Topological Change in Mean Convex Mean Curvature Flow

Consider the mean curvature flow of an (n+1)-dimensional, compact, mean convex region in Euclidean space (or, if n<7, in a Riemannian manifold). We prove that elements of the m-th homotopy group of the complementary region can die only if there is a shrinking S^k x R^(n-k) singularity for some k less than or equal to m. We also prove that for each m from 1 to n, there is a nonempty open set of compact, mean convex regions K in R^(n+1) with smooth boundary for which the resulting mean curvature flow has a shrinking S^m x R^(n-m) singularity.Comment: 19 pages. This version includes a new section proving that certain kinds of mean curvature flow singularities persist under arbitrary small perturbations of the initial surface. Newest update (Oct 2013) fixes some bibliographic reference

Higher signature Delaunay decompositions

A Delaunay decomposition is a cell decomposition in R^d for which each cell is inscribed in a Euclidean ball which is empty of all other vertices. This article introduces a generalization of the Delaunay decomposition in which the Euclidean balls in the empty ball condition are replaced by other families of regions bounded by certain quadratic hypersurfaces. This generalized notion is adaptable to geometric contexts in which the natural space from which the point set is sampled is not Euclidean, but rather some other flat semi-Riemannian geometry, possibly with degenerate directions. We prove the existence and uniqueness of the decomposition and discuss some of its basic properties. In the case of dimension d = 2, we study the extent to which some of the well-known optimality properties of the Euclidean Delaunay triangulation generalize to the higher signature setting. In particular, we describe a higher signature generalization of a well-known description of Delaunay decompositions in terms of the intersection angles between the circumscribed circles.Comment: 25 pages, 6 figure

PALP - a User Manual

This article provides a complete user's guide to version 2.1 of the toric geometry package PALP by Maximilian Kreuzer and others. In particular, previously undocumented applications such as the program nef.x are discussed in detail. New features of PALP 2.1 include an extension of the program mori.x which can now compute Mori cones and intersection rings of arbitrary dimension and can also take specific triangulations of reflexive polytopes as input. Furthermore, the program nef.x is enhanced by an option that allows the user to enter reflexive Gorenstein cones as input. The present documentation is complemented by a Wiki which is available online.Comment: 71 pages, to appear in "Strings, Gauge Fields, and the Geometry Behind - The Legacy of Maximilian Kreuzer". PALP Wiki available at http://palp.itp.tuwien.ac.at/wiki/index.php/Main_Pag

The extrinsic nature of the Hausdorff distance of optimal triangulations of manifolds

Fejes Tóth [5] and Schneider [9] studied approximations of smooth convex hypersurfaces in Euclidean space by piecewise flat triangular meshes with a given number of vertices on the hypersurface that are optimal with respect to Hausdorff distance. They proved that this Hausdorff distance decreases inversely proportional with m 2/(d−1), where m is the number of vertices and d is the dimension of Euclidean space. Moreover the pro-portionality constant can be expressed in terms of the Gaussian curvature, an intrinsic quantity. In this short note, we prove the extrinsic nature of this constant for manifolds of sufficiently high codimension. We do so by constructing an family of isometric embeddings of the flat torus in Euclidean space

Ambient and intrinsic triangulations and topological methods in cosmology

The thesis consist of two parts, one part concerns triangulations the other the structure of the universe. 1 Images in films such as Shrek or Frozen and in computer games are often made using small triangles. Subdividing a figure (such as Shrek) into small triangles is called triangulating. This may be done in two different ways. The first method makes use of straight triangles and is used most often. Because computer power is limited, we want to use as few triangles as possible, while maintaining the quality of the image. This means that one has to choose the triangles in a clever manner. Much is known about the choice of triangles if the surface is convex (egg-shaped). This thesis contributes to our understanding of non-convex surfaces. The second and new method uses curved triangles that follow the surface. The triangles we use are determined by the intrinsic geometry of the surface and are called intrinsic triangles. 2 Shortly after the Big Bang the universe was very hot and dense. Quantum mechanical effects introduced structure into the matter distribution in the early universe. The universe expanded according the laws of General Relativity and the matter cooled down. After the matter in the universe had cooled down, clusters of galaxies formed out of the densest regions. These clusters of galaxies are connected by stringy structures consisting of galaxies. This thesis contributes to the understanding of this intricate structure

Ambient and intrinsic triangulations and topological methods in cosmology

The thesis consist of two parts, one part concerns triangulations the other the structure of the universe. 1 Images in films such as Shrek or Frozen and in computer games are often made using small triangles. Subdividing a figure (such as Shrek) into small triangles is called triangulating. This may be done in two different ways. The first method makes use of straight triangles and is used most often. Because computer power is limited, we want to use as few triangles as possible, while maintaining the quality of the image. This means that one has to choose the triangles in a clever manner. Much is known about the choice of triangles if the surface is convex (egg-shaped). This thesis contributes to our understanding of non-convex surfaces. The second and new method uses curved triangles that follow the surface. The triangles we use are determined by the intrinsic geometry of the surface and are called intrinsic triangles. 2 Shortly after the Big Bang the universe was very hot and dense. Quantum mechanical effects introduced structure into the matter distribution in the early universe. The universe expanded according the laws of General Relativity and the matter cooled down. After the matter in the universe had cooled down, clusters of galaxies formed out of the densest regions. These clusters of galaxies are connected by stringy structures consisting of galaxies. This thesis contributes to the understanding of this intricate structure