Related papers: Ptolemy relations for punctured discs
We present a simplification of the recursive algorithm for the evaluation of intersection numbers for differential $n$-forms, by combining the advantages emerging from the choice of delta-forms as generators of relative twisted cohomology…
We generalize Dynnikov coordinate system previosly defined on the standard punctured disk to an orientable surface of genus-1 with n punctures and one boundary component.
Much recent work has been done on the local Fourier transforms for connections on the punctured formal disk. Specifically, the local Fourier transforms have been introduced, shown to induce certain equivalences of categories, and explicit…
This is a sequel to the second and third author's Mixed Dimer Configuration Model in Type $D$ Cluster Algebras where we extend our model to work for quivers that contain oriented cycles. Namely, we extend a combinatorial model for…
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…
In this thesis we present several original contributions to the study of: - DG categories and their invariants; - Neeman's well-generated (algebraic) triangulated categories; - Fomin-Zelevinsky's cluster algebras approach via representation…
We give a sequence of equivalent formulations of the $ADE$ and $\hat A\hat D\hat E$ height models defined on a random triangulated surface: random surfaces immersed in Dynkin diagrams, chains of coupled random matrices, Coulomb gases, and…
We revisit a construction principle of Fredholm operators using Hilbert complexes of densely defined, closed linear operators and apply this to particular choices of differential operators. The resulting index is then computed with the help…
The paper considers the sequence of the Motzkin words, which is constructed according to formal features of natural numbers. We investigate the decomposition of well-formed parentheses into the matched pairs of parentheses (analogous to…
Tree-level scattering amplitudes of particles have a geometrical description in terms of intersection numbers of pairs of twisted differential forms on the moduli space of Riemann spheres with punctures. We customize a catalog of twisted…
We construct a finite dimensional quiver algebra from the non-simply laced type $B$ Dynkin diagram, which we call the type $B$ zigzag algebra. This leads to a faithful categorical action of the type $B$ braid group $\mathcal{A}(B)$, acting…
A classic result of Conway and Coxeter on frieze patterns has been generalized to a bijection between $p$-angulations of regular polygons and frieze patterns of type $\Lambda_p$. One of the features of Conway-Coxeter theory is a…
Interpolation of classes of differentiated functions given on a finite interval by trigonometric splines using the phantom node method is considered. This method consists in supplementing a given sequence of values of an approximate…
Fractional differential and integral operators, Dirichlet averages, and splines of complex order are three seemingly distinct mathematical subject areas addressing different questions and employing different methodologies. It is the purpose…
In recent decades, identities similar to the one in the Ptolemy's theorem started to pop up in many fields in connection to the notion of cluster algebras introduced and studied since 2000 by Fomin and Zelevinsky. In this brief note we will…
This paper explores the cluster algebra structure of the moduli space $\mathscr{A}_{\mathrm{SL}_{n+1},\mathbb{S}}$ of twisted $\mathrm{SL}_{n+1}$-local systems on a surface. We derive general recurrence relations for cluster variables…
Using twisted Fock spaces, we formulate and study two twisted versions of the n-point correlation functions of Bloch-Okounkov, and then identify them with q-expectation values of certain functions on the set of (odd) strict partitions. We…
It was shown by Fock, Goncharov and Fomin, Shapiro, Thurston that some cluster algebras arise from triangulated orientable suraces. Subsequently Dupont and Palesi generalised this construction to include unpunctured non-orientable surfaces,…
We introduce the notion of extended admissible dissections of a marked surface, building upon the notion of an admissible dissection of a marked surface by Amiot--Plamondon--Schroll. For each extended admissible dissection we construct a…
A cluster expansion is proposed, that applies to both continuous and discrete systems. The assumption for its convergence involves an extension of the neat Kotecky-Preiss criterion. Expressions and estimates for correlation functions are…