Related papers: Cluster algebras: Notes for the CDM-03 conference
In this paper, we will present the author's interpretation and embellishment of five lectures on cluster theory given by Kiyoshi Igusa during the Spring semester of 2022 at Brandeis University. They are meant to be used as an introduction…
Cluster algebras were introduced by Fomin-Zelevinsky in 2002 in order to give a combinatorial framework for phenomena occurring in the context of algebraic groups. Cluster algebras also have links to a wide range of other subjects,…
In this work, I develop a new view of presentation theory for C*-algebras, both unital and non-unital, heavily grounded in classical notions from algebra. In particular, I introduce Tietze transformations for these presentations, which lead…
These lectures cover various aspects of the statistical description of cosmological density fields. Observationally, this consists of the point process defined by galaxies, and the challenge is to relate this to the continuous density field…
These are extended notes of a course given at Tulane University for the 2015 Clifford Lectures. Their aim is to present structure results for group schemes of finite type over a field, with applications to Picard varieties and automorphism…
We present a combinatorial model for cluster algebras of type $D_n$ in terms of centrally symmetric pseudotriangulations of a regular $2n$-gon with a small disk in the centre. This model provides convenient and uniform interpretations for…
The cluster analysis of very large objects is an important problem, which spans several theoretical as well as applied branches of mathematics and computer science. Here we suggest a novel approach: under assumption of local convergence of…
Clustering is a fundamental machine learning task which has been widely studied in the literature. Classic clustering methods follow the assumption that data are represented as features in a vectorized form through various representation…
We give a uniform geometric realization for the cluster algebra of an arbitrary finite type with principal coefficients at an arbitrary acyclic seed. This algebra is realized as the coordinate ring of a certain reduced double Bruhat cell in…
In 2011 Musiker, Schiffler and Williams obtained expansion formulae for cluster algebras from orientable surfaces. For singly and doubly notched arcs these formulae required the notion of $\gamma$-symmetric perfect matchings and…
We apply the new theory of cluster algebras of Fomin and Zelevinsky to study some combinatorial problems arising in Lie theory. This is joint work with Geiss and Schr\"oer (3, 4, 5, 6), and with Hernandez (8, 9).
Cluster algebras are a recent topic of study and have been shown to be a useful tool to characterize structures in several knowledge fields. An important problem is to establish whether or not a given cluster algebra is of finite type.…
Graph clustering is a fundamental problem in unsupervised learning, with numerous applications in computer science and in analysing real-world data. In many real-world applications, we find that the clusters have a significant high-level…
This is an appendix to the Handbook of Tilting Theory, edited by Angeleri-Huegel, Happel and Krause, to be published soon. Part 1 of the appendix provides an outline of the core of tilting theory. Part 2 is devoted to topics where tilting…
We clarify the natural cluster algebra of type A that exists in a residual and tropical form in the kinematical space as suggested in 1711.09102 by the use of triangulations, mutations and associahedron on the definition of scattering…
This contribution gives a short review of recent theoretical advances in most topics of nuclear cluster physics concentrating, however, around {$\alpha$} particle clustering. Along the route, the point of view will be critical mentioning…
In the cluster algebra literature, the notion of a graded cluster algebra has been implicit since the origin of the subject. In this work, we wish to bring this aspect of cluster algebra theory to the foreground and promote its study. We…
This is a graduate-level introduction to C*-algebras, Hilbert C*-modules, vector bundles, and induced representations of groups and C*-algebras, with applications to quantization theory, phase space localization, and configuration space…
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…
These notes contain a survey of some aspects of the theory of graded differential algebras and of noncommutative differential calculi as well as of some applications connected with physics. They also give a description of several new…