Dimers on Cylinders over Dynkin Diagrams and Cluster Algebras

This dissertation describes a general setting for dimer models on cylinders over Dynkin diagrams which in type A reduces to the well-studied case of dimer models on a disc. We prove that all Berenstein--Fomin--Zelevinsky quivers for Schubert cells in a symmetric Kac--Moody algebra give rise to dimer models on the cylinder over the corresponding Dynkin diagram. We also give an independent proof of a result of Buan, Iyama, Reiten and Smith that the corresponding superpotentials are rigid using the dimer model structure of the quivers

The Deligne-Simpson problem for connections on Gm\mathbb{G}_m with a maximally ramified singularity

The classical additive Deligne-Simpson problem is the existence problem for Fuchsian connections with residues at the singular points in specified adjoint orbits. Crawley-Boevey found the solution in 2003 by reinterpreting the problem in terms of quiver varieties. A more general version of this problem, solved by Hiroe, allows additional unramified irregular singularities. We apply the theory of fundamental and regular strata due to Bremer and Sage to formulate a version of the Deligne-Simpson problem in which certain ramified singularities are allowed. These allowed singular points are called toral singularities; they are singularities whose leading term with respect to a lattice chain filtration is regular semisimple. We solve this problem in the important special case of connections on $\mathbb{G}_m$ with a maximally ramified singularity at $0$ and possibly an additional regular singular point at infinity. We also give a complete characterization of all such connections which are rigid, under the additional hypothesis of unipotent monodromy at infinity.Comment: 27 pages. Minor correction

Management of tubular oesophageal duplication - A novel approach

Background and aim: Oesophageal duplication is a rare congenital malformation which occurs due to embryogenesis errors. It accounts for 15% of all digestive tract duplications, most common being ileal duplications. Here we report a case of tubular Oesophageal duplication and a novel surgical approach for its management.6 year male child, 11 kgs, presented with progressive dysphagia to solids > liquids since birth. He had stunted growth, height and weight were less than 50th percentile. On investigations, Barium swallow and CECT neck with thorax was done to diagnose the pathology and define anatomy. It reported that the child had a congenital partial duplication of the proximal and mid esophagus with midline fold/membrane approximately 6 cm from C5 to D7 level with a dilated lower esophagus and a large hiatal hernia with focal stricture in the right lumen of proximal end. Surgical management was done by excision of oesophageal septum by endoscopic staplers via a gastrotomy under endoscopic guidance per orally and via gastrotomy. Results: Gradually diet started and well tolerated, to be kept on endoscopic surveillance. Conclusions: This is a novel approach to manage oesophageal tubular duplication with minimal risk of mediastinal complications that would occur otherwise by a thoracic approach