---
OA_place: publisher
OA_type: hybrid
PlanS_conform: '1'
_id: '20260'
abstract:
- lang: eng
text: The medial axis of a set consists of the points in the ambient space without
a unique closest point in the original set. Since its introduction, the medial
axis has been used extensively in many applications as a method of computing a
skeleton topologically equivalent to the original set. Unfortunately, one limiting
factor in the use of the medial axis of a smooth manifold is that it is not necessarily
topologically stable under small perturbations of the manifold. To counter these
instabilities, various prunings of the medial axis have been proposed in the computational
geometry community. Here, we examine one type of pruning, called burning. Because
of the good experimental results it was hoped that the burning method of simplifying
the medial axis would be stable. In this work, we show a simple example that dashes
such hopes. Based on Bing’s house with two rooms, we demonstrate an isotopy of
a shape where the medial axis goes from collapsible to non-collapsible. More precisely,
we consider the standard deformation retract from the closed ball to Bing’s house
with two rooms, but stop just short of the point where Bing’s house becomes two
dimensional. This way we obtain an isotopy from the 3-ball to a thickened version
of Bing’s house. Under this isotopy, the medial axis goes from collapsible to
non-collapsible. We stress that this isotopy can be made generic, in the sense
of singularity theory, as developed by Arnol’d and Thom.
acknowledgement: "We thank André Lieutier, David Letscher, Ellen Gasparovic, Kathryn
Leonard, and Tao Ju for early discussions on this work. We also thank Lu Liu, Yajie
Yan, and Tao Ju for sharing code to generate the examples. We further thank Abigail
Thompson for discussion on the conjecture and James Damon for sharing his insight
in singularity theory. We thank the reviewers for their detailed reviews, which
helped to improve the exposition.\r\nOpen access funding provided by Institute of
Science and Technology (IST Austria). Partially supported by the DFG Collaborative
Research Center TRR 109, ‘Discretization in Geometry and Dynamics’ and the European
Research Council (ERC), grant no. 788183, ‘Alpha Shape Theory Extended’. The first
author was supported in part by the National Science Foundation through grants DBI-1759807,
CCF-1907612, and CCF-2444309. The fourth author was supported by the European Union’s
Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie
grant agreement No. 754411, the Austrian science fund (FWF) M-3073, ANR grant StratMesh,
ANR-24-CE48-1899, and the welcome package from IDEX of the Université Côte d’Azur,
ANR-15-IDEX-01."
article_processing_charge: Yes (via OA deal)
article_type: original
author:
- first_name: Erin Wolf
full_name: Chambers, Erin Wolf
last_name: Chambers
- first_name: Christopher D
full_name: Fillmore, Christopher D
id: 35638A5C-AAC7-11E9-B0BF-5503E6697425
last_name: Fillmore
- first_name: Elizabeth R
full_name: Stephenson, Elizabeth R
id: 2D04F932-F248-11E8-B48F-1D18A9856A87
last_name: Stephenson
orcid: 0000-0002-6862-208X
- first_name: Mathijs
full_name: Wintraecken, Mathijs
id: 307CFBC8-F248-11E8-B48F-1D18A9856A87
last_name: Wintraecken
orcid: 0000-0002-7472-2220
citation:
ama: Chambers EW, Fillmore CD, Stephenson ER, Wintraecken M. Burning or collapsing
the medial axis is unstable. La Matematica. 2025;4:811-828. doi:10.1007/s44007-025-00170-0
apa: Chambers, E. W., Fillmore, C. D., Stephenson, E. R., & Wintraecken, M.
(2025). Burning or collapsing the medial axis is unstable. La Matematica.
Springer Nature. https://doi.org/10.1007/s44007-025-00170-0
chicago: Chambers, Erin Wolf, Christopher D Fillmore, Elizabeth R Stephenson, and
Mathijs Wintraecken. “Burning or Collapsing the Medial Axis Is Unstable.” La
Matematica. Springer Nature, 2025. https://doi.org/10.1007/s44007-025-00170-0.
ieee: E. W. Chambers, C. D. Fillmore, E. R. Stephenson, and M. Wintraecken, “Burning
or collapsing the medial axis is unstable,” La Matematica, vol. 4. Springer
Nature, pp. 811–828, 2025.
ista: Chambers EW, Fillmore CD, Stephenson ER, Wintraecken M. 2025. Burning or collapsing
the medial axis is unstable. La Matematica. 4, 811–828.
mla: Chambers, Erin Wolf, et al. “Burning or Collapsing the Medial Axis Is Unstable.”
La Matematica, vol. 4, Springer Nature, 2025, pp. 811–28, doi:10.1007/s44007-025-00170-0.
short: E.W. Chambers, C.D. Fillmore, E.R. Stephenson, M. Wintraecken, La Matematica
4 (2025) 811–828.
corr_author: '1'
date_created: 2025-08-31T22:01:33Z
date_published: 2025-12-01T00:00:00Z
date_updated: 2026-04-07T11:42:48Z
day: '01'
ddc:
- '510'
department:
- _id: HeEd
doi: 10.1007/s44007-025-00170-0
ec_funded: 1
file:
- access_level: open_access
checksum: e2043259194bfcdf3d74c4da8a5a853f
content_type: application/pdf
creator: dernst
date_created: 2025-12-30T07:52:58Z
date_updated: 2025-12-30T07:52:58Z
file_id: '20885'
file_name: 2025_LaMatematica_Chambers.pdf
file_size: 2678640
relation: main_file
success: 1
file_date_updated: 2025-12-30T07:52:58Z
has_accepted_license: '1'
intvolume: ' 4'
language:
- iso: eng
month: '12'
oa: 1
oa_version: Published Version
page: 811-828
project:
- _id: 266A2E9E-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '788183'
name: Alpha Shape Theory Extended
- _id: 260C2330-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '754411'
name: ISTplus - Postdoctoral Fellowships
- _id: fc390959-9c52-11eb-aca3-afa58bd282b2
grant_number: M03073
name: Learning and triangulating manifolds via collapses
publication: La Matematica
publication_identifier:
eissn:
- 2730-9657
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
related_material:
record:
- id: '21021'
relation: dissertation_contains
status: public
scopus_import: '1'
status: public
title: Burning or collapsing the medial axis is unstable
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: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 4
year: '2025'
...
---
DOAJ_listed: '1'
OA_place: publisher
OA_type: gold
PlanS_conform: '1'
_id: '20293'
abstract:
- lang: eng
text: Motivated by questions arising at the intersection of information theory and
geometry, we compare two dissimilarity measures between finite categorical distributions.
One is the well-known Jensen–Shannon divergence, which is easy to compute and
whose square root is a proper metric. The other is what we call the minmax divergence,
which is harder to compute. Just like the Jensen–Shannon divergence, it arises
naturally from the Kullback–Leibler divergence. The main contribution of this
paper is a proof showing that the minmax divergence can be tightly approximated
by the Jensen–Shannon divergence. The bounds suggest that the square root of the
minmax divergence is a metric, and we prove that this is indeed true in the one-dimensional
case. The general case remains open. Finally, we consider analogous questions
in the context of another Bregman divergence and the corresponding Burbea–Rao
(Jensen–Bregman) divergence.
acknowledgement: "This research received partial funding from the European Research
Council (ERC) under\r\nthe European Union’s Horizon 2020 research and innovation
programme, grant no. 788183, the\r\nWittgenstein Prize, Austrian Science Fund (FWF),
grant no. Z 342-N31, the DFG Collaborative\r\nResearch Center TRR 109, ‘Discretization
in Geometry and Dynamics’, Austrian Science Fund (FWF), grant no. I 02979-N35, and
the 2022 Google Research Scholar Award for project ‘Algorithms for Topological Analysis
of Neural Networks’. The APC was waived."
article_number: '854'
article_processing_charge: Yes
article_type: original
author:
- first_name: Arseniy
full_name: Akopyan, Arseniy
id: 430D2C90-F248-11E8-B48F-1D18A9856A87
last_name: Akopyan
orcid: 0000-0002-2548-617X
- first_name: Herbert
full_name: Edelsbrunner, Herbert
id: 3FB178DA-F248-11E8-B48F-1D18A9856A87
last_name: Edelsbrunner
orcid: 0000-0002-9823-6833
- first_name: Ziga
full_name: Virk, Ziga
id: 2E36B656-F248-11E8-B48F-1D18A9856A87
last_name: Virk
- first_name: Hubert
full_name: Wagner, Hubert
id: 379CA8B8-F248-11E8-B48F-1D18A9856A87
last_name: Wagner
citation:
ama: Akopyan A, Edelsbrunner H, Virk Z, Wagner H. Tight bounds between the Jensen–Shannon
divergence and the minmax divergence. Entropy. 2025;27(8). doi:10.3390/e27080854
apa: Akopyan, A., Edelsbrunner, H., Virk, Z., & Wagner, H. (2025). Tight bounds
between the Jensen–Shannon divergence and the minmax divergence. Entropy.
