Tsinghua University. The N=2 superconformal algebra, its deformations and its chiral ring are reviewed. A…
We study topological states of matter in quasicrystals, which do not rely on crystalline orders. In the absence of a bandstructure description and spin-orbit coupling, we show that a three-dimensional quasicrystal can nevertheless form a…
We study static and dynamical properties that distinguish two dimensional crystals constrained to lie on a curved substrate from their flat space counterparts. A generic mechanism of dislocation unbinding in the presence of varying Gaussian…
Given a trivalent graph in the 3-dimensional Euclidean space, we call it a discrete surface because it has a tangent space at each vertex determined by its neighbor vertices. To abstract a continuum object hidden in the discrete surface, we…
Using tools and concepts from contact topology we show that non-vanishing twist implies conservation of the layer structure in cholesteric liquid crystals. This leads to a number of additional topological invariants for cholesteric…
This paper is devoted to the problem of choosing the most suitable model of a geometrical system for describing the real crystallographic space. It has been shown that all 230 crystallographic groups used to describe the crystalline…
A three-dimensional continuum dislocation theory for single crystals containing curved dislocations is proposed. A set of governing equations and boundary conditions is derived for the true placement, plastic slips, and loop functions in…
With the rapid development of topological states in crystals, the study of topological states has been extended to quasicrystals in recent years. In this review, we summarize the recent progress of topological states in quasicrystals,…
The aim of this paper is twofold. On the one hand, it provides a review of the links between random tensor models, seen as quantum gravity theories, and the PL-manifolds representation by means of edge-colored graphs (crystallization…
Stable array of point defects was generated in nematic liquid crystal. Point defects with topological charge of +1 and -1 (hedgehogs) were generated in vertically aligned liquid crystal cell. The hedgehogs were arranged in square or…