Related papers: Cluster algebras: Notes for the CDM-03 conference
Matrix mutation appears in the definition of cluster algebras of Fomin and Zelevinsky. We give a representation theoretic interpretation of matrix mutation, using tilting theory in cluster categories of hereditary algebras. Using this, we…
I summarize what is know about the evolution of galaxies in clusters from the observational point of view presenting results at high (z~1), intermediate (z~0.5) and low (z~0) redshifts. I comment on the comparison between observations and…
In this paper we give a graph theoretic combinatorial interpretation for the cluster variables that arise in most cluster algebras of finite type. In particular, we provide a family of graphs such that a weighted enumeration of their…
It is well known that the relation-extensions of tilted algebras are cluster-tilted algebras. In this paper, we extend the result to silted algebras and prove some extension of silted algebras are cluster-tilted algebras.
We introduce a class of non-commutative algebras that carry a non-commutative (geometric) cluster structure which are generated by identical copies of generalized Weyl algebras. Equivalent conditions for the finiteness of the set of the…
Clustering is an unsupervised learning problem that aims to partition unlabelled data points into groups with similar features. Traditional clustering algorithms provide limited insight into the groups they find as their main focus is…
These are notes of a mini-course given at Dennisfest in June 2001. The goal of these notes is to give a self-contained survey of deformation quantization, operad theory, and graph homology. Some new results related to "String Topology" and…
We are learning much about how structure forms, in particular how clusters as nodes in the cosmic web evolve and accrete matter, and about the physical processes within these objects. In the next decade, the study of clusters will enable us…
Given a finite dimensional algebra $C$ (over an algebraically closed field) of global dimension at most two, we define its relation-extension algebra to be the trivial extension $C\ltimes \Ext_C^2(DC,C)$ of $C$ by the $C$-$C$-bimodule…
These are lecture notes mainly aimed at graduate students on selected aspects of generalized geometry: in particular generalized complex and Kaehler structures and generalized holomorphic bundles. They are based on lectures given in March…
It is well known that most of the common clustering objectives are NP-hard to optimize. In practice, however, clustering is being routinely carried out. One approach for providing theoretical understanding of this seeming discrepancy is to…
It is well known that most of the common clustering objectives are NP-hard to optimize. In practice, however, clustering is being routinely carried out. One approach for providing theoretical understanding of this seeming discrepancy is to…
We continue the study of cluster algebras initiated in math.RT/0104151 and math.RA/0208229. We develop a new approach based on the notion of an upper cluster algebra, defined as an intersection of certain Laurent polynomial rings.…
Generalized Cluster Algebras (GCA) are generalizations of Cluster Algebras (CA) with higher-order exchange relations. Previously, Chekhov-Shapiro conjectured that every GCA can be embedded into a CA. In this paper, we prove a modified…
We use the representation theory of preprojective algebras to construct and study certain cluster algebras related to semisimple algebraic groups.
This paper continues the study of cluster algebras initiated in math.RT/0104151. Its main result is the complete classification of the cluster algebras of finite type, i.e., those with finitely many clusters. This classification turns out…
In this introductory talk we will establish connections between the statistical analysis of galaxy clustering in cosmology and recent work in mainstream spatial statistics. The lecture will review the methods of spatial statistics used by…
We continue the work started in parts (I) and (II). In this part we classify which continuous type A quivers are derived equivalent and introduce the new continuous cluster category with E-clusters, which are a generalization of clusters.…
We construct the intermediate coverings of cluster-tilted algebras by defining the generalized cluster categories. These generalized cluster categories are Calabi-Yau triangulated categories with fraction CY-dimension and have also cluster…
These are lectures notes for the introductory graduate courses on geometric complexity theory (GCT) in the computer science department, the university of Chicago. Part I consists of the lecture notes for the course given by the first author…