MDPI. https://doi.org/10.3390/e27080854
chicago: Akopyan, Arseniy, Herbert Edelsbrunner, Ziga Virk, and Hubert Wagner. “Tight
Bounds between the Jensen–Shannon Divergence and the Minmax Divergence.” Entropy.
MDPI, 2025. https://doi.org/10.3390/e27080854.
ieee: A. Akopyan, H. Edelsbrunner, Z. Virk, and H. Wagner, “Tight bounds between
the Jensen–Shannon divergence and the minmax divergence,” Entropy, vol.
27, no. 8. MDPI, 2025.
ista: Akopyan A, Edelsbrunner H, Virk Z, Wagner H. 2025. Tight bounds between the
Jensen–Shannon divergence and the minmax divergence. Entropy. 27(8), 854.
mla: Akopyan, Arseniy, et al. “Tight Bounds between the Jensen–Shannon Divergence
and the Minmax Divergence.” Entropy, vol. 27, no. 8, 854, MDPI, 2025, doi:10.3390/e27080854.
short: A. Akopyan, H. Edelsbrunner, Z. Virk, H. Wagner, Entropy 27 (2025).
corr_author: '1'
date_created: 2025-09-07T22:01:33Z
date_published: 2025-08-01T00:00:00Z
date_updated: 2025-09-30T14:32:31Z
day: '01'
ddc:
- '500'
department:
- _id: HeEd
doi: 10.3390/e27080854
ec_funded: 1
external_id:
isi:
- '001557476000001'
pmid:
- '40870326'
file:
- access_level: open_access
checksum: 65c5399c4015d9c8abb8c7a96f3d7836
content_type: application/pdf
creator: dernst
date_created: 2025-09-08T07:55:48Z
date_updated: 2025-09-08T07:55:48Z
file_id: '20309'
file_name: 2025_Entropy_Akopyan.pdf
file_size: 379340
relation: main_file
success: 1
file_date_updated: 2025-09-08T07:55:48Z
has_accepted_license: '1'
intvolume: ' 27'
isi: 1
issue: '8'
language:
- iso: eng
month: '08'
oa: 1
oa_version: Published Version
pmid: 1
project:
- _id: 266A2E9E-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '788183'
name: Alpha Shape Theory Extended
- _id: 268116B8-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: Z00342
name: Mathematics, Computer Science
- _id: 2561EBF4-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: I02979-N35
name: Persistence and stability of geometric complexes
publication: Entropy
publication_identifier:
eissn:
- 1099-4300
publication_status: published
publisher: MDPI
quality_controlled: '1'
scopus_import: '1'
status: public
title: Tight bounds between the Jensen–Shannon divergence and the minmax divergence
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: journal_article
user_id: 317138e5-6ab7-11ef-aa6d-ffef3953e345
volume: 27
year: '2025'
...
---
OA_place: publisher
OA_type: hybrid
PlanS_conform: '1'
_id: '20323'
abstract:
- lang: eng
text: We establish several results combining discrete Morse theory and microlocal
sheaf theory in the setting of finite posets and simplicial complexes. Our primary
tool is a computationally tractable description of the bounded derived category
of sheaves on a poset with the Alexandrov topology. We prove that each bounded
complex of sheaves on a finite poset admits a unique (up to isomorphism of complexes)
minimal injective resolution, and we provide algorithms for computing minimal
injective resolution of an injective complex, as well as several useful functors
between derived categories of sheaves. For the constant sheaf on a simplicial
complex, we give asymptotically tight bounds on the complexity of computing the
minimal injective resolution using those algorithms. Our main result is a novel
definition of the discrete microsupport of a bounded complex of sheaves on a finite
poset. We detail several foundational properties of the discrete microsupport,
as well as a microlocal generalization of the discrete homological Morse theorem
and Morse inequalities.
acknowledgement: This project has received funding from the European Research Council
(ERC) under the European Union’s Horizon 2020 research and innovation programme,
grant no. 788183, from the Wittgenstein Prize, Austrian Science Fund (FWF), grant
no. Z342-N31, and from the DFG Collaborative Research Center TRR 109, ‘Discretization
in Geometry and Dynamics’, Austrian Science Fund (FWF), grant no. I 02979-N35
article_number: '108068'
article_processing_charge: Yes (via OA deal)
article_type: original
arxiv: 1
author:
- first_name: Adam
full_name: Brown, Adam
id: 70B7FDF6-608D-11E9-9333-8535E6697425
last_name: Brown
- first_name: Ondrej
full_name: Draganov, Ondrej
id: 2B23F01E-F248-11E8-B48F-1D18A9856A87
last_name: Draganov
orcid: 0000-0003-0464-3823
citation:
ama: Brown A, Draganov O. Discrete microlocal Morse theory. Journal of Pure and
Applied Algebra. 2025;229(10). doi:10.1016/j.jpaa.2025.108068
apa: Brown, A., & Draganov, O. (2025). Discrete microlocal Morse theory. Journal
of Pure and Applied Algebra. Elsevier. https://doi.org/10.1016/j.jpaa.2025.108068
chicago: Brown, Adam, and Ondrej Draganov. “Discrete Microlocal Morse Theory.” Journal
of Pure and Applied Algebra. Elsevier, 2025. https://doi.org/10.1016/j.jpaa.2025.108068.
ieee: A. Brown and O. Draganov, “Discrete microlocal Morse theory,” Journal of
Pure and Applied Algebra, vol. 229, no. 10. Elsevier, 2025.
ista: Brown A, Draganov O. 2025. Discrete microlocal Morse theory. Journal of Pure
and Applied Algebra. 229(10), 108068.
mla: Brown, Adam, and Ondrej Draganov. “Discrete Microlocal Morse Theory.” Journal
of Pure and Applied Algebra, vol. 229, no. 10, 108068, Elsevier, 2025, doi:10.1016/j.jpaa.2025.108068.
short: A. Brown, O. Draganov, Journal of Pure and Applied Algebra 229 (2025).
corr_author: '1'
date_created: 2025-09-10T05:40:09Z
date_published: 2025-10-01T00:00:00Z
date_updated: 2025-12-30T07:55:21Z
day: '01'
ddc:
- '510'
department:
- _id: HeEd
doi: 10.1016/j.jpaa.2025.108068
ec_funded: 1
external_id:
arxiv:
- '2209.14993'
file:
- access_level: open_access
checksum: 39bcad462278c9322ef810af7db67f56
content_type: application/pdf
creator: dernst
date_created: 2025-12-30T07:55:08Z
date_updated: 2025-12-30T07:55:08Z
file_id: '20886'
file_name: 2025_JourPureAppliedAlgebra_Brown.pdf
file_size: 3090836
relation: main_file
success: 1
file_date_updated: 2025-12-30T07:55:08Z
has_accepted_license: '1'
intvolume: ' 229'
issue: '10'
language:
- iso: eng
month: '10'
oa: 1
oa_version: Published Version
project:
- _id: 266A2E9E-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '788183'
name: Alpha Shape Theory Extended
- _id: 268116B8-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: Z00342
name: Mathematics, Computer Science
- _id: 2561EBF4-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: I02979-N35
name: Persistence and stability of geometric complexes
publication: Journal of Pure and Applied Algebra
publication_identifier:
issn:
- 0022-4049
publication_status: published
publisher: Elsevier
quality_controlled: '1'
related_material:
record:
- id: '18981'
relation: earlier_version
status: public
scopus_import: '1'
status: public
title: Discrete microlocal Morse theory
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: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 229
year: '2025'
...
