---
OA_place: publisher
OA_type: gold
_id: '22000'
abstract:
- lang: eng
text: 'Simplicial approximation provides a framework for constructing simplicial
complexes that are homotopy equivalent to a given manifold, provided a CW structure
is explicitly known. However, its conventional implementation quickly becomes
intractable on a computer: barycentric subdivision produces poorly shaped simplices,
and the star condition introduces many vertices. To address these limitations,
this article develops a subdivision scheme based on spherical Delaunay triangulations,
which attains better refinement properties than barycentric subdivisions. Moreover,
the star condition is reframed as two independent problems, one geometric and
the other combinatorial, respectively tackled in the language of locally equiconnected
spaces and the list homomorphism problem, allowing an exponential reduction in
the number of vertices. Via a prototype implementation, we obtain simplicial complexes
homotopy equivalent to Grassmannians and Stiefel manifolds up to dimension 5.'
article_number: 93:1-93:22
article_processing_charge: Yes
arxiv: 1
author:
- first_name: Raphaël
full_name: Tinarrage, Raphaël
id: 40ebcc9d-905f-11ef-bf0a-dc475da8a04e
last_name: Tinarrage
orcid: 0000-0002-1404-1095
citation:
ama: 'Tinarrage R. Simplicial approximation to CW complexes with spherical Delaunay
triangulations. In: 42nd International Symposium on Computational Geometry.
Vol 367. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2026. doi:10.4230/LIPIcs.SoCG.2026.93'
apa: 'Tinarrage, R. (2026). Simplicial approximation to CW complexes with spherical
Delaunay triangulations. In 42nd International Symposium on Computational Geometry
(Vol. 367). New Brunswick, NJ, United States: Schloss Dagstuhl - Leibniz-Zentrum
für Informatik. https://doi.org/10.4230/LIPIcs.SoCG.2026.93'
chicago: Tinarrage, Raphaël. “Simplicial Approximation to CW Complexes with Spherical
Delaunay Triangulations.” In 42nd International Symposium on Computational
Geometry, Vol. 367. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2026.
https://doi.org/10.4230/LIPIcs.SoCG.2026.93.
ieee: R. Tinarrage, “Simplicial approximation to CW complexes with spherical Delaunay
triangulations,” in 42nd International Symposium on Computational Geometry,
New Brunswick, NJ, United States, 2026, vol. 367.
ista: 'Tinarrage R. 2026. Simplicial approximation to CW complexes with spherical
Delaunay triangulations. 42nd International Symposium on Computational Geometry.
SoCG: Symposium on Computational Geometry vol. 367, 93:1-93:22.'
mla: Tinarrage, Raphaël. “Simplicial Approximation to CW Complexes with Spherical
Delaunay Triangulations.” 42nd International Symposium on Computational Geometry,
vol. 367, 93:1-93:22, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2026,
doi:10.4230/LIPIcs.SoCG.2026.93.
short: R. Tinarrage, in:, 42nd International Symposium on Computational Geometry,
Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2026.
conference:
end_date: 2026-06-05
location: New Brunswick, NJ, United States
name: 'SoCG: Symposium on Computational Geometry'
start_date: 2026-06-02
corr_author: '1'
das_tickbox: '0'
date_created: 2026-06-14T22:01:43Z
date_published: 2026-05-27T00:00:00Z
date_updated: 2026-06-22T11:28:26Z
day: '27'
ddc:
- '500'
department:
- _id: UlWa
doi: 10.4230/LIPIcs.SoCG.2026.93
external_id:
arxiv:
- '2112.07573'
file:
- access_level: open_access
checksum: a468edad327962309688aa78678138da
content_type: application/pdf
creator: dernst
date_created: 2026-06-22T07:53:13Z
date_updated: 2026-06-22T07:53:13Z
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file_name: 2026_LIPIcSSoCG_Tinarrage.pdf
file_size: 1436035
relation: main_file
success: 1
file_date_updated: 2026-06-22T07:53:13Z
has_accepted_license: '1'
intvolume: ' 367'
keyword:
- Triangulation of manifolds
- Simplicial approximation
- CW complexes
- Delaunay complexes
- List homomorphism problem
- Topological Data Analysis
language:
- iso: eng
month: '05'
oa: 1
oa_version: Published Version
publication: 42nd International Symposium on Computational Geometry
publication_identifier:
eissn:
- 1868-8969
isbn:
- '9783959774185'
publication_status: published
publisher: Schloss Dagstuhl - Leibniz-Zentrum für Informatik
quality_controlled: '1'
related_material:
link:
- relation: software
url: https://doi.org/10.5281/zenodo.19251455
researchdata_availability: no
scopus_import: '1'
status: public
supplementarymaterial: yes
title: Simplicial approximation to CW complexes with spherical Delaunay triangulations
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type: conference
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 367
year: '2026'
...
