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The area of constrained clustering has been extensively explored by researchers and used by practitioners. Constrained clustering formulations exist for popular algorithms such as k-means, mixture models, and spectral clustering but have…
We elaborate on the recent proposal that intersection numbers of certain cohomology classes on the moduli space of genus-zero Riemann surfaces with punctures compute tree-level scattering amplitudes in quantum field theories. The relevant…
Recent articles have shown the connection between representation theory of quivers and the theory of cluster algebras. In this article, we prove that some cluster algebras of type ADE can be recovered from the data of the corresponding…
We study the relation between the cluster automorphisms and the quasi-automorphisms of a cluster algebra $\mathcal{A}$. We proof that under some mild condition, satisfied for example by every skew-symmetric cluster algebra, the…
These are lecture notes from a mini-course given at the CIMPA in Mar del Plata, Argentina, in March 2016. The aim of the course was to introduce cluster characters for 2-Calabi-Yau triangulated categories and present their main properties.…
These expository notes are dedicated to the study of the topology of configuration spaces of manifolds. We give detailed computations of many invariants, including the fundamental group of the configuration spaces of $\mathbb{R}^2$, the…
Cluster varieties are geometric objects that have recently found applications in several areas of mathematics and mathematical physics. This thesis studies the geometry of a large class of cluster varieties associated to compact oriented…
The concept of cluster tilting gives a higher analogue of classical Auslander correspondence between representation-finite algebras and Auslander algebras. The $n$-Auslander-Reiten translation functor $\tau_n$ plays an important role in the…
We complete classification of mutation-finite cluster algebras by extending the technique derived by Fomin, Shapiro, and Thurston to skew-symmetrizable case. We show that for every mutation-finite skew-symmetrizable matrix a diagram…
We describe a new way to relate an acyclic, skew-symmetrizable cluster algebra to the representation theory of a finite dimensional hereditary algebra. This approach is designed to explain the c-vectors of the cluster algebra. We obtain a…
Markov numbers, i.e. positive integers appearing in solutions to $x^2 + y^2 + z^2 = 3xyz$, can be viewed as specializations of cluster variables. The second author and Matsushita gave a generalization of the Markov equation, $x^2 + y^2 +…
We provide a cluster-algebraic approach to the computation of the recently introduced generalised biadjoint scalar amplitudes related to Grassmannians ${\rm Gr}(k,n)$. A finite cluster algebra provides a natural triangulation for the…
Let $G$ be a simply connected simple algebraic group over $\mathbb{C}$ of type $B_r$, $B$ and $B_-$ be its two opposite Borel subgroups, and $W$ be the associated Weyl group. For $u$, $v\in W$, it is known that the coordinate ring ${\mathbb…
The area of constrained clustering has been extensively explored by researchers and used by practitioners. Constrained clustering formulations exist for popular algorithms such as k-means, mixture models, and spectral clustering but have…
We generalise surface cluster algebras to the case of infinite surfaces where the surface contains finitely many accumulation points of boundary marked points. To connect different triangulations of an infinite surface, we consider infinite…
We study cluster algebras with principal and arbitrary coefficient systems that are associated to unpunctured surfaces. We give a direct formula for the Laurent polynomial expansion of cluster variables in these cluster algebras in terms of…
We continue our investigation on cluster algebras arising from cluster tubes. Let $\mathcal{C}$ be a cluster tube of rank $n+1$. For an arbitrary basic maximal rigid object $T$ of $\mathcal{C}$, one may associate a skew-symmetrizable…
We construct frieze patterns of type D_N with entries which are numbers of matchings between vertices and triangles of corresponding triangulations of a punctured disc. For triangulations corresponding to orientations of the Dynkin diagram…
We study vector spaces associated to a family of generalized Euler integrals. Their dimension is given by the Euler characteristic of a very affine variety. Motivated by Feynman integrals from particle physics, this has been investigated…
Using recursion formulas for vertex operator algebra higher genus characters with formal parameters identified with local coordinates around marked points on a Riemann surface of arbitrary genus, we introduce the notion of a vertex operator…