---
OA_place: repository
OA_type: green
_id: '20490'
abstract:
- lang: eng
text: "We study flips in hypertriangulations of planar points sets. Here a level-k
hypertriangulation of n\r\n points in the plane is a subdivision induced by the
projection of a k-hypersimplex, which is the convex hull of the barycenters of
the (k-1)-dimensional faces of the standard (n-1)-simplex. In particular, we introduce
four types of flips and prove that the level-2 hypertriangulations are connected
by these flips.\r\n"
acknowledgement: Work by all authors but the second is supported by the European Research
Council (ERC), grant no. 788183, by the Wittgenstein Prize, Austrian Science Fund
(FWF), grant no. Z 342-N31, and by the DFG Collaborative Research Center TRR 109,
Austrian Science Fund (FWF), grant no. I 02979-N35. Work by the second author is
partially supported by the Alexander von Humboldt Foundation and by the Simons Foundation
. The second author thanks Jesús A. De Loera for useful discussions on flips and
non-flips and Pavel Galashin and Alexey Balitskiy for useful discussions on plabic
graphs.
article_number: '104248'
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Herbert
full_name: Edelsbrunner, Herbert
id: 3FB178DA-F248-11E8-B48F-1D18A9856A87
last_name: Edelsbrunner
orcid: 0000-0002-9823-6833
- first_name: Alexey
full_name: Garber, Alexey
last_name: Garber
- first_name: Mohadese
full_name: Ghafari, Mohadese
last_name: Ghafari
- first_name: Teresa
full_name: Heiss, Teresa
id: 4879BB4E-F248-11E8-B48F-1D18A9856A87
last_name: Heiss
orcid: 0000-0002-1780-2689
- first_name: Morteza
full_name: Saghafian, Morteza
id: f86f7148-b140-11ec-9577-95435b8df824
last_name: Saghafian
citation:
ama: Edelsbrunner H, Garber A, Ghafari M, Heiss T, Saghafian M. Flips in two-dimensional
hypertriangulations. European Journal of Combinatorics. 2025;132. doi:10.1016/j.ejc.2025.104248
apa: Edelsbrunner, H., Garber, A., Ghafari, M., Heiss, T., & Saghafian, M. (2025).
Flips in two-dimensional hypertriangulations. European Journal of Combinatorics.
Elsevier. https://doi.org/10.1016/j.ejc.2025.104248
chicago: Edelsbrunner, Herbert, Alexey Garber, Mohadese Ghafari, Teresa Heiss, and
Morteza Saghafian. “Flips in Two-Dimensional Hypertriangulations.” European
Journal of Combinatorics. Elsevier, 2025. https://doi.org/10.1016/j.ejc.2025.104248.
ieee: H. Edelsbrunner, A. Garber, M. Ghafari, T. Heiss, and M. Saghafian, “Flips
in two-dimensional hypertriangulations,” European Journal of Combinatorics,
vol. 132. Elsevier, 2025.
ista: Edelsbrunner H, Garber A, Ghafari M, Heiss T, Saghafian M. 2025. Flips in
two-dimensional hypertriangulations. European Journal of Combinatorics. 132, 104248.
mla: Edelsbrunner, Herbert, et al. “Flips in Two-Dimensional Hypertriangulations.”
European Journal of Combinatorics, vol. 132, 104248, Elsevier, 2025, doi:10.1016/j.ejc.2025.104248.
short: H. Edelsbrunner, A. Garber, M. Ghafari, T. Heiss, M. Saghafian, European
Journal of Combinatorics 132 (2025).
corr_author: '1'
date_created: 2025-10-19T22:01:31Z
date_published: 2025-10-10T00:00:00Z
date_updated: 2025-12-01T12:57:29Z
day: '10'
department:
- _id: HeEd
doi: 10.1016/j.ejc.2025.104248
ec_funded: 1
external_id:
arxiv:
- '2212.11380'
isi:
- '001599061500002'
intvolume: ' 132'
isi: 1
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://doi.org/10.48550/arXiv.2212.11380
month: '10'
oa: 1
oa_version: Preprint
project:
- _id: 266A2E9E-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '788183'
name: Alpha Shape Theory Extended
- _id: 268116B8-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: Z00342
name: Mathematics, Computer Science
- _id: 2561EBF4-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: I02979-N35
name: Persistence and stability of geometric complexes
publication: European Journal of Combinatorics
publication_identifier:
issn:
- 0195-6698
publication_status: epub_ahead
publisher: Elsevier
quality_controlled: '1'
scopus_import: '1'
status: public
title: Flips in two-dimensional hypertriangulations
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 132
year: '2025'
...
---
OA_place: repository
OA_type: green
_id: '20585'
abstract:
- lang: eng
text: Motivated by applications in medical sciences, we study finite chromatic sets
in Euclidean space from a topological perspective. Based on the persistent homology
for images, kernels and cokernels, we design provably stable homological quantifiers
that describe the geometric micro- and macro-structure of how the color classes
mingle. These can be efficiently computed using chromatic variants of Delaunay
and alpha complexes, and code that does these computations is provided.
acknowledgement: "This project has received funding from the European Research\r\nCouncil
(ERC) under the European Union’s Horizon 2020 research and innovation\r\nprogramme,
grant no. 788183, from the Wittgenstein Prize, Austrian Science Fund\r\n(FWF), grant
no. Z 342-N31, and from the DFG Collaborative Research Center TRR\r\n109, ‘Discretization
in Geometry and Dynamics’, Austrian Science Fund (FWF),\r\ngrant no. I 02979-N35."
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Sebastiano
full_name: Cultrera di Montesano, Sebastiano
id: 34D2A09C-F248-11E8-B48F-1D18A9856A87
last_name: Cultrera di Montesano
orcid: 0000-0001-6249-0832
- first_name: Ondrej
full_name: Draganov, Ondrej
id: 2B23F01E-F248-11E8-B48F-1D18A9856A87
last_name: Draganov
orcid: 0000-0003-0464-3823
- first_name: Herbert
full_name: Edelsbrunner, Herbert
id: 3FB178DA-F248-11E8-B48F-1D18A9856A87
last_name: Edelsbrunner
orcid: 0000-0002-9823-6833
- first_name: Morteza
full_name: Saghafian, Morteza
id: f86f7148-b140-11ec-9577-95435b8df824
last_name: Saghafian
citation:
ama: Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M. Chromatic
alpha complexes. Foundations of Data Science. 2025;8:30-62. doi:10.3934/fods.2025003
apa: Cultrera di Montesano, S., Draganov, O., Edelsbrunner, H., & Saghafian,
M. (2025). Chromatic alpha complexes. Foundations of Data Science. American
Institute of Mathematical Sciences. https://doi.org/10.3934/fods.2025003
chicago: Cultrera di Montesano, Sebastiano, Ondrej Draganov, Herbert Edelsbrunner,
and Morteza Saghafian. “Chromatic Alpha Complexes.” Foundations of Data Science.
American Institute of Mathematical Sciences, 2025. https://doi.org/10.3934/fods.2025003.
ieee: S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, and M. Saghafian,
“Chromatic alpha complexes,” Foundations of Data Science, vol. 8. American
Institute of Mathematical Sciences, pp. 30–62, 2025.
ista: Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M. 2025. Chromatic
alpha complexes. Foundations of Data Science. 8, 30–62.
mla: Cultrera di Montesano, Sebastiano, et al. “Chromatic Alpha Complexes.” Foundations
of Data Science, vol. 8, American Institute of Mathematical Sciences, 2025,
pp. 30–62, doi:10.3934/fods.2025003.
short: S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, M. Saghafian, Foundations
of Data Science 8 (2025) 30–62.
corr_author: '1'
date_created: 2025-11-02T23:01:33Z
date_published: 2025-03-01T00:00:00Z
date_updated: 2025-11-04T12:25:47Z
day: '01'
department:
- _id: HeEd
doi: 10.3934/fods.2025003
ec_funded: 1
external_id:
arxiv:
- '2212.03128'
intvolume: ' 8'
language:
- iso: eng
month: '03'
oa_version: Preprint
page: 30-62
project:
- _id: 266A2E9E-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '788183'
name: Alpha Shape Theory Extended
- _id: 268116B8-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: Z00342
name: Mathematics, Computer Science
- _id: 2561EBF4-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: I02979-N35
name: Persistence and stability of geometric complexes
publication: Foundations of Data Science
publication_identifier:
eissn:
- 2639-8001
publication_status: epub_ahead
publisher: American Institute of Mathematical Sciences
quality_controlled: '1'
related_material:
record:
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relation: earlier_version
status: public
scopus_import: '1'
status: public
title: Chromatic alpha complexes
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 8
year: '2025'
...
