Discrete Morse theory for random complexes

A. Nikitenko, Discrete Morse Theory for Random Complexes , IST Austria, 2017.

Download
OA 2.32 MB

Thesis | Published | English
Department
Series Title
IST Austria Thesis
Abstract
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.
Publishing Year
Date Published
2017-10-27
Page
86
IST-REx-ID

Cite this

Nikitenko A. Discrete Morse Theory for Random Complexes . IST Austria; 2017. doi:10.15479/AT:ISTA:th_873
Nikitenko, A. (2017). Discrete Morse theory for random complexes . IST Austria. https://doi.org/10.15479/AT:ISTA:th_873
Nikitenko, Anton. Discrete Morse Theory for Random Complexes . IST Austria, 2017. https://doi.org/10.15479/AT:ISTA:th_873.
A. Nikitenko, Discrete Morse theory for random complexes . IST Austria, 2017.
Nikitenko A. 2017. Discrete Morse theory for random complexes , IST Austria, 86p.
Nikitenko, Anton. Discrete Morse Theory for Random Complexes . IST Austria, 2017, doi:10.15479/AT:ISTA:th_873.
All files available under the following license(s):
Creative Commons License:
CC-BYCreative Commons Attribution 4.0 International Public License (CC-BY 4.0)
Main File(s)
File Name
Access Level
OA Open Access
Last Uploaded
2019-04-09T14:54:51Z

Source File
Access Level
Restricted Closed Access
Last Uploaded
2019-08-13T08:31:21Z

Export

Marked Publications

Open Data IST Research Explorer

Search this title in

Google Scholar