Annals of mathematics studies

Definitions and basic properties of the Voronoi diagram generalizations of the Voronoi diagram algorithms for computing Voronoi diagrams poisson Voronoi diagrams spatial interpolation models of spatial processes point pattern analysis locational optimization through Voronoi diagrams.

Spatial tessellations. Concepts and Applications of Voronoi diagrams

On the stability of persistent entropy and new summary functions for Topological Data Analysis

Persistent homology computes the multiscale topology of a data set by using a sequence of discrete complexes. In this paper, we propose that persistent homology may be a useful tool for studying the structure of the landscape of string vacua. As a scaled-down version of the program, we use persistent homology to characterize distributions of Type IIB flux vacua on moduli space for three examples: the rigid Calabi-Yau, a hypersurface in weighted projective space, and the symmetric six-torus T6 = (T2)3. These examples suggest that persistence pairing and multiparameter persistence contain useful information for characterization of the landscape in addition to the usual information contained in standard persistent homology. We also study how restricting to special vacua with phenomenologically interesting low-energy properties affects the topology of a distribution.

https://link.springer.com/content/pdf/10.1007/JHEP03(2019)054.pdf

Topological Data Analysis for the String Landscape

This study presents a numerical analysis of the topology of a set of cosmologically interesting 3D Gaussian random fields in terms of their Betti numbers β_0, β_1, and β_2. We show that Betti numbers entail a considerably richer characterization of the topology of the primordial density field. Of particular interest is that the Betti numbers specify which topological features – islands, cavities, or tunnels – define the spatial structure of the field. A principal characteristic of Gaussian fields is that the three Betti numbers dominate the topology at different density ranges. At extreme density levels, the topology is dominated by a single class of features. At low levels this is a Swiss-cheeselike topology dominated by isolated cavities, and, at high levels, a predominantly Meatball-like topology composed of isolated objects. At moderate density levels, two Betti numbers define a more Sponge-like topology. At mean density, the description of topology even needs three Betti numbers, quantifying a field consisting of several disconnected complexes, not of one connected and percolating overdensity. A second important aspect of Betti number statistics is that they are sensitive to the power spectrum. They reveal a monotonic trend, in which at a moderate density range, a lower spectral index corresponds to a considerably higher (relative) population of cavities and islands. We also assess the level of complementary information that the Betti numbers represent, in addition to conventional measures such as Minkowski functionals. To this end, we include an extensive description of the Gaussian Kinematic Formula, which represents a major theoretical underpinning for this discussion.

Topology and geometry of Gaussian random fields I: On Betti numbers, Euler characteristic, and Minkowski functionals

We introduce a multiscale topological description of the Megaparsec web-like cosmic matter distribution. Betti numbers and topological persistence offer a powerful means of describing the rich connectivity structure of the cosmic web and of its multiscale arrangement of matter and galaxies. Emanating from algebraic topology and Morse theory, Betti numbers and persistence diagrams represent an extension and deepening of the cosmologically familiar topological genus measure and the related geometric Minkowski functionals. In addition to a description of the mathematical background, this study presents the computational procedure for computing Betti numbers and persistence diagrams for density field filtrations. The field may be computed starting from a discrete spatial distribution of galaxies or simulation particles. The main emphasis of this study concerns an extensive and systematic exploration of the imprint of different web-like morphologies and different levels of multiscale clustering in the corresponding computed Betti numbers and persistence diagrams. To this end, we use Voronoi clustering models as templates for a rich variety of web-like configurations and the fractal-like Soneira-Peebles models exemplify a range of multiscale configurations. We have identified the clear imprint of cluster nodes, filaments, walls, and voids in persistence diagrams, along with that of the nested hierarchy of structures in multiscale point distributions. We conclude by outlining the potential of persistent topology for understanding the connectivity structure of the cosmic web, in large simulations of cosmic structure formation and in the challenging context of the observed galaxy distribution in large galaxy surveys.

/pdf/the-topology-of-the-cosmic-web-in-terms-of-persistent-betti-3jna8vj599.pdf

The Topology of the Cosmic Web in Terms of Persistent Betti Numbers

This paper presents a rather simple tracing algorithm to sample and mesh an m-dimensional sub-manifold of Rd for arbitrary m and d. We extend the work of Dobkin et al. to submanifolds of arbitrary dimension and codimension. The algorithm is practical and has been thoroughly investigated from both theoretical and experimental perspectives. The paper provides a full description and analysis of the data structure and of the tracing algorithm. The main contributions are : 1. We unify and complement the knowledge about Coxeter and Freudenthal-Kuhn triangulations. 2. We introduce an elegant and compact data structure to store Coxeter or Freudenthal-Kuhn triangulations and describe output sensitive algorithms to compute faces and cofaces or any simplex in the triangulation. 3. We present a manifold tracing algorithm based on the above data structure. We provide a detailled complexity analysis along with experimental results that show that the algorithm can handle cases that are far ahead of the state-of-the-art.

Sampling and Meshing Submanifolds in High Dimension

We consider the piecewise linear approximation of saddle functions of the form $f(x,y)=ax^2-by^2$ under the L-infinity error norm. We show that interpolating approximations are not optimal. One can get slightly smaller errors by allowing the vertices of the approximation to move away from the graph of the function.

Optimal Triangulation of Saddle Surfaces

We consider the piecewise linear approximation of saddle functions of the form $$f(x,y)=ax^2-by^2$$
 under the $$L_{\infty }$$
 error norm. We show that interpolating approximations are not optimal. One can get slightly smaller errors by allowing the vertices of the approximation to move away from the graph of the function.

/pdf/optimal-triangulation-of-saddle-surfaces-3ph8esyywn.pdf

Mathijs Wintraecken

Papers

The Topology of the Cosmic Web in Terms of Persistent Betti Numbers

Sampling and Meshing Submanifolds in High Dimension

Optimal Triangulation of Saddle Surfaces

Optimal triangulation of saddle surfaces