---
OA_place: publisher
OA_type: hybrid
PlanS_conform: '1'
_id: '20657'
abstract:
- lang: eng
text: 'The Upper Bound Theorem for convex polytopes implies that the p-th Betti
number of the Čech complex of any set of N points in ℝ^d and any radius satisfies
β_p = O(N^m), with m = min{p+1, ⌈d/2⌉}. We construct sets in even and odd dimensions,
which prove that this upper bound is asymptotically tight. For example, we describe
a set of N = 2(n+1) points in ℝ³ and two radii such that the first Betti number
of the Čech complex at one radius is (n+1)² - 1, and the second Betti number of
the Čech complex at the other radius is n². '
acknowledgement: The first author is supported by the European Research Council (ERC),
grant no. 788183, and by the DFG Collaborative Research Center TRR 109, Austrian
Science Fund (FWF), grant no. I 02979-N35. The second author is supported by the
European Research Council (ERC), grant “GeoScape” and by the Hungarian Science Foundation
(NKFIH), grant K-131529. Both authors are supported by the Wittgenstein Prize, Austrian
Science Fund (FWF), grant no. Z 342-N31.
article_processing_charge: Yes (via OA deal)
article_type: original
arxiv: 1
author:
- first_name: Herbert
full_name: Edelsbrunner, Herbert
id: 3FB178DA-F248-11E8-B48F-1D18A9856A87
last_name: Edelsbrunner
orcid: 0000-0002-9823-6833
- first_name: János
full_name: Pach, János
id: E62E3130-B088-11EA-B919-BF823C25FEA4
last_name: Pach
citation:
ama: Edelsbrunner H, Pach J. Maximum Betti numbers of Čech complexes. Discrete
& Computational Geometry. 2025. doi:10.1007/s00454-025-00796-5
apa: Edelsbrunner, H., & Pach, J. (2025). Maximum Betti numbers of Čech complexes.
Discrete & Computational Geometry. Springer Nature. https://doi.org/10.1007/s00454-025-00796-5
chicago: Edelsbrunner, Herbert, and János Pach. “Maximum Betti Numbers of Čech Complexes.”
Discrete & Computational Geometry. Springer Nature, 2025. https://doi.org/10.1007/s00454-025-00796-5.
ieee: H. Edelsbrunner and J. Pach, “Maximum Betti numbers of Čech complexes,” Discrete
& Computational Geometry. Springer Nature, 2025.
ista: Edelsbrunner H, Pach J. 2025. Maximum Betti numbers of Čech complexes. Discrete
& Computational Geometry.
mla: Edelsbrunner, Herbert, and János Pach. “Maximum Betti Numbers of Čech Complexes.”
Discrete & Computational Geometry, Springer Nature, 2025, doi:10.1007/s00454-025-00796-5.
short: H. Edelsbrunner, J. Pach, Discrete & Computational Geometry (2025).
corr_author: '1'
date_created: 2025-11-19T09:44:58Z
date_published: 2025-11-10T00:00:00Z
date_updated: 2025-12-01T15:19:21Z
day: '10'
ddc:
- '510'
department:
- _id: HeEd
doi: 10.1007/s00454-025-00796-5
ec_funded: 1
external_id:
arxiv:
- '2310.14801'
isi:
- '001610592600001'
has_accepted_license: '1'
isi: 1
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://doi.org/10.1007/s00454-025-00796-5
month: '11'
oa: 1
oa_version: Published Version
project:
- _id: 266A2E9E-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '788183'
name: Alpha Shape Theory Extended
- _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: Discrete & Computational Geometry
publication_identifier:
eissn:
- 1432-0444
issn:
- 0179-5376
publication_status: epub_ahead
publisher: Springer Nature
quality_controlled: '1'
related_material:
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relation: earlier_version
status: public
scopus_import: '1'
status: public
title: Maximum Betti numbers of Čech complexes
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: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2025'
...
---
OA_place: repository
OA_type: green
_id: '20658'
abstract:
- lang: eng
text: The medial axis of a smoothly embedded surface in R^3 consists of all points
for which the Euclidean distance function on the surface has at least two global
minima. We generalize this notion to the mid-sphere axis, which consists of all
points for which the Euclidean distance function has two interchanging saddles
that swap their partners in the pairing by persistent homology. It offers a discrete-algebraic
multi-scale approach to computing ridge-like structures on the surface. As a proof
of concept, an algorithm that computes stair-case approximations of the mid-sphere
axis is provided.
alternative_title:
- LNCS
article_processing_charge: No
arxiv: 1
author:
- first_name: Herbert
full_name: Edelsbrunner, Herbert
id: 3FB178DA-F248-11E8-B48F-1D18A9856A87
last_name: Edelsbrunner
orcid: 0000-0002-9823-6833
- first_name: Elizabeth R
full_name: Stephenson, Elizabeth R
id: 2D04F932-F248-11E8-B48F-1D18A9856A87
last_name: Stephenson
orcid: 0000-0002-6862-208X
- first_name: Martin H
full_name: Thoresen, Martin H
id: 47CB1472-F248-11E8-B48F-1D18A9856A87
last_name: Thoresen
citation:
ama: 'Edelsbrunner H, Stephenson ER, Thoresen MH. The mid-sphere cousin of the medial
axis transform. In: 4th International Joint Conference on Discrete Geometry
and Mathematical Morphology. Vol 16296. Springer Nature; 2025:133-147. doi:10.1007/978-3-032-09544-2_10'
apa: 'Edelsbrunner, H., Stephenson, E. R., & Thoresen, M. H. (2025). The mid-sphere
cousin of the medial axis transform. In 4th International Joint Conference
on Discrete Geometry and Mathematical Morphology (Vol. 16296, pp. 133–147).
Groningen, The Netherlands: Springer Nature. https://doi.org/10.1007/978-3-032-09544-2_10'
chicago: Edelsbrunner, Herbert, Elizabeth R Stephenson, and Martin H Thoresen. “The
Mid-Sphere Cousin of the Medial Axis Transform.” In 4th International Joint
Conference on Discrete Geometry and Mathematical Morphology, 16296:133–47.
Springer Nature, 2025. https://doi.org/10.1007/978-3-032-09544-2_10.
ieee: H. Edelsbrunner, E. R. Stephenson, and M. H. Thoresen, “The mid-sphere cousin
of the medial axis transform,” in 4th International Joint Conference on Discrete
Geometry and Mathematical Morphology, Groningen, The Netherlands, 2025, vol.
16296, pp. 133–147.
ista: 'Edelsbrunner H, Stephenson ER, Thoresen MH. 2025. The mid-sphere cousin of the medial
axis transform. 4th International Joint Conference on Discrete Geometry and Mathematical
Morphology. DGMM: Discrete Geometry and Mathematical Morphology, LNCS, vol. 16296,
133–147.'
mla: Edelsbrunner, Herbert, et al. “The Mid-Sphere Cousin of the Medial Axis Transform.”
4th International Joint Conference on Discrete Geometry and Mathematical Morphology,
vol. 16296, Springer Nature, 2025, pp. 133–47, doi:10.1007/978-3-032-09544-2_10.
short: H. Edelsbrunner, E.R. Stephenson, M.H. Thoresen, in:, 4th International Joint
Conference on Discrete Geometry and Mathematical Morphology, Springer Nature,
2025, pp. 133–147.
conference:
end_date: 2025-11-06
location: Groningen, The Netherlands
name: 'DGMM: Discrete Geometry and Mathematical Morphology'
start_date: 2025-11-03
date_created: 2025-11-23T23:01:37Z
date_published: 2025-11-01T00:00:00Z
date_updated: 2025-11-24T10:05:11Z
day: '01'
department:
- _id: HeEd
doi: 10.1007/978-3-032-09544-2_10
external_id:
arxiv:
- '2504.14743'
intvolume: ' 16296'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://doi.org/10.48550/arXiv.2504.14743
month: '11'
oa: 1
oa_version: Preprint
page: 133-147
publication: 4th International Joint Conference on Discrete Geometry and Mathematical
Morphology
publication_identifier:
eissn:
- 1611-3349
isbn:
- '9783032095435'
issn:
- 0302-9743
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: The mid-sphere cousin of the medial axis transform
type: conference
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 16296
year: '2025'
...
