Planar maps, circle patterns and 2d gravity

Via circle pattern techniques, random planar triangulations (with angle variables) are mapped onto Delaunay triangulations in the complex plane. The uniform measure on triangulations is mapped onto a conformally invariant spatial point process. We show that this measure can be expressed as: (1) a sum over 3-spanning-trees partitions of the edges of the Delaunay triangulations; (2) the volume form of a K\"ahler metric over the space of Delaunay triangulations, whose prepotential has a simple formulation in term of ideal tessellations of the 3d hyperbolic space; (3) a discretized version (involving finite difference complex derivative operators) of Polyakov's conformal Fadeev-Popov determinant in 2d gravity; (4) a combination of Chern classes, thus also establishing a link with topological 2d gravity.Comment: Misprints corrected and a couple of footnotes added. 42 pages, 17 figure

Quivers, Tilings, Branes and Rhombi

We describe a simple algorithm that computes the recently discovered brane tilings for a given generic toric singular Calabi-Yau threefold. This therefore gives AdS/CFT dual quiver gauge theories for D3-branes probing the given non-compact manifold. The algorithm solves a longstanding problem by computing superpotentials for these theories directly from the toric diagram of the singularity. We study the parameter space of a-maximization; this study is made possible by identifying the R-charges of bifundamental fields as angles in the brane tiling. We also study Seiberg duality from a new perspective.Comment: 36 pages, 40 figures, JHEP

TESELA: a new Virtual Observatory tool to determine blank fields for astronomical observations

The observation of blank fields, regions of the sky devoid of stars down to a given threshold magnitude, constitutes one of the typical important calibration procedures required for the proper reduction of astronomical data obtained in imaging mode. This work describes a method, based on the use of the Delaunay triangulation on the surface of a sphere, that allows the easy generation of blank fields catalogues. In addition to that, a new tool named TESELA, accessible through the WEB, has been created to facilitate the user to retrieve, and visualise using the VO-tool Aladin, the blank fields available near a given position in the sky.Comment: Accepted for publication in Monthly Notices of the Royal Astronomical Society. 11 pages, 10 figures. Related Web tool accessible at http://sdc.cab.inta-csic.es/tesel

Two triangulations methods based on edge refinement

In this paper two curvature adaptive methods of surface triangulation are presented. Both methods are based on edge refinement to obtain a triangulation compatible with the curvature requirements. The first method applies an incremental and constrained Delaunay triangulation and uses curvature bounds to determine if an edge of the triangulation is admissible. The second method uses this function also in the edge refinement process, i.e. in the computation of the location of a refining point, and in the re-triangulation needed after the insertion of this refining point. Results are presented, comparing both approachesPostprint (published version

The local maxima of maximal injectivity radius among hyperbolic surfaces

The function on the Teichmueller space of complete, orientable, finite-area hyperbolic surfaces of a fixed topological type that assigns to a hyperbolic surface its maximal injectivity radius has no local maxima that are not global maxima.Comment: Dramatically shortened (now 12 pp.), and introductory material simplified, following a referee's suggestion