Related papers: Tiling the Euclidean and Hyperbolic planes with ri…
We show that the Charney--Davis strict hyperbolization procedure can preserve stable tangent bundles, answering a question of Charney and Davis. The key input is the construction of many hyperbolizing pieces, obtained using separability…
The present paper is a new version of the arXiv paper revisiting the proof given in a previous paper of the author published in 2008 proving that the general tiling problem of the hyperbolic plane is undecidable by proving a slightly…
To a given tiling a non commutative space and the corresponding C*-algebra are constructed. This includes the definition of a topology on the groupoid induced by translations of the tiling. The algebra is also the algebra of observables for…
This thesis is concerned with a realisation of topological theories in terms of statistical models known as Calabi-Yau crystals. The thesis starts with an introduction and review of topological field and string theories. Subsequently…
In this paper we present new results regarding the periodicity of outer billiards in the hyperbolic plane around polygonal tables which are tiles in regular two-piece tilings of the hyperbolic plane.
We investigate toric varieties defined by arrangements of hyperplanes and call them strongly symmetric. The smoothness of such a toric variety translates to the fact that the arrangement is crystallographic. As a result, we obtain a…
We demonstrate that higher-order electric susceptibilities in crystals can be enhanced and understood through nontrivial topological invariants and quantum geometry, using one-dimensional $\pi$-conjugated chains as representative model…
We show that Euclidean geometry in suitably high dimension can be expressed as a theory of orthogonality of subspaces with fixed dimensions and fixed dimension of their meet.
We briefly review the standard methods used to construct quasiperiodic tilings, such as the projection, the inflation, and the grid method. A number of sample Mathematica programs, implementing the different approaches for one- and…
We study infinite dimensional tilting modules over a concealed canonical algebra of domestic or tubular type. In the domestic case, such tilting modules are constructed by using the technique of universal localization, and they can be…
Given any knot k, there exists a hyperbolic knot tilde k with arbitrarily large volume such that the knot group pi k is a quotient of pi tilde k by a map that sends meridian to meridian and longitude to longitude. The knot tilde k can be…
The Heesch problem 'grades' polygons that fail to tile the plane in terms of the number of layers (or corollas) of copies of it that can be formed around a central unit. We study the different topology of ' walls', which we define to be…
We construct a limit aperiodic coloring of hyperbolic groups. Also we construct limit strongly aperiodic strictly balanced tilings of the Davis complex for all Coxeter groups.
We study the cohomology and hence $K$-theory of the aperiodic tilings formed by the so called 'cut and project' method, i.e., patterns in $d$ dimensional Euclidean space which arise as sections of higher dimensional, periodic structures.…
We classify the dihedral edge-to-edge tilings of the sphere by squares and rhombi.
This article examines the tilings of a strip with equilateral triangles. The number of ways in which the lattices can be covered with a combination of tiles of the two types of triangles is related to Pell's numbers. Additionally, the…
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…
There are three complete plane geometries of constant curvature: spherical, Euclidean and hyperbolic geometry. We explain how a closed oriented surface can carry a geometry which locally looks like one of these. Focussing on the hyperbolic…
Regluing is a topological operation that helps to construct topological models for rational functions on the boundaries of certain hyperbolic components. It also has a holomorphic interpretation, with the flavor of infinite dimensional…
In 1885, Fedorov discovered that a convex domain can form a lattice tiling of the Euclidean plane if and only if it is a parallelogram or a centrally symmetric hexagon. This paper proves the following results: Besides parallelograms and…