---
OA_place: publisher
OA_type: hybrid
_id: '20729'
abstract:
- lang: eng
text: 'Persistence modules (defined as a sequence of vector spaces and linear maps
between them) are a key tool in topological data analysis. They are easy to interpret
and fast to compute. However, when considering persistence maps (i.e. maps between
persistence modules), these properties are lost. We propose a new invariant for
persistence maps consisting of a partial matching such that: it is easy to interpret,
it is more discriminative than the image of the persistence map, and can be calculated
with cubical complexity.'
acknowledgement: Álvaro Torras-Casas contract is funded by the French Agence Nationale
de la Recherche through the project reference ANR-22-CPJ1-0047-01. Rocio Gonzalez-Diaz
is partially funded by the European Union under grant agreement no. 101070028-2
(REXASI-PRO).
article_processing_charge: Yes (in subscription journal)
author:
- first_name: Rocio
full_name: Gonzalez-Diaz, Rocio
last_name: Gonzalez-Diaz
- first_name: Manuel
full_name: Soriano Trigueros, Manuel
id: 15ebd7cf-15bf-11ee-aebd-bb4bb5121ea8
last_name: Soriano Trigueros
orcid: 0000-0003-2449-1433
- first_name: Alvaro
full_name: Torras-Casas, Alvaro
last_name: Torras-Casas
citation:
ama: 'Gonzalez-Diaz R, Soriano Trigueros M, Torras-Casas A. Additive partial matchings
for persistent homology. In: Proceedings of the 2025 International Symposium
on Symbolic and Algebraic Computation. Association for Computing Machinery;
2025:188-196. doi:10.1145/3747199.3747561'
apa: 'Gonzalez-Diaz, R., Soriano Trigueros, M., & Torras-Casas, A. (2025). Additive
partial matchings for persistent homology. In Proceedings of the 2025 International
Symposium on Symbolic and Algebraic Computation (pp. 188–196). Guanajuato,
Mexico: Association for Computing Machinery. https://doi.org/10.1145/3747199.3747561'
chicago: Gonzalez-Diaz, Rocio, Manuel Soriano Trigueros, and Alvaro Torras-Casas.
“Additive Partial Matchings for Persistent Homology.” In Proceedings of the
2025 International Symposium on Symbolic and Algebraic Computation, 188–96.
Association for Computing Machinery, 2025. https://doi.org/10.1145/3747199.3747561.
ieee: R. Gonzalez-Diaz, M. Soriano Trigueros, and A. Torras-Casas, “Additive partial
matchings for persistent homology,” in Proceedings of the 2025 International
Symposium on Symbolic and Algebraic Computation, Guanajuato, Mexico, 2025,
pp. 188–196.
ista: 'Gonzalez-Diaz R, Soriano Trigueros M, Torras-Casas A. 2025. Additive partial
matchings for persistent homology. Proceedings of the 2025 International Symposium
on Symbolic and Algebraic Computation. ISSAC: International Symposium on Symbolic
and Algebraic Computation, 188–196.'
mla: Gonzalez-Diaz, Rocio, et al. “Additive Partial Matchings for Persistent Homology.”
Proceedings of the 2025 International Symposium on Symbolic and Algebraic Computation,
Association for Computing Machinery, 2025, pp. 188–96, doi:10.1145/3747199.3747561.
short: R. Gonzalez-Diaz, M. Soriano Trigueros, A. Torras-Casas, in:, Proceedings
of the 2025 International Symposium on Symbolic and Algebraic Computation, Association
for Computing Machinery, 2025, pp. 188–196.
conference:
end_date: 2025-08-01
location: Guanajuato, Mexico
name: 'ISSAC: International Symposium on Symbolic and Algebraic Computation'
start_date: 2025-07-28
corr_author: '1'
date_created: 2025-12-07T23:02:01Z
date_published: 2025-11-10T00:00:00Z
date_updated: 2025-12-09T13:46:42Z
day: '10'
ddc:
- '510'
department:
- _id: HeEd
doi: 10.1145/3747199.3747561
file:
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checksum: 1c299cca165a20e2518afe4fda63dbf1
content_type: application/pdf
creator: dernst
date_created: 2025-12-09T13:43:17Z
date_updated: 2025-12-09T13:43:17Z
file_id: '20751'
file_name: 2025_ISSAC_GonzalezDiaz.pdf
file_size: 761617
relation: main_file
success: 1
file_date_updated: 2025-12-09T13:43:17Z
has_accepted_license: '1'
language:
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month: '11'
oa: 1
oa_version: Published Version
page: 188-196
publication: Proceedings of the 2025 International Symposium on Symbolic and Algebraic
Computation
publication_identifier:
isbn:
- '9798400720758'
publication_status: published
publisher: Association for Computing Machinery
quality_controlled: '1'
scopus_import: '1'
status: public
title: Additive partial matchings for persistent homology
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: conference
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2025'
...
---
OA_place: publisher
OA_type: hybrid
_id: '13182'
abstract:
- lang: eng
text: "We characterize critical points of 1-dimensional maps paired in persistent
homology\r\ngeometrically and this way get elementary proofs of theorems about
the symmetry\r\nof persistence diagrams and the variation of such maps. In particular,
we identify\r\nbranching points and endpoints of networks as the sole source of
asymmetry and\r\nrelate the cycle basis in persistent homology with a version
of the stable marriage\r\nproblem. Our analysis provides the foundations of fast
algorithms for maintaining a\r\ncollection of sorted lists together with its persistence
diagram."
acknowledgement: Open access funding provided by Austrian Science Fund (FWF). This
project has received funding from the European Research Council (ERC) under the
European Union’s Horizon 2020 research and innovation programme, grant no. 788183,
from the Wittgenstein Prize, Austrian Science Fund (FWF), Grant No. Z 342-N31, and
from the DFG Collaborative Research Center TRR 109, ‘Discretization in Geometry
and Dynamics’, Austrian Science Fund (FWF), Grant No. I 02979-N35. The authors of
this paper thank anonymous reviewers for their constructive criticism and Monika
Henzinger for detailed comments on an earlier version of this paper.
article_processing_charge: Yes (via OA deal)
article_type: original
author:
- first_name: Ranita
full_name: Biswas, Ranita
id: 3C2B033E-F248-11E8-B48F-1D18A9856A87
last_name: Biswas
orcid: 0000-0002-5372-7890
- first_name: Sebastiano
full_name: Cultrera Di Montesano, Sebastiano
id: 34D2A09C-F248-11E8-B48F-1D18A9856A87
last_name: Cultrera Di Montesano
orcid: 0000-0001-6249-0832
- first_name: Herbert
full_name: Edelsbrunner, Herbert
id: 3FB178DA-F248-11E8-B48F-1D18A9856A87
last_name: Edelsbrunner
orcid: 0000-0002-9823-6833
- first_name: Morteza
full_name: Saghafian, Morteza
id: f86f7148-b140-11ec-9577-95435b8df824
last_name: Saghafian
citation:
ama: Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. Geometric characterization
of the persistence of 1D maps. Journal of Applied and Computational Topology.
2024;8:1101-1119. doi:10.1007/s41468-023-00126-9
apa: Biswas, R., Cultrera di Montesano, S., Edelsbrunner, H., & Saghafian, M.
(2024). Geometric characterization of the persistence of 1D maps. Journal of
Applied and Computational Topology. Springer Nature. https://doi.org/10.1007/s41468-023-00126-9
chicago: Biswas, Ranita, Sebastiano Cultrera di Montesano, Herbert Edelsbrunner,
and Morteza Saghafian. “Geometric Characterization of the Persistence of 1D Maps.”
