Quasi-isomorphisms of cluster algebras and the combinatorics of webs (extended abstract)

International audienceWe provide bijections between the cluster variables (and clusters) in two families of cluster algebras which have received considerable attention. These cluster algebras are the ones associated with certain Grassmannians of k-planes, and those associated with certain spaces of decorated SLk-local systems in the disk in the work of Fock and Goncharov. When k is 3, this bijection can be described explicitly using the combinatorics of Kuperberg's basis of non-elliptic webs. Using our bijection and symmetries of these cluster algebras, we provide evidence for conjectures of Fomin and Pylyavskyy concerning cluster variables in Grassmannians of 3-planes. We also prove their conjecture that there are infinitely many indecomposable nonarborizable webs in the Grassmannian of 3-planes in 9-dimensional space

A 22-year-old woman with recurrent gastrointestinal bleeding since childhood

CASE PRESENTATIONA 22-year-old woman was referred to our unit for capsule endoscopy because of recurrent iron deficiency anaemia and gastrointestinal bleeding since childhood. Cutaneous vascular lesions had been surgically removed when she was a child.During the preceding year she required blood transfusions every month and had a total of 9 iron infusions. Gastroscopy and colono-scopy at another hospital had shown small vascular lesions in the stomach, duodenum and colon. Clinical examination revealed pallor. Blood investigations showed a haemoglobin level of 9.6 g/dl and a MCV of 67 fl.The capsule endoscopy (PillCam, Given Imaging) was performed. Fifteen lesions similar to those shown in figures 1 and 2 were noted through-out small bowel. She underwent a laparotomy with intra-operative entero-scopy; the lesions were removed and the histology is shown in Figure 3

Design and development of Taeneb City Guide - from paper maps and guidebooks to electronic guides

This paper reports the design, development and feedback from the initial trial of the Taeneb City Guide project developing tourist information software on Personal Digital Assistant (PDA) handheld computers. Based on the users' requirements for electronic tourists guides already published in the literature, the paper focuses on the three main technology features of the systems, which would give the advantage over the existing paper publication: query-able dynamic map interface, dynamic information content and community review systems and users' forum. The paper also reports the results of an initial trial of a City Guide for Glasgow conducted as part of the EMAC 03 conference

Positroid cluster structures from relabeled plabic graphs

The Grassmannian is a disjoint union of open positroid varieties $P_v$, certain smooth irreducible subvarieties whose definition is motivated by total positivity. The coordinate ring of $P_v$ is a cluster algebra, and each reduced plabic graph $G$ for $P_v$ determines a cluster. We study the effect of relabeling the boundary vertices of $G$ by a permutation $r$. Under suitable hypotheses on the permutation, we show that the relabeled graph $G^r$ determines a cluster for a different open positroid variety $P_w$. As a key step of the proof, we show that $P_v$ and $P_w$ are isomorphic by a nontrivial twist isomorphism. Our constructions yield many cluster structures on each open positroid variety $P_w$, given by plabic graphs with appropriately relabeled boundary. We conjecture that the seeds in all of these cluster structures are related by a combination of mutations and Laurent monomial transformations involving frozen variables, and establish this conjecture for (open) Schubert and opposite Schubert varieties. As an application, we also show that for certain reduced plabic graphs $G$, the "source" cluster and the "target" cluster are related by mutation and Laurent monomial rescalings.Comment: 45 pages, comments welcome! v2: minor change

Progressing Science Education : Constructing the Scientific Research Programme into the Contingent Nature of Learning Science. Keith S. Taber London: Springer (2009) pp. 402 Hbk. £136.00 ISBN 9789048124305

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Relative cluster categories and Higgs categories with infinite-dimensional morphism spaces

Cluster algebras *with coefficients* are important since they appear in nature as coordinate algebras of varieties like Grassmannians, double Bruhat cells, unipotent cells, ... . The approach of Geiss-Leclerc-Schr\"oer often yields Frobenius exact categories which allow to categorify such cluster algebras. In previous work, the third-named author has constructed Higgs categories and relative cluster categories in the relative Jacobi-finite setting (arXiv:2109.03707). Higgs categories generalize the Frobenius categories used by Geiss-Leclerc-Schr\"oer. In this article, we construct the Higgs category and the relative cluster category in the relative Jacobi-infinite setting under suitable hypotheses. This covers for example the case of Jensen-King-Su's Grassmannian cluster category. As in the relative Jacobi-finite case, the Higgs category is no longer exact but still extriangulated in the sense of Nakaoka-Palu. We also construct a cluster character refining Plamondon's. In the appendix, Chris Fraser and the second-named author categorify quasi-cluster morphisms using Frobenius categories. A recent application of this result is due to Matthew Pressland, who uses it to prove a conjecture by Muller-Speyer.Comment: 44 pages, with an appendix by Chris Fraser and Bernhard Keller; v2: Corrections in abstract, references and addres

From dimers to webs

We formulate a higher-rank version of the boundary measurement map for weighted planar bipartite networks in the disk. It sends a network to a linear combination of $\textnormal {SL}_r$-webs and is built upon the $r$-fold dimer model on the network. When $r$ equals 1, our map is a reformulation of Postnikov's boundary measurement used to coordinatize positroid strata. When $r$ equals 2 or 3, it is a reformulation of the $\textnormal {SL}_2$- and $\textnormal {SL}_3$-web immanants defined by the second author. The basic result is that the higher-rank map factors through Postnikov's map. As an application, we deduce generators and relations for the space of $\textnormal {SL}_r$-webs, re-proving a result of Cautis-Kamnitzer-Morrison. We establish compatibility between our map and restriction to positroid strata and thus between webs and total positivity