相关论文: Penrose tilings, infinite friezes, and the $A_\inf…
We study a category $\mathcal{C}_2$ of $\mathbb{Z}$-graded MCM modules over the $A_\infty$ curve singularity and demonstrate it has infinite type $A$ cluster combinatorics. In particular, we show that this Frobenius category (or a suitable…
We provide a characterization of infinite frieze patterns of positive integers via triangulations of an infinite strip in the plane. In the periodic case, these triangulations may be considered as triangulations of annuli. We also give a…
Paquette and Y{\i}ld{\i}r{\i}m recently introduced triangulated categories of arcs in completed infinity-gons, which are discs with an infinite closed set of marked points on their boundary. These categories have many features in common…
This is a survey on recent developments in Cohen-Macaulay representations via tilting and cluster tilting theory. We explain triangle equivalences between the singularity categories of Gorenstein rings and the derived (or cluster)…
We show that the well known two-dimensional Penrose tiling admits an infinite number of independent scaling factors and an infinite number of inflation centers.
We provide a classification of positive integral friezes on marked bordered surfaces in the style of Conway and Coxeter. More precisely, we prove that positive integral friezes are in one-to-one correspondence with ideal triangulations…
It is known that any infinite frieze comes from a triangulation of an annulus by Baur, Parsons and Tschabold. In this paper we show that each periodic infinite frieze determines a triangulation of an annulus in essentially a unique way.…
We first study the (canonical) orbit category of the bounded derived category of finite dimensional representations of a quiver with no infinite path, and we pay more attention on the case where the quiver is of infinite Dynkin type. In…
Considering quantales as generalised noncommutative spaces, we address as an example a quantale Pen based on the Penrose tilings of the plane. We study in general the representations of involutive quantales on those of binary relations, and…
Penrose hyperbolic tilings are tilings of the hyperbolic plane which admit, up to affine transformations a finite number of prototiles. In this paper, we give a complete description of the C*-algebras and of the K-theory for such tilings.…
We construct a non-commutative scheme that behaves as if it is the space of Penrose tilings of the plane. Let k be a field and B=k<x,y>(y^2). We consider B as the homogeneous coordinate ring of a non-commutative projective scheme. The…
We define an elementary relatively $\mathbb Z/4$ graded Lagrangian-Floer chain complex for restricted immersions of compact 1-manifolds into the pillowcase, and apply it to the intersection diagram obtained by taking traceless $SU(2)$…
We present new results about fully leafed induced subtrees in Penrose P2 tilings. We first determine the graph structure of these subtrees and show that they are caterpillars, up to an appendix of at most six tiles. We then study…
Recently there has been significant progress in classifying integer friezes and $\text{SL}_2$-tilings. Typically, combinatorial methods are employed, involving triangulations of regions and inventive counting techniques. Here we develop a…
This paper investigates a certain 2-Calabi-Yau triangulated category D whose Auslander-Reiten quiver is ZA_{\infty}. We show that the cluster tilting subcategories of D form a so-called cluster structure, and we classify these subcategories…
A new infinite class of Chern-Simons theories is presented using brane tilings. The new class reproduces all known cases so far and introduces many new models that are dual to M2 brane theories which probe a toric non-compact CY 4-fold. The…
We study constrained percolation models on planar lattices including the $[m,4,n,4]$ lattice and the square tilings of the hyperbolic plane, satisfying certain local constraints on faces of degree 4, and investigate the existence of…
We introduce a new family of nonperiodic tilings, based on a substitution rule that generalizes the pinwheel tiling of Conway and Radin. In each tiling the tiles are similar to a single triangular prototile. In a countable number of cases,…
Tropical friezes are the tropical analogues of Coxeter-Conway frieze patterns. In this note, we study them using triangulated categories. A tropical frieze on a 2-Calabi-Yau triangulated category $\mathcal{C}$ is a function satisfying a…
Finite frieze patterns with entries in $\mathbb{Z}[\lambda_{p_1},\ldots,\lambda_{p_s}]$ where $\{p_1,\ldots,p_s\} \subseteq \mathbb{Z}_{\geq 3}$ and $\lambda_p = 2 \cos(\pi/p)$ were shown to have a connection to dissected polygons by Holm…