Journal of Applied and Computational Topology. Springer Nature, 2024. https://doi.org/10.1007/s41468-023-00126-9.
ieee: R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, and M. Saghafian, “Geometric
characterization of the persistence of 1D maps,” Journal of Applied and Computational
Topology, vol. 8. Springer Nature, pp. 1101–1119, 2024.
ista: Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. 2024. Geometric
characterization of the persistence of 1D maps. Journal of Applied and Computational
Topology. 8, 1101–1119.
mla: Biswas, Ranita, et al. “Geometric Characterization of the Persistence of 1D
Maps.” Journal of Applied and Computational Topology, vol. 8, Springer
Nature, 2024, pp. 1101–19, doi:10.1007/s41468-023-00126-9.
short: R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, M. Saghafian, Journal
of Applied and Computational Topology 8 (2024) 1101–1119.
corr_author: '1'
date_created: 2023-07-02T22:00:44Z
date_published: 2024-10-01T00:00:00Z
date_updated: 2026-04-07T12:58:47Z
day: '01'
ddc:
- '000'
department:
- _id: HeEd
doi: 10.1007/s41468-023-00126-9
ec_funded: 1
external_id:
pmid:
- '39678706'
file:
- access_level: open_access
checksum: d493df5088c222b88d9ca46b623ad0ee
content_type: application/pdf
creator: dernst
date_created: 2025-01-09T07:39:41Z
date_updated: 2025-01-09T07:39:41Z
file_id: '18783'
file_name: 2024_JourApplCompTopo_Biswas.pdf
file_size: 476896
relation: main_file
success: 1
file_date_updated: 2025-01-09T07:39:41Z
has_accepted_license: '1'
intvolume: ' 8'
language:
- iso: eng
month: '10'
oa: 1
oa_version: Published Version
page: 1101-1119
pmid: 1
project:
- _id: 266A2E9E-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '788183'
name: Alpha Shape Theory Extended
- _id: 0aa4bc98-070f-11eb-9043-e6fff9c6a316
grant_number: I4887
name: Persistent Homology, Algorithms and Stochastic Geometry
- _id: 268116B8-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: Z00342
name: Mathematics, Computer Science
publication: Journal of Applied and Computational Topology
publication_identifier:
eissn:
- 2367-1734
issn:
- 2367-1726
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
related_material:
record:
- id: '15094'
relation: dissertation_contains
status: public
scopus_import: '1'
status: public
title: Geometric characterization of the persistence of 1D maps
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: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 8
year: '2024'
...
---
_id: '14345'
abstract:
- lang: eng
text: For a locally finite set in R2, the order-k Brillouin tessellations form an
infinite sequence of convex face-to-face tilings of the plane. If the set is coarsely
dense and generic, then the corresponding infinite sequences of minimum and maximum
angles are both monotonic in k. As an example, a stationary Poisson point process
in R2 is locally finite, coarsely dense, and generic with probability one. For
such a set, the distributions of angles in the Voronoi tessellations, Delaunay
mosaics, and Brillouin tessellations are independent of the order and can be derived
from the formula for angles in order-1 Delaunay mosaics given by Miles (Math.
Biosci. 6, 85–127 (1970)).
acknowledgement: Work by all authors but A. Garber is supported by the European Research
Council (ERC), Grant No. 788183, by the Wittgenstein Prize, Austrian Science Fund
(FWF), Grant No. Z 342-N31, and by the DFG Collaborative Research Center TRR 109,
Austrian Science Fund (FWF), Grant No. I 02979-N35. Work by A. Garber is partially
supported by the Alexander von Humboldt Foundation.
article_processing_charge: Yes (via OA deal)
article_type: original
arxiv: 1
author:
- first_name: Herbert
full_name: Edelsbrunner, Herbert
id: 3FB178DA-F248-11E8-B48F-1D18A9856A87
last_name: Edelsbrunner
orcid: 0000-0002-9823-6833
- first_name: Alexey
full_name: Garber, Alexey
last_name: Garber
- first_name: Mohadese
full_name: Ghafari, Mohadese
last_name: Ghafari
- first_name: Teresa
full_name: Heiss, Teresa
id: 4879BB4E-F248-11E8-B48F-1D18A9856A87
last_name: Heiss
orcid: 0000-0002-1780-2689
- first_name: Morteza
full_name: Saghafian, Morteza
id: f86f7148-b140-11ec-9577-95435b8df824
last_name: Saghafian
citation:
ama: Edelsbrunner H, Garber A, Ghafari M, Heiss T, Saghafian M. On angles in higher
order Brillouin tessellations and related tilings in the plane. Discrete and
Computational Geometry. 2024;72:29-48. doi:10.1007/s00454-023-00566-1
apa: Edelsbrunner, H., Garber, A., Ghafari, M., Heiss, T., & Saghafian, M. (2024).
On angles in higher order Brillouin tessellations and related tilings in the plane.
Discrete and Computational Geometry. Springer Nature. https://doi.org/10.1007/s00454-023-00566-1
chicago: Edelsbrunner, Herbert, Alexey Garber, Mohadese Ghafari, Teresa Heiss, and
Morteza Saghafian. “On Angles in Higher Order Brillouin Tessellations and Related
Tilings in the Plane.” Discrete and Computational Geometry. Springer Nature,
2024. https://doi.org/10.1007/s00454-023-00566-1.
ieee: H. Edelsbrunner, A. Garber, M. Ghafari, T. Heiss, and M. Saghafian, “On angles
in higher order Brillouin tessellations and related tilings in the plane,” Discrete
and Computational Geometry, vol. 72. Springer Nature, pp. 29–48, 2024.
ista: Edelsbrunner H, Garber A, Ghafari M, Heiss T, Saghafian M. 2024. On angles
in higher order Brillouin tessellations and related tilings in the plane. Discrete
and Computational Geometry. 72, 29–48.
mla: Edelsbrunner, Herbert, et al. “On Angles in Higher Order Brillouin Tessellations
and Related Tilings in the Plane.” Discrete and Computational Geometry,
vol. 72, Springer Nature, 2024, pp. 29–48, doi:10.1007/s00454-023-00566-1.
short: H. Edelsbrunner, A. Garber, M. Ghafari, T. Heiss, M. Saghafian, Discrete
and Computational Geometry 72 (2024) 29–48.
corr_author: '1'
date_created: 2023-09-17T22:01:10Z
date_published: 2024-07-01T00:00:00Z
date_updated: 2025-04-23T08:41:59Z
day: '01'
ddc:
- '510'
department:
- _id: HeEd
doi: 10.1007/s00454-023-00566-1
ec_funded: 1
external_id:
arxiv:
- '2204.01076'
isi:
- '001060727600004'
pmid:
- '39610762'
file:
- access_level: open_access
checksum: b207b4e00f904e8ea8a30e24f0251f79
content_type: application/pdf
creator: dernst
date_created: 2024-07-22T09:43:19Z
date_updated: 2024-07-22T09:43:19Z
file_id: '17301'
file_name: 2024_DiscreteComputGeom_Edelsbrunner.pdf
file_size: 892019
relation: main_file
success: 1
file_date_updated: 2024-07-22T09:43:19Z
has_accepted_license: '1'
intvolume: ' 72'
isi: 1
language:
- iso: eng
month: '07'
oa: 1
oa_version: Published Version
page: 29-48
pmid: 1
project:
- _id: 266A2E9E-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '788183'
name: Alpha Shape Theory Extended
- _id: 268116B8-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: Z00342
name: Mathematics, Computer Science
- _id: 2561EBF4-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: I02979-N35
name: Persistence and stability of geometric complexes
publication: Discrete and Computational Geometry
publication_identifier:
eissn:
- 1432-0444
issn:
- 0179-5376
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: On angles in higher order Brillouin tessellations and related tilings in the
plane
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: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 72
year: '2024'
...
---
_id: '14888'
abstract:
- lang: eng
text: 'A face in a curve arrangement is called popular if it is bounded by the same
curve multiple times. Motivated by the automatic generation of curved nonogram
puzzles, we investigate possibilities to eliminate the popular faces in an arrangement
by inserting a single additional curve. This turns out to be NP-hard; however,
it becomes tractable when the number of popular faces is small: We present a probabilistic
FPT-approach in the number of popular faces.'
acknowledgement: 'This work was initiated at the 16th European Research Week on Geometric
Graphs in Strobl in 2019. A.W. is supported by the Austrian Science Fund (FWF):
W1230. S.T. has been funded by the Vienna Science and Technology Fund (WWTF) [10.47379/ICT19035].