---
OA_place: publisher
_id: '18979'
abstract:
- lang: eng
text: "Topological Data Analysis (TDA) is a discipline utilizing the mathematical
field of topology to study data, most prominently collections of point sets. This
thesis summarizes three projects related to computations in TDA.\r\n\r\nThe first
one establishes a variant of TDA for chromatic point sets, where each point is
given a color. For example, we are given positions of cells within a tumor microenvironment,
and color the cancerous cells red, and the immune cells blue.\r\n\r\nThe aim is
then to give a quantitative description of how the two or more sets of points
spatially interact. Building on image, kernel and cokernel variants of persistent
homology, we suggest six-packs of persistent diagrams as such a descriptor.\r\n\r\nWe
describe a construction of a chromatic alpha complex, which enables efficient
computation of several variants of the six-packs. We give topological descriptions
of natural subcomplexes of the chromatic alpha complex, and show that the radii
of the simplices form a discrete Morse function. Finally, we provide an implementation
of the presented chromatic TDA pipeline.\r\n\r\nThe second part aims to translate
a powerful tool of sheaf theory to elementary terms using labeled matrices. The
goal is to enable their use in computational settings. We show that derived categories
of sheaves over finite posets have, up to isomorphism, unique objects---minimal
injective resolutions---and give a concrete algorithm to compute them. We further
describe simple algorithms to compute derived pushforwards and pullbacks for monotonic
maps, and their proper variants for inclusions, and demonstrate their tractability
by providing an implementation. Finally, we suggest a discrete definition of microsupport
and show desirable properties inspired by discrete Morse theory.\r\n\r\nIn the
last part, we present a collection of observations about collapses. We give a
characterization of collapsibility in terms of unitriangular submatrices of the
boundary matrix, a cotree-tree decomposition, and the optimal solution to a variant
of the Procrustes problem. We establish relation between dual collapses and relative
Morse theory and pose several open questions. Finally, focusing on complexes embedded
in the three-dimensional Euclidean space, we describe a relation between the collapsibility
and the triviality of a polygonal knot."
acknowledgement: "The research presented in this thesis was funded with the Wittgenstein
Prize,\r\nAustrian Science Fund (FWF), grant no. Z 342-N31, and from the DFG Collaborative
Research\r\nCenter TRR 109, ‘Discretization in Geometry and Dynamics’, Austrian
Science Fund (FWF),\r\ngrant no. I 02979-N35.\r\n"
alternative_title:
- ISTA Thesis
article_processing_charge: No
author:
- first_name: Ondrej
full_name: Draganov, Ondrej
id: 2B23F01E-F248-11E8-B48F-1D18A9856A87
last_name: Draganov
orcid: 0000-0003-0464-3823
citation:
ama: Draganov O. Structures and computations in topological data analysis. 2025.
doi:10.15479/at:ista:18979
apa: Draganov, O. (2025). Structures and computations in topological data analysis.
Institute of Science and Technology Austria. https://doi.org/10.15479/at:ista:18979
chicago: Draganov, Ondrej. “Structures and Computations in Topological Data Analysis.”
Institute of Science and Technology Austria, 2025. https://doi.org/10.15479/at:ista:18979.
ieee: O. Draganov, “Structures and computations in topological data analysis,” Institute
of Science and Technology Austria, 2025.
ista: Draganov O. 2025. Structures and computations in topological data analysis.
Institute of Science and Technology Austria.
mla: Draganov, Ondrej. Structures and Computations in Topological Data Analysis.
