---
OA_place: publisher
_id: '6287'
abstract:
- lang: eng
  text: 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.
alternative_title:
- ISTA Thesis
article_processing_charge: No
author:
- first_name: Anton
  full_name: Nikitenko, Anton
  id: 3E4FF1BA-F248-11E8-B48F-1D18A9856A87
  last_name: Nikitenko
  orcid: 0000-0002-0659-3201
citation:
  ama: Nikitenko A. Discrete Morse theory for random complexes . 2017. doi:<a href="https://doi.org/10.15479/AT:ISTA:th_873">10.15479/AT:ISTA:th_873</a>
  apa: Nikitenko, A. (2017). <i>Discrete Morse theory for random complexes </i>. Institute
    of Science and Technology Austria. <a href="https://doi.org/10.15479/AT:ISTA:th_873">https://doi.org/10.15479/AT:ISTA:th_873</a>
  chicago: Nikitenko, Anton. “Discrete Morse Theory for Random Complexes .” Institute
    of Science and Technology Austria, 2017. <a href="https://doi.org/10.15479/AT:ISTA:th_873">https://doi.org/10.15479/AT:ISTA:th_873</a>.
  ieee: A. Nikitenko, “Discrete Morse theory for random complexes ,” Institute of
    Science and Technology Austria, 2017.
  ista: Nikitenko A. 2017. Discrete Morse theory for random complexes . Institute
    of Science and Technology Austria.
  mla: Nikitenko, Anton. <i>Discrete Morse Theory for Random Complexes </i>. Institute
    of Science and Technology Austria, 2017, doi:<a href="https://doi.org/10.15479/AT:ISTA:th_873">10.15479/AT:ISTA:th_873</a>.
  short: A. Nikitenko, Discrete Morse Theory for Random Complexes , Institute of Science
    and Technology Austria, 2017.
corr_author: '1'
date_created: 2019-04-09T15:04:32Z
date_published: 2017-10-27T00:00:00Z
date_updated: 2026-04-08T14:19:31Z
day: '27'
ddc:
- '514'
- '516'
- '519'
degree_awarded: PhD
department:
- _id: HeEd
doi: 10.15479/AT:ISTA:th_873
file:
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has_accepted_license: '1'
language:
- iso: eng
license: https://creativecommons.org/licenses/by/4.0/
month: '10'
oa: 1
oa_version: Published Version
page: '86'
publication_identifier:
  issn:
  - 2663-337X
publication_status: published
publisher: Institute of Science and Technology Austria
pubrep_id: '873'
related_material:
  record:
  - id: '87'
    relation: part_of_dissertation
    status: public
  - id: '5678'
    relation: part_of_dissertation
    status: public
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    relation: part_of_dissertation
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status: public
supervisor:
- first_name: Herbert
  full_name: Edelsbrunner, Herbert
  id: 3FB178DA-F248-11E8-B48F-1D18A9856A87
  last_name: Edelsbrunner
  orcid: 0000-0002-9823-6833
title: 'Discrete Morse theory for random complexes '
tmp:
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  legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
  name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
  short: CC BY (4.0)
type: dissertation
user_id: ba8df636-2132-11f1-aed0-ed93e2281fdd
year: '2017'
...