A preliminary version of this work has been presented at the 38th European Workshop
on Computational Geometry (EuroCG 2022) in Perugia [9]. A full version of this paper,
which includes appendices but is otherwise identical, is available as a technical
report [10].'
alternative_title:
- LNCS
article_processing_charge: No
arxiv: 1
author:
- first_name: Phoebe
full_name: De Nooijer, Phoebe
last_name: De Nooijer
- first_name: Soeren
full_name: Terziadis, Soeren
last_name: Terziadis
- first_name: Alexandra
full_name: Weinberger, Alexandra
last_name: Weinberger
- first_name: Zuzana
full_name: Masárová, Zuzana
id: 45CFE238-F248-11E8-B48F-1D18A9856A87
last_name: Masárová
orcid: 0000-0002-6660-1322
- first_name: Tamara
full_name: Mchedlidze, Tamara
last_name: Mchedlidze
- first_name: Maarten
full_name: Löffler, Maarten
last_name: Löffler
- first_name: Günter
full_name: Rote, Günter
last_name: Rote
citation:
ama: 'De Nooijer P, Terziadis S, Weinberger A, et al. Removing popular faces in curve
arrangements. In: 31st International Symposium on Graph Drawing and Network
Visualization. Vol 14466. Springer Nature; 2024:18-33. doi:10.1007/978-3-031-49275-4_2'
apa: 'De Nooijer, P., Terziadis, S., Weinberger, A., Masárová, Z., Mchedlidze, T.,
Löffler, M., & Rote, G. (2024). Removing popular faces in curve arrangements.
In 31st International Symposium on Graph Drawing and Network Visualization
(Vol. 14466, pp. 18–33). Isola delle Femmine, Palermo, Italy: Springer Nature.
https://doi.org/10.1007/978-3-031-49275-4_2'
chicago: De Nooijer, Phoebe, Soeren Terziadis, Alexandra Weinberger, Zuzana Masárová,
Tamara Mchedlidze, Maarten Löffler, and Günter Rote. “Removing Popular Faces in Curve
Arrangements.” In 31st International Symposium on Graph Drawing and Network
Visualization, 14466:18–33. Springer Nature, 2024. https://doi.org/10.1007/978-3-031-49275-4_2.
ieee: P. De Nooijer et al., “Removing popular faces in curve arrangements,”
in 31st International Symposium on Graph Drawing and Network Visualization,
Isola delle Femmine, Palermo, Italy, 2024, vol. 14466, pp. 18–33.
ista: 'De Nooijer P, Terziadis S, Weinberger A, Masárová Z, Mchedlidze T, Löffler
M, Rote G. 2024. Removing popular faces in curve arrangements. 31st International
Symposium on Graph Drawing and Network Visualization. GD: Graph Drawing and Network
Visualization, LNCS, vol. 14466, 18–33.'
mla: De Nooijer, Phoebe, et al. “Removing Popular Faces in Curve Arrangements.”
31st International Symposium on Graph Drawing and Network Visualization,
vol. 14466, Springer Nature, 2024, pp. 18–33, doi:10.1007/978-3-031-49275-4_2.
short: P. De Nooijer, S. Terziadis, A. Weinberger, Z. Masárová, T. Mchedlidze, M.
Löffler, G. Rote, in:, 31st International Symposium on Graph Drawing and Network
Visualization, Springer Nature, 2024, pp. 18–33.
conference:
end_date: 2023-09-22
location: Isola delle Femmine, Palermo, Italy
name: 'GD: Graph Drawing and Network Visualization'
start_date: 2023-09-20
date_created: 2024-01-28T23:01:43Z
date_published: 2024-01-06T00:00:00Z
date_updated: 2025-09-04T11:52:35Z
day: '06'
department:
- _id: UlWa
- _id: HeEd
doi: 10.1007/978-3-031-49275-4_2
external_id:
arxiv:
- '2202.12175'
isi:
- '001207942000002'
intvolume: ' 14466'
isi: 1
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://doi.org/10.48550/arXiv.2202.12175
month: '01'
oa: 1
oa_version: Preprint
page: 18-33
publication: 31st International Symposium on Graph Drawing and Network Visualization
publication_identifier:
eissn:
- 1611-3349
isbn:
- '9783031492747'
issn:
- 0302-9743
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: Removing popular faces in curve arrangements
type: conference
user_id: 317138e5-6ab7-11ef-aa6d-ffef3953e345
volume: 14466
year: '2024'
...
---
_id: '15012'
abstract:
- lang: eng
text: We solve a problem of Dujmović and Wood (2007) by showing that a complete
convex geometric graph on n vertices cannot be decomposed into fewer than n-1
star-forests, each consisting of noncrossing edges. This bound is clearly tight.
We also discuss similar questions for abstract graphs.
acknowledgement: János Pach’s Research partially supported by European Research Council
(ERC), grant “GeoScape” No. 882971 and by the Hungarian Science Foundation (NKFIH),
grant K-131529. Work by Morteza Saghafian is partially supported by the European
Research Council (ERC), grant No. 788183, and by the Wittgenstein Prize, Austrian
Science Fund (FWF), grant No. Z 342-N31.
alternative_title:
- LNCS
article_processing_charge: No
arxiv: 1
author:
- first_name: János
full_name: Pach, János
id: E62E3130-B088-11EA-B919-BF823C25FEA4
last_name: Pach
- first_name: Morteza
full_name: Saghafian, Morteza
id: f86f7148-b140-11ec-9577-95435b8df824
last_name: Saghafian
- first_name: Patrick
full_name: Schnider, Patrick
last_name: Schnider
citation:
ama: 'Pach J, Saghafian M, Schnider P. Decomposition of geometric graphs into star-forests.
In: 31st International Symposium on Graph Drawing and Network Visualization.
Vol 14465. Springer Nature; 2024:339-346. doi:10.1007/978-3-031-49272-3_23'
apa: 'Pach, J., Saghafian, M., & Schnider, P. (2024). Decomposition of geometric
graphs into star-forests. In 31st International Symposium on Graph Drawing
and Network Visualization (Vol. 14465, pp. 339–346). Isola delle Femmine,
Palermo, Italy: Springer Nature. https://doi.org/10.1007/978-3-031-49272-3_23'
chicago: Pach, János, Morteza Saghafian, and Patrick Schnider. “Decomposition of Geometric
Graphs into Star-Forests.” In 31st International Symposium on Graph Drawing
and Network Visualization, 14465:339–46. Springer Nature, 2024. https://doi.org/10.1007/978-3-031-49272-3_23.
ieee: J. Pach, M. Saghafian, and P. Schnider, “Decomposition of geometric graphs
into star-forests,” in 31st International Symposium on Graph Drawing and Network
Visualization, Isola delle Femmine, Palermo, Italy, 2024, vol. 14465, pp.
339–346.
ista: 'Pach J, Saghafian M, Schnider P. 2024. Decomposition of geometric graphs
into star-forests. 31st International Symposium on Graph Drawing and Network Visualization.
GD: Graph Drawing and Network Visualization, LNCS, vol. 14465, 339–346.'
mla: Pach, János, et al. “Decomposition of Geometric Graphs into Star-Forests.”
31st International Symposium on Graph Drawing and Network Visualization,
vol. 14465, Springer Nature, 2024, pp. 339–46, doi:10.1007/978-3-031-49272-3_23.
short: J. Pach, M. Saghafian, P. Schnider, in:, 31st International Symposium on
Graph Drawing and Network Visualization, Springer Nature, 2024, pp. 339–346.
conference:
end_date: 2023-09-22
location: Isola delle Femmine, Palermo, Italy
name: 'GD: Graph Drawing and Network Visualization'
start_date: 2023-09-20
date_created: 2024-02-18T23:01:03Z
date_published: 2024-01-01T00:00:00Z
date_updated: 2026-04-16T09:12:37Z
day: '01'
department:
- _id: HeEd
doi: 10.1007/978-3-031-49272-3_23
ec_funded: 1
external_id:
arxiv:
- '2306.13201'
isi:
- '001207939600023'
intvolume: ' 14465'
isi: 1
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://doi.org/10.48550/arXiv.2306.13201
month: '01'
oa: 1
oa_version: Preprint
page: 339-346
project:
- _id: 266A2E9E-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '788183'
name: Alpha Shape Theory Extended
- _id: 268116B8-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: Z00342
name: Mathematics, Computer Science
publication: 31st International Symposium on Graph Drawing and Network Visualization
publication_identifier:
eisbn:
- '9783031492723'
eissn:
- 1611-3349
isbn:
- '9783031492716'
issn:
- 0302-9743
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
related_material:
record:
- id: '21253'
relation: later_version
status: public
scopus_import: '1'
status: public
title: Decomposition of geometric graphs into star-forests
type: conference
user_id: ba8df636-2132-11f1-aed0-ed93e2281fdd
volume: 14465
year: '2024'
...
---
OA_place: repository
_id: '15091'
abstract:
- lang: eng
text: "Motivated by applications in the medical sciences, we study finite chromatic\r\nsets
in Euclidean space from a topological perspective. Based on the persistent\r\nhomology
for images, kernels and cokernels, we design provably stable\r\nhomological quantifiers
that describe the geometric micro- and macro-structure\r\nof how the color classes
mingle. These can be efficiently computed using\r\nchromatic variants of Delaunay
and alpha complexes, and code that does these\r\ncomputations is provided."
article_number: '2212.03128'
article_processing_charge: No
arxiv: 1
author:
- first_name: Sebastiano
full_name: Cultrera di Montesano, Sebastiano
id: 34D2A09C-F248-11E8-B48F-1D18A9856A87
last_name: Cultrera di Montesano
orcid: 0000-0001-6249-0832
- first_name: Ondrej
full_name: Draganov, Ondrej
id: 2B23F01E-F248-11E8-B48F-1D18A9856A87
last_name: Draganov
orcid: 0000-0003-0464-3823
- first_name: Herbert
full_name: Edelsbrunner, Herbert
id: 3FB178DA-F248-11E8-B48F-1D18A9856A87
last_name: Edelsbrunner
orcid: 0000-0002-9823-6833
- first_name: Morteza
full_name: Saghafian, Morteza
id: f86f7148-b140-11ec-9577-95435b8df824
last_name: Saghafian
citation:
ama: Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M. Chromatic
alpha complexes. arXiv. doi:10.48550/arXiv.2212.03128
apa: Cultrera di Montesano, S., Draganov, O., Edelsbrunner, H., & Saghafian,
M. (n.d.). Chromatic alpha complexes. arXiv. https://doi.org/10.48550/arXiv.2212.03128
chicago: Cultrera di Montesano, Sebastiano, Ondrej Draganov, Herbert Edelsbrunner,
and Morteza Saghafian. “Chromatic Alpha Complexes.” ArXiv, n.d. https://doi.org/10.48550/arXiv.2212.03128.
ieee: S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, and M. Saghafian,
“Chromatic alpha complexes,” arXiv. .
ista: Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M. Chromatic
alpha complexes. arXiv, 2212.03128.
mla: Cultrera di Montesano, Sebastiano, et al. “Chromatic Alpha Complexes.” ArXiv,
2212.03128, doi:10.48550/arXiv.2212.03128.
short: S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, M. Saghafian, ArXiv
(n.d.).
corr_author: '1'
date_created: 2024-03-08T10:13:59Z
date_published: 2024-02-07T00:00:00Z
date_updated: 2026-04-07T12:58:47Z
day: '07'
department:
- _id: HeEd
doi: 10.48550/arXiv.2212.03128
external_id:
arxiv:
- '2212.03128'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/2212.03128
month: '02'
oa: 1
oa_version: Preprint
publication: arXiv
publication_status: draft
related_material:
record:
- id: '20585'
relation: later_version
status: public
- id: '18979'
relation: dissertation_contains
status: public
- id: '15094'
relation: dissertation_contains
status: public
status: public
title: Chromatic alpha complexes
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: preprint
user_id: 8b945eb4-e2f2-11eb-945a-df72226e66a9
year: '2024'
...
---
_id: '15093'
abstract:
- lang: eng
text: We present a dynamic data structure for maintaining the persistent homology
of a time series of real numbers. The data structure supports local operations,
including the insertion and deletion of an item and the cutting and concatenating
of lists, each in time O(log n + k), in which n counts the critical items and
k the changes in the augmented persistence diagram. To achieve this, we design
a tailor-made tree structure with an unconventional representation, referred to
as banana tree, which may be useful in its own right.
acknowledgement: The first and second authors are funded by the European Research Council under the
European Union’s Horizon 2020 research and innovation programme, ERC grant no. 788183,“Alpha
Shape Theory Extended (Alpha)”, by the Wittgenstein Prize, FWF grant no. Z 342-N31,
and by the DFG Collaborative Research Center TRR 109, FWF grant no. I 02979-N35.The
third author received funding by the European Research Council under the European
Union’s Horizon 2020research and innovation programme, ERC grant no. 101019564, “The Design of Modern Fully Dynamic DataStructures
(MoDynStruct)”, and by the Austrian Science Fund through the Wittgenstein Prize
with FWF grant no. Z 422-N, and also by FWF grant no. I 5982-N, and by FWF grant
no. P 33775-N, with additional funding from the netidee SCIENCE Stiftung, 2020–2024. The
fourth author is funded by the Vienna Graduate School on Computational Optimization,
FWF project no. W1260-N35.
article_processing_charge: No
arxiv: 1
author:
- first_name: Sebastiano
full_name: Cultrera di Montesano, Sebastiano
id: 34D2A09C-F248-11E8-B48F-1D18A9856A87
last_name: Cultrera di Montesano
orcid: 0000-0001-6249-0832
- first_name: Herbert
full_name: Edelsbrunner, Herbert
id: 3FB178DA-F248-11E8-B48F-1D18A9856A87
last_name: Edelsbrunner
orcid: 0000-0002-9823-6833
- first_name: Monika H
full_name: Henzinger, Monika H
id: 540c9bbd-f2de-11ec-812d-d04a5be85630
last_name: Henzinger
orcid: 0000-0002-5008-6530
- first_name: Lara
full_name: Ost, Lara
last_name: Ost
citation:
ama: 'Cultrera di Montesano S, Edelsbrunner H, Henzinger M, Ost L. Dynamically maintaining
the persistent homology of time series. In: Woodruff DP, ed. Proceedings of
the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA). Society
for Industrial and Applied Mathematics; 2024:243-295. doi:10.1137/1.9781611977912.11'
apa: 'Cultrera di Montesano, S., Edelsbrunner, H., Henzinger, M., & Ost, L.
(2024). Dynamically maintaining the persistent homology of time series. In D.
P. Woodruff (Ed.), Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete
Algorithms (SODA) (pp. 243–295). Alexandria, VA, USA: Society for Industrial
and Applied Mathematics. https://doi.org/10.1137/1.9781611977912.11'
chicago: Cultrera di Montesano, Sebastiano, Herbert Edelsbrunner, Monika Henzinger,
and Lara Ost. “Dynamically Maintaining the Persistent Homology of Time Series.”
In Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms
(SODA), edited by David P. Woodruff, 243–95. Society for Industrial and Applied
Mathematics, 2024. https://doi.org/10.1137/1.9781611977912.11.
ieee: S. Cultrera di Montesano, H. Edelsbrunner, M. Henzinger, and L. Ost, “Dynamically
maintaining the persistent homology of time series,” in Proceedings of the
2024 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), Alexandria,
VA, USA, 2024, pp. 243–295.
ista: 'Cultrera di Montesano S, Edelsbrunner H, Henzinger M, Ost L. 2024. Dynamically
maintaining the persistent homology of time series. Proceedings of the 2024 Annual
ACM-SIAM Symposium on Discrete Algorithms (SODA). SODA: Symposium on Discrete
Algorithms, 243–295.'
mla: Cultrera di Montesano, Sebastiano, et al. “Dynamically Maintaining the Persistent
Homology of Time Series.” Proceedings of the 2024 Annual ACM-SIAM Symposium
on Discrete Algorithms (SODA), edited by David P. Woodruff, Society for Industrial
and Applied Mathematics, 2024, pp. 243–95, doi:10.1137/1.9781611977912.11.
short: S. Cultrera di Montesano, H. Edelsbrunner, M. Henzinger, L. Ost, in:, D.P.
Woodruff (Ed.), Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete
Algorithms (SODA), Society for Industrial and Applied Mathematics, 2024, pp. 243–295.
conference:
end_date: 2024-01-10
location: Alexandria, VA, USA
name