Cluster varieties from Legendrian knots

Many interesting spaces --- including all positroid strata and wild character varieties --- are moduli of constructible sheaves on a surface with microsupport in a Legendrian link. We show that the existence of cluster structures on these spaces may be deduced in a uniform, systematic fashion by constructing and taking the sheaf quantizations of a set of exact Lagrangian fillings in correspondence with isotopy representatives whose front projections have crossings with alternating orientations. It follows in turn that results in cluster algebra may be used to construct and distinguish exact Lagrangian fillings of Legendrian links in the standard contact three space.Comment: 47 page

Moving Polygon Methods for Incompressible Fluid Dynamics

Hybrid particle-mesh numerical approaches are proposed to solve incompressible fluid flows. The methods discussed in this work consist of a collection of particles each wrapped in their own polygon mesh cell, which then move through the domain as the flow evolves. Variables such as pressure, velocity, mass, and momentum are located either on the mesh or on the particles themselves, depending on the specific algorithm described, and each will be shown to have its own advantages and disadvantages. This work explores what is required to obtain local conservation of mass, momentum, and convergence for the velocity and pressure in a particle-mesh CFD simulation method. Current particle methods are explored and analyzed for their benefits and deficiencies, and newly developed methods are described with results and analysis. A new method for generating locally orthogonal polygonal meshes from a set of generator points is presented in which polygon areas are a constraint. The area constraint property is particularly useful for particle methods where moving polygons track a discrete portion of material. Voronoi polygon meshes have some very attractive mathematical and numerical properties for numerical computation, so a generalization of Voronoi polygon meshes is formulated that enforces a polygon area constraint. Area constrained moving polygonal meshes allow one to develop hybrid particle-mesh numerical methods that display some of the most attractive features of each approach. It is shown that this mesh construction method can continuously reconnect a moving, unstructured polygonal mesh in a pseudo-Lagrangian fashion without change in cell area/volume, and the method\u27s ability to simulate various physical scenarios is shown. The advantages are identified for incompressible fluid flow calculations, with demonstration cases that include material discontinuities of all three phases of matter and large density jumps

A global procedure for flow and heat transfer simulation with complex obstacles on curvilinear grids

A global methodology for simulating multiphase flows and heat transfers interacting with complex objects or interfaces is presented. Elliptic equations or Navier-Stokes equations are resolved on a fixed structured curvilinear grid and the solid objects are initially represented by triangular surface elements. Several difficulties arise as soon as these two non-conforming grids have to interact in the same physical problem, and accurate methods are presented for each issues: Lagrangian/Eulerian grid projection, immersed boundary of interface problems, and finally visualization. Hence, a new fast point-in-solid method for curvilinear grid is presented to project Lagrangian shapes on Eulerian grid. A new immersed boundary and interface method is presented, the Algebraic Immersed Interface method. Several validation and application problems are presented to demonstrate the interest and accuracy of the method

Smoothed particle hydrodynamics for fluid-solid coupling: modelling fixed and mobile boundaries

Smoothed Particle Hydrodynamics (SPH) is a meshless Lagrangian numerical method which has undergone extensive development in recent years. SPH has properties which make it especially suited to certain problem types with which traditional methods have struggled. This includes fluid-solid coupled problems, which are particularly relevant for the modelling of coastal dynamics - a domain which is becoming increasingly relevant owing to the changing global infrastructure and climate. SPH, however, is a relatively young method, and suffers drawbacks which have been dubbed its grand challenges. The main one of interest for this work, is the handling of boundary conditions within the method. This thesis aims to present a discussion on SPH in the context of modelling boundaries, both fixed and mobile. Improvements were made to the so-called semi-analytical boundary method, solving the mathematical and numerical problems associated with the method, and presenting the work in such a way that it can easily beported to existing SPH models. Further discussions and adjustments were also made to the boundary method, attempting to tackle some of its inconsistencies and render it more viable for typical use cases. Finally, the coupling of the DEM method for modelling mobile boundaries was addressed, with a particular focus on mesoscale modelling of solids at a similar resolution to the fluid domain. This was previously an unviable problem, but with improvements to the solid fraction calculation presented here, this is no longer the case. This presents the opportunity for full range mesoscale modelling in SPH.Open Acces

Cluster Algebras and Related Topics

Cluster algebras are a class of commutative algebras intoduced by Fomin and Zelevinsky in 2000. Their original purpose was to obtain a combinatorial approach to Lusztig’s dual canonical bases of quantum groups and to total positivity. Since then numerous connections between other areas of mathematics have been discovered. The aim of this workshop was to further strengthen these connections and to develop interactions