Related papers: Cluster algebras and triangulated surfaces. Part I…
We give a module-theoretic interpretation of Schiffler's expansion formula which is defined combinatorially in terms of complete (T,r)-paths in order to get the expansion of the cluster variables in the cluster algebra of a marked surface…
We investigate topological T-duality in the framework of non-abelian gerbes and higher gauge groups. We show that this framework admits the gluing of locally defined T-duals, in situations where no globally defined ("geometric") T-duals…
We prove a multiplication formula for cluster characters induced by generating extensions in a gentle algebra A, generalizing a result of Cerulli Irelli, Esposito, Franzen, Reineke. In the case where A is the gentle algebra of a…
Spectral clustering is one of the most popular clustering methods for finding clusters in a graph, which has found many applications in data mining. However, the input graph in those applications may have many missing edges due to error in…
There is a growing interest in characterizing circular data found in biological systems. Such data are wide ranging and varied, from signal phase in neural recordings to nucleotide sequences in round genomes. Traditional clustering…
We establish a novel relation between the cluster categories associated with marked surfaces and the topological Fukaya categories of the surfaces. We consider a generalization of the triangulated cluster category of the surface by a…
The iteration dynamics of the coupled cluster equations exhibits a synergistic relationship among the cluster amplitudes. The iteration scheme may be viewed as a multivariate discrete-time propagation of nonlinearly coupled equations, which…
Cluster algebras, introduced by Fomin and Zelevinsky through the process of quiver mutation, have become central objects in modern algebra and geometry, linking combinatorial constructions with diverse mathematical domains such as…
This is an introductory survey on cluster algebras and their (additive) categorification using derived categories of Ginzburg algebras. After a gentle introduction to cluster combinatorics, we review important examples of coordinate rings…
We develop a version of cluster algebra extending the ring of Laurent polynomials by adding Grassmann variables. These algebras can be described in terms of `extended quivers' which are oriented hypergraphs. We describe mutations of such…
We prove several new results on the combinatorial structures of the unit spheres of the norms induced by Thurston's metric on the tangent and cotangent spaces of the Teichm{\"u}ller space of a closed surface of negative Euler…
Cluster categories and cluster algebras encode two dimensional structures. For instance, the Auslander--Reiten quiver of a cluster category can be drawn on a surface, and there is a class of cluster algebras determined by surfaces with…
Bordered Floer homology associates to a parametrized oriented surface a certain differential graded algebra. We study the properties of this algebra under splittings of the surface. To the circle we associate a differential graded…
Higher Auslander algebras were introduced by Iyama generalizing classical concepts from representation theory of finite dimensional algebras. Recently these higher analogues of classical representation theory have been increasingly studied.…
We continue our investigation on denominator conjecture of Fomin and Zelevinsky for cluster algebras via geometric models initialed in \cite{FG22}. In this paper, we confirm the denominator conjecture for cluster algebras of finite type.…
In a categorification of skew-symmetric cluster algebras, each cluster variable corresponds with an indecomposable module over the associated Jacobian algebra. Buan, Marsh and Reiten studied when the denominator vector of each cluster…
We introduce a framework for $\mathbb{Z}$-gradings on cluster algebras (and their quantum analogues) that are compatible with mutation. To do this, one chooses the degrees of the (quantum) cluster variables in an initial seed subject to a…
We characterize completey (give a necessary and suffcient condition using special neat embeddings)for a relation algebra to belong to the amalgamation, strong amalgamation, and superamalgamation base of the class of representable algebras.…
The cluster category is a triangulated category introduced for its combinatorial similarities with cluster algebras. We prove that a cluster algebra A of finite type can be realized as a Hall algebra, called the exceptional Hall algebra, of…
Affine cluster varieties are covered up to codimension 2 by open algebraic tori. We put forth a general conjecture (based on earlier conversation between Vivek Shende and the last author) characterizing their deep locus, i.e. the complement…