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Discrete Morse theory for random complexes

By: Nikitenko, Anton.
Material type: materialTypeLabelBookPublisher: IST Austria 2017
Contents:
Abstract Acknowledgments List of publications 1 Introduction 2 Results 3 Blaschke- Petkantschin formulas 4 Constants 5 Poisson-Delaunay, Poisson-Čech and weighted Poisson-Delaunay complexes 6 Poisson-Delaunay complexes of higher order 7 Random inscribed polytops 8 Future directions Bibliography Index
Summary: The main objects considered in the present work are simplicial and CW-complexes with vertices forming a random point cloud. In particular, we consider a Poisson point process in R^n and study Delaunay and Voronoi complexes of the first and higher orders and weighted Delaunay complexes obtained as sections of Delaunay complexes, as well as the Čech complex. Further, we examine theDelaunay complex of a Poisson point process on the sphere S^n, as well as of a uniform point cloud, which is equivalent to the convex hull, providing a connection to the theory of random polytopes. Each of the complexes in question can be endowed with a radius function, which maps its cells to the radii of appropriately chosen circumspheres, called the radius of the cell. Applying and developing discrete Morse theory for these functions, joining it together with probabilistic and sometimes analytic machinery, and developing several integral geometric tools, we aim at getting the distributions of circumradii of typical cells. For all considered complexes, we are able to generalize and obtain up to constants the distribution of radii of typical intervals of all types. In low dimensions the constants can be computed explicitly, thus providing the explicit expressions for the expected numbers of cells. In particular, it allows to find the expected density of simplices of every dimension for a Poisson point process in R^4, whereas the result for R^3 was known already in 1970's.
List(s) this item appears in: IST Austria Thesis 2018
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Thesis

Abstract
Acknowledgments
List of publications
1 Introduction
2 Results
3 Blaschke- Petkantschin formulas
4 Constants
5 Poisson-Delaunay, Poisson-Čech and weighted Poisson-Delaunay complexes
6 Poisson-Delaunay complexes of higher order
7 Random inscribed polytops
8 Future directions
Bibliography
Index

The main objects considered in the present work are simplicial and
CW-complexes with vertices forming a random point cloud.
In particular, we consider a Poisson point process in R^n
and study Delaunay and Voronoi complexes of the first and higher
orders and weighted Delaunay complexes obtained as sections
of Delaunay complexes, as well as the Čech complex.
Further, we examine theDelaunay complex of a Poisson
point process on the sphere S^n, as well as
of a uniform point cloud, which is equivalent
to the convex hull, providing a connection to the
theory of random polytopes.

Each of the complexes in question can be endowed with a radius function,
which maps its cells to the radii of appropriately chosen circumspheres,
called the radius of the cell.
Applying and developing discrete Morse theory for these functions,
joining it together with probabilistic and sometimes analytic machinery,
and developing several integral geometric tools,
we aim at getting the distributions of circumradii of typical cells.
For all considered complexes, we are able to generalize and obtain up to constants
the distribution of radii of typical intervals of all types.
In low dimensions the constants
can be computed explicitly, thus providing the explicit
expressions for the expected numbers of cells.
In particular, it allows to find the expected density of simplices
of every dimension for a Poisson point process in R^4,
whereas the result for R^3 was known already in 1970's.

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