Related papers: Minimal Matchings for dP3 Cluster Variables

We study a twisted version of Fraser, Lam, and Le's higher boundary measurement map, using face weights instead of edge weights, thereby providing Laurent polynomial expansions, in Pl\"ucker coordinates, for twisted web immanants for…

We consider T-systems and Y-systems arising from cluster mutations applied to quivers that have the property of being periodic under a sequence of mutations. The corresponding nonlinear recurrences for cluster variables (coefficient-free…

Di Francesco introduced Aztec triangles as combinatorial objects for which their domino tilings are equinumerous with certain sets of configurations of the twenty-vertex model that are the main focus of his article. We generalize Di…

Let $D$ be the Auslander algebra of $\mathbb{C}[t]/(t^n)$, which is quasi-hereditary, and $\mathcal{F}_\Delta$ the subcategory of good $D$-modules. For any $\mathsf{J}\subseteq[1, n-1]$, we construct a subcategory…

We construct cluster algebras the variables and coefficients of which satisfy the discrete mKdV equation, the discrete Toda equation and other integrable bilinear equations, several of which lead to q-discrete Painlev\'e equations. These…

We introduce a natural correspondence between artinian monomial almost complete intersections in three variables and punctured hexagonal regions. We use this correspondence to investigate the algebras for the presence of the weak Lefschetz…

We present a proof of a conjecture about the relationship between Baxter permutations and pairs of alternating sign matrices that are produced from domino tilings of Aztec diamonds. It is shown that if and only if a tiling corresponds to a…

In 2003, Fomin and Zelevinsky proved that finite type cluster algebras can be classified by Dynkin diagrams. Then in 2013, Barot and Marsh defined the presentation of a reflection group associated to a Dynkin diagram in terms of an…

Biclustering is used for simultaneous clustering of the observations and variables when there is no group structure known \textit{a priori}. It is being increasingly used in bioinformatics, text analytics, etc. Previously, biclustering has…

Modular invariants of families of curves are Arakelov invariants in arithmetic algebraic geometry. All the known uniform lower bounds of these invariants are not sharp. In this paper, we aim to give explicit lower bounds of modular…

Topological data analysis is an emerging field that applies the study of topological invariants to data. Perhaps the simplest of these invariants is the number of connected components or clusters. In this work, we explore a topological…

Recent stringent experiment data of neutrino oscillations induces partial symmetries such as $Z_{2}$, $Z_{2}\times CP$ to derive lepton mixing patterns. New partial symmetries expressed with elements of group algebras are studied. A…

In this PhD thesis we develop the frame work of triple crossing diagram maps (TCD maps), which describes constrained configurations of points in projective spaces and discrete dynamics on these configurations. We are able to capture the…

In this paper, we introduce the enough $g$-pairs property for a principal coefficients cluster algebra, which can be understood as a strong version of the sign-coherence of the $G$-matrices. Then we prove that any skew-symmetrizable…

We continue the exploration of various appearances of cluster algebras in scattering amplitudes and related topics in physics. The cluster configuration spaces generalize the familiar moduli space ${\mathcal M}_{0,n}$ to finite-type cluster…

We study degenerations of cluster type varieties and pairs. Our first theorem proves that degenerations of toric pairs are finite quotients of toric pairs. In a similar vein, under some mild conditions, we prove that degenerations of…

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…

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.…

This paper studies clustering for possibly high dimensional data (e.g. images, time series, gene expression data, and many other settings), and rephrase it as low rank matrix estimation in the PAC-Bayesian framework. Our approach leverages…

We consider discrete orthogonal polynomial ensembles which are discrete analogues of the orthogonal polynomial ensembles in random matrix theory. These ensembles occur in certain problems in combinatorial probability and can be thought of…