Survey of two-dimensional acute triangulations

AbstractWe give a brief introduction to the topic of two-dimensional acute triangulations, mention results on related areas, survey existing achievements–with emphasis on recent activity–and list related open problems, both concrete and conceptual

Collapsibility of CAT(0) spaces

Collapsibility is a combinatorial strengthening of contractibility. We relate this property to metric geometry by proving the collapsibility of any complex that is CAT(0) with a metric for which all vertex stars are convex. This strengthens and generalizes a result by Crowley. Further consequences of our work are: (1) All CAT(0) cube complexes are collapsible. (2) Any triangulated manifold admits a CAT(0) metric if and only if it admits collapsible triangulations. (3) All contractible d-manifolds ($d \ne 4$) admit collapsible CAT(0) triangulations. This discretizes a classical result by Ancel--Guilbault.Comment: 27 pages, 3 figures. The part on collapsibility of convex complexes has been removed and forms a new paper, called "Barycentric subdivisions of convexes complex are collapsible" (arXiv:1709.07930). The part on enumeration of manifolds has also been removed and forms now a third paper, called "A Cheeger-type exponential bound for the number of triangulated manifolds" (arXiv:1710.00130

Regge's Einstein-Hilbert Functional on the Double Tetrahedron

The double tetrahedron is the triangulation of the three-sphere gotten by gluing together two congruent tetrahedra along their boundaries. As a piecewise flat manifold, its geometry is determined by its six edge lengths, giving a notion of a metric on the double tetrahedron. We study notions of Einstein metrics, constant scalar curvature metrics, and the Yamabe problem on the double tetrahedron, with some reference to the possibilities on a general piecewise flat manifold. The main tool is analysis of Regge's Einstein-Hilbert functional, a piecewise flat analogue of the Einstein-Hilbert (or total scalar curvature) functional on Riemannian manifolds. We study the Einstein-Hilbert-Regge functional on the space of metrics and on discrete conformal classes of metrics

Acute and nonobtuse triangulations of polyhedral surfaces

In this paper, we prove the existence of acute triangulations for general polyhedral surfaces. We also show how to obtain nonobtuse subtriangulations of triangulated polyhedral surfaces.Massachusetts Institute of Technology (UROP Program

PocketPicker: analysis of ligand binding-sites with shape descriptors

Background Identification and evaluation of surface binding-pockets and occluded cavities are initial steps in protein structure-based drug design. Characterizing the active site's shape as well as the distribution of surrounding residues plays an important role for a variety of applications such as automated ligand docking or in situ modeling. Comparing the shape similarity of binding site geometries of related proteins provides further insights into the mechanisms of ligand binding. Results We present PocketPicker, an automated grid-based technique for the prediction of protein binding pockets that specifies the shape of a potential binding-site with regard to its buriedness. The method was applied to a representative set of protein-ligand complexes and their corresponding apo-protein structures to evaluate the quality of binding-site predictions. The performance of the pocket detection routine was compared to results achieved with the existing methods CAST, LIGSITE, LIGSITEcs, PASS and SURFNET. Success rates PocketPicker were comparable to those of LIGSITEcs and outperformed the other tools. We introduce a descriptor that translates the arrangement of grid points delineating a detected binding-site into a correlation vector. We show that this shape descriptor is suited for comparative analyses of similar binding-site geometry by examining induced-fit phenomena in aldose reductase. This new method uses information derived from calculations of the buriedness of potential binding-sites. Conclusions The pocket prediction routine of PocketPicker is a useful tool for identification of potential protein binding-pockets. It produces a convenient representation of binding-site shapes including an intuitive description of their accessibility. The shape-descriptor for automated classification of binding-site geometries can be used as an additional tool complementing elaborate manual inspections

On a proper acute triangulation of a polyhedral surface

AbstractLet Σ be a polyhedral surface in R3 with n edges. Let L be the length of the longest edge in Σ, δ be the minimum value of the geodesic distance from a vertex to an edge that is not incident to the vertex, and θ be the measure of the smallest face angle in Σ. We prove that Σ can be triangulated into at most CLn/(δθ) planar and rectilinear acute triangles, where C is an absolute constant

Apollonian structure in the Abelian sandpile

The Abelian sandpile process evolves configurations of chips on the integer lattice by toppling any vertex with at least 4 chips, distributing one of its chips to each of its 4 neighbors. When begun from a large stack of chips, the terminal state of the sandpile has a curious fractal structure which has remained unexplained. Using a characterization of the quadratic growths attainable by integer-superharmonic functions, we prove that the sandpile PDE recently shown to characterize the scaling limit of the sandpile admits certain fractal solutions, giving a precise mathematical perspective on the fractal nature of the sandpile.Comment: 27 Pages, 7 Figure