Related papers: Cluster algebras and triangulated surfaces. Part I…
In a vertex algebra setting, we consider non-local screening operators associated to the basis of any non-integral lattice. We have previously shown that, under certain restrictions, these screening operators satisfy the relations of a…
We show that cluster algebras do not contain non-trivial units and that all cluster variables are irreducible elements. Both statements follow from Fomin and Zelevinsky's Laurent phenomenon. As an application we give a criterion for a…
We introduce a tensor-based clustering method to extract sparse, low-dimensional structure from high-dimensional, multi-indexed datasets. This framework is designed to enable detection of clusters of data in the presence of structural…
Clustering is a popular unsupervised learning tool often used to discover groups within a larger population such as customer segments, or patient subtypes. However, despite its use as a tool for subgroup discovery and description - few…
In this paper we try to look at the compactification of Teichmuller spaces from a tropical viewpoint. We describe a general construction for the compactification of algebraic varieties, using their amoebas, and we describe the boundary via…
We describe in elementary geometrical terms Teichm\" uller spaces of decorated and holed surfaces. We construct explicit global coordinates on them as well as on the spaces of measured laminations with compact and closed support…
We show that having any planar (cyclic or acyclic) directed network on a disc with the only condition that all $n_1+m$ sources are separated from all $n_2+m$ sinks, we can construct a cluster-algebra realization of elements of an affine…
Clustering is a fundamental learning task widely used as a first step in data analysis. For example, biologists use cluster assignments to analyze genome sequences, medical records, or images. Since downstream analysis is typically…
The geometrical features of the (non-convex) loss landscape of neural network models are crucial in ensuring successful optimization and, most importantly, the capability to generalize well. While minimizers' flatness consistently…
Graph clustering groups entities -- the vertices of a graph -- based on their similarity, typically using a complex distance function over a large number of features. Successful integration of clustering approaches in automated…
We consider, for each exchange matrix B, a category of geometric cluster algebras over B and coefficient specializations between the cluster algebras. The category also depends on an underlying ring R, usually the integers, rationals, or…
We introduce tabular algebras, which are simultaneous generalizations of cellular algebras (in the sense of Graham-Lehrer) and table algebras (in the sense of Arad-Blau). We show that if a tabular algebra is equipped with a certain kind of…
Conformal nets provide a mathematical formalism for conformal field theory. Associated to a conformal net with finite index, we give a construction of the `bundle of conformal blocks', a representation of the mapping class groupoid of…
A general method is described for detecting and analysing galaxy systems. The multivariate geometrical structure of the sample is studied by using an extension of the method which we introduced in a previous paper. The method is based on an…
The recent investigation of the gauge structure of extended geometry is generalised to situations when ancillary transformations appear in the commutator of two generalised diffeomorphisms. The relevant underlying algebraic structure turns…
The cluster-tilted algebras have been introduced by Buan, Marsh and Reiten, they are the endomorphism rings of cluster-tilting objects $T$ in cluster categories; we call such an algebra cluster-concealed in case $T$ is obtained from a…
In a cluster algebra, a subset of initial cluster variables can be specialised in such a way that all elements of the resulting algebra become polynomial in the remaining variables.
This paper concerns cluster algebras with principal coefficients A(S,M) associated to bordered surfaces (S,M), and is a companion to a concurrent work of the authors with Schiffler [MSW2]. Given any (generalized) arc or loop in the surface…
In this paper, we present a novel non-parametric clustering technique. Our technique is based on the notion that each latent cluster is comprised of layers that surround its core, where the external layers, or border points, implicitly…
We discuss in this paper combinatorial aspects of boundary loop models, that is models of self-avoiding loops on a strip where loops get different weights depending on whether they touch the left, the right, both or no boundary. These…