Institute of Science and Technology Austria, 2025, doi:10.15479/at:ista:18979.
short: O. Draganov, Structures and Computations in Topological Data Analysis, Institute
of Science and Technology Austria, 2025.
corr_author: '1'
date_created: 2025-01-31T17:04:40Z
date_published: 2025-02-03T00:00:00Z
date_updated: 2026-04-07T11:47:30Z
day: '03'
ddc:
- '514'
- '004'
degree_awarded: PhD
department:
- _id: GradSch
- _id: HeEd
doi: 10.15479/at:ista:18979
file:
- access_level: closed
checksum: af6567e5d35e5eb330b8925ae37f1998
content_type: application/zip
creator: odragano
date_created: 2025-01-31T16:58:30Z
date_updated: 2025-01-31T16:58:30Z
file_id: '18983'
file_name: Thesis.zip
file_size: 11899491
relation: source_file
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checksum: c3fef68e35b9dc2020b2ca6006da6343
content_type: application/pdf
creator: odragano
date_created: 2025-02-04T16:22:07Z
date_updated: 2025-02-04T16:22:07Z
file_id: '19000'
file_name: Thesis.pdf
file_size: 8857514
relation: main_file
file_date_updated: 2025-02-04T16:22:07Z
has_accepted_license: '1'
keyword:
- topological data analysis
- chromatic point set
- alpha complex
- persistent homology
- six pack
- sheaf
- microlocal discrete Morse
- injective resolution
- collapse
- knot
- discrete Morse theory
language:
- iso: eng
month: '02'
oa: 1
oa_version: Published Version
page: '140'
project:
- _id: 2561EBF4-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: I02979-N35
name: Persistence and stability of geometric complexes
- _id: 268116B8-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: Z00342
name: Mathematics, Computer Science
publication_identifier:
issn:
- 2663-337X
publication_status: published
publisher: Institute of Science and Technology Austria
related_material:
record:
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relation: part_of_dissertation
status: public
- id: '18981'
relation: part_of_dissertation
status: public
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: Structures and computations in topological data analysis
tmp:
image: /images/cc_by.png
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: '2025'
...
---
OA_place: publisher
_id: '18667'
abstract:
- lang: eng
text: "Many chemical and physical properties of materials are determined by the
material’s shape,\r\nfor example the size of its pores and the width of its tunnels.
This makes materials science\r\na prime application area for geometrical and topological
methods. Nevertheless many\r\nmethods in topological data analysis have not been
satisfyingly extended to the needs of\r\nmaterials science. This thesis provides
new methods and new mathematical theorems\r\ntargeted at those specific needs
by answering four different research questions. While the\r\nmotivation for each
of the research questions arises from materials science, the methods\r\nare versatile
and can be applied in different areas as well. \r\n\r\nThe first research question
is concerned with image data, for example a three-dimensional\r\ncomputed tomography
(CT) scan of a material, like sand or stone. There are two commonly\r\nused topologies
for digital images and depending on the application either of them might be\r\nrequired.
However, software for computing the topological data analysis method persistence\r\nhomology,
usually supports only one of the two topologies. We answer the question how to\r\ncompute
persistent homology of an image with respect to one of the two topologies using\r\nsoftware
that is intended for the other topology. \r\n\r\nThe second research question
is concerned with image data as well, and asks how much\r\nof the topological
information of an image is lost when the resolution is coarsened. As\r\ncomputer
tomography scanners are more expensive the higher the resolution, it is an\r\nimportant
question in materials science to know which resolution is enough to get satisfying\r\npersistent
homology. We give theoretical bounds on the information loss based on different\r\ngeometrical
properties of the object to be scanned. In addition, we conduct experiments on\r\nsand
and stone CT image data. \r\n\r\nThe third research question is motivated by comparing
crystalline materials efficiently. As\r\nthe atoms within a crystal repeat periodically,
crystalline materials are either modeled by\r\nunmanageable infinite periodic
point sets, or by one of their fundamental domains, which is\r\nunstable under
perturbation. Therefore a fingerprint of crystalline materials is needed, with\r\nappropriate
properties such that comparing the crystals can be eased by comparing the\r\nfingerprints
instead. We define the density fingerprint and prove the necessary properties.
\r\n\r\nThe fourth research question is motivated by studying the hole-structure
or connectedness,\r\ni.e. persistent homology or merge trees, of crystalline materials.
