# Discrete Morse theory for random complexes

##### By: Nikitenko, Anton.

Material type: BookPublisher: IST Austria 2017Item type | Current location | Call number | Status | Date due | Barcode | Item holds |
---|---|---|---|---|---|---|

Book | Library | Available | AT-ISTA#001531 |

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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