Related papers: Frieze patterns and combinatorics of curve singula…
Let $C$ be a generic complex plane plane curve with a given Newton polygon $P$. We compute the number of its inflection points and bitangents (equivalently, the number of singularities of the projectively dual curve $C^\vee$). We also prove…
We introduce a novel combinatorial method to study $Q^{**}$-transformations of group presentations or, equivalently, 3-deformations of CW-complexes of dimension 2. Our procedure is based on a refinement of discrete Morse theory that gives a…
In this note we present some results that were already conjectured in the work [9] by Bildhauer, Fuchs and Weickert, where they have investigated analytical aspects of coupled variational models with applications to mathematical imaging.…
We consider frieze sequences corresponding to sequences of cluster mutations for affine D and E type quivers. We show that the cluster variables satisfy linear recurrences with periodic coefficients, which imply the constant coefficient…
The primary objects of study in the ``knot theory of complex plane curves'' are C-links: links (or knots) cut out of a 3-sphere in the complex plane by complex plane transverse and totally tangential. Transverse C-links are naturally…
Frieze patterns have an interesting combinatorial structure, which has proven very useful in the study of cluster algebras. We introduce $(k,n)$-frieze patterns, a natural generalisation of the classical notion. A generalisation of the…
The dimensions of the graded quotients of the cohomology of a plane curve complement with respect to the Hodge filtration are described in terms of simple geometrical invariants. The case of curves with ordinary singularities is discussed…
For a cluster algebra $\mathcal{A}$ over $\mathbb{Q}$ of geometric type, a $\textit{frieze}$ of $\mathcal{A}$ is defined to be a $\mathbb{Q}$-algebra homomorphism from $\mathcal{A}$ to $\mathbb{Q}$ that takes positive integer values on all…
This thesis deals with three different aspects of the combinatorics of permutations. In the first two papers, two flavours of pattern avoiding permutations are examined; and in the third paper Young tableaux, which are closely related to…
Originally studied by Conway and Coxeter, friezes appeared in various recreational mathematics publications in the 1970s. More recently, in 2015, Baur, Parsons, and Tschabold constructed periodic infinite friezes and related them to…
The (usual) Caldero-Chapoton map is a map from the set of objects of a category to a Laurent polynomial ring over the integers. In the case of a cluster category, it maps "reachable" indecomposable objects to the corresponding cluster…
We present an intersection-theoretical approach to the invariants of plane curve singularities $\mu$, $\delta$, $r$ related by the Milnor formula $2\delta=\mu+r-1$. Using Newton transformations we give formulae for $\mu$, $\delta$, $r$…
A fundamental problem in spherical distance geometry aims to recover an $n$-tuple of points on a 2-sphere in $\mathbb{R}^3$, viewed up to oriented isometry, from $O(n)$ input measurements. We solve this problem using algorithms that employ…
In this paper, we extend the method developed in [17, 18] to curves in the Minkowski plane. The method proposes a way to study deformations of plane curves taking into consideration their geometry as well as their singularities. We deal in…
The notion of $SL_2$-tiling is a generalization of that of classical Coxeter-Conway frieze pattern. We classify doubly antiperiodic $SL_2$-tilings that contain a rectangular domain of positive integers. Every such $SL_2$-tiling corresponds…
The McKay correspondence has had much success in studying resolutions of 3-fold quotient singularities through a wide range of tools coming from geometry, combinatorics, and representation theory. We develop a computational perspective in…
We present a collection of results on the evolution by curvature of networks of planar curves. We discuss in particular the existence of a solution and the analysis of singularities.
Much of the analysis of F-theory-based Standard Models boils down to computing cohomologies of line bundles on matter curves. By varying parameters one can degenerate such matter curves to singular ones, typically with many nodes, where the…
We show the existence of toric resolution tower for an irreducible curve singularity which is explicitly described by Tschirnhausen polynomials. We deduce for a smooth affine plane curve from its topology restrictions for its singularity at…
Frieze patterns are numerical arrangements that satisfy a local arithmetic rule. These arrangements are actively studied in connection to the theory of cluster algebras. In the setting of cluster algebras, the notion of a frieze pattern can…