A common way to deal\r\nwith periodicity is to take a fundamental domain and identify
opposite boundaries to form a\r\ntorus. However, computing persistent homology
or merge trees on that torus loses some\r\nof the information materials scientists
are interested in and is additionally not stable under\r\ncertain noise. We therefore
decorate the merge tree stemming from the torus with additional\r\ninformation
describing the density and growth rate of the periodic copies of a component\r\nwithin
a growing spherical window. We prove all desired properties, like stability and
efficient\r\ncomputability."
acknowledgement: "I was supported by the European Research Council (ERC) Horizon 2020
project\r\n“Alpha Shape Theory Extended” No. 788183 and by the Pöttinger Scholarship.
In addition,\r\nI am very thankful for having been able to attend the second Workshop
for Women in\r\nComputational Topology in July 2019, funded by the Mathematical
Sciences Institute at\r\nANU, the US National Science Foundation through the award
CCF-1841455, the Australian\r\nMathematical Sciences Institute and the Association
for Women in Mathematics. Two of the\r\nprojects presented in this thesis started
there. One of them reached completion thanks to\r\nfunding from the MSRI Summer
Research in Mathematics program awarded to me and my\r\ncollaborators in 2020."
alternative_title:
- ISTA Thesis
article_processing_charge: No
author:
- first_name: Teresa
full_name: Heiss, Teresa
id: 4879BB4E-F248-11E8-B48F-1D18A9856A87
last_name: Heiss
orcid: 0000-0002-1780-2689
citation:
ama: Heiss T. New methods for applying topological data analysis to materials science.
2024. doi:10.15479/at:ista:18667
apa: Heiss, T. (2024). New methods for applying topological data analysis to
materials science. Institute of Science and Technology Austria. https://doi.org/10.15479/at:ista:18667
chicago: Heiss, Teresa. “New Methods for Applying Topological Data Analysis to Materials
Science.” Institute of Science and Technology Austria, 2024. https://doi.org/10.15479/at:ista:18667.
ieee: T. Heiss, “New methods for applying topological data analysis to materials
science,” Institute of Science and Technology Austria, 2024.
ista: Heiss T. 2024. New methods for applying topological data analysis to materials
science. Institute of Science and Technology Austria.
mla: Heiss, Teresa. New Methods for Applying Topological Data Analysis to Materials
Science. Institute of Science and Technology Austria, 2024, doi:10.15479/at:ista:18667.
short: T. Heiss, New Methods for Applying Topological Data Analysis to Materials
Science, Institute of Science and Technology Austria, 2024.
corr_author: '1'
date_created: 2024-12-17T16:17:55Z
date_published: 2024-12-17T00:00:00Z
date_updated: 2026-07-07T13:43:27Z
day: '17'
ddc:
- '514'
- '516'
- '004'
degree_awarded: PhD
department:
- _id: GradSch
- _id: HeEd
doi: 10.15479/at:ista:18667
ec_funded: 1
file:
- access_level: open_access
checksum: 247bb057aed2fba1cd4711917aaa2d77
content_type: application/pdf
creator: theiss
date_created: 2024-12-19T10:24:46Z
date_updated: 2024-12-19T10:24:46Z
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creator: theiss
date_created: 2024-12-19T10:24:50Z
date_updated: 2024-12-19T10:24:50Z
file_id: '18687'
file_name: PhD_Thesis.zip
file_size: 17197731
relation: source_file
file_date_updated: 2024-12-19T10:24:50Z
has_accepted_license: '1'
keyword:
- persistent homology
- topological data analysis
- periodic
- crystalline materials
- images
- fingerprint
language:
- iso: eng
month: '12'
oa: 1
oa_version: Published Version
page: '111'
project:
- _id: 266A2E9E-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '788183'
name: Alpha Shape Theory Extended
publication_identifier:
isbn:
- 978-3-99078-052-7
issn:
- 2663-337X
publication_status: published
publisher: Institute of Science and Technology Austria
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- id: '18673'
relation: part_of_dissertation
status: public
- id: '9345'
relation: part_of_dissertation
status: public
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: New methods for applying topological data analysis to materials science
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type: dissertation
user_id: ba8df636-2132-11f1-aed0-ed93e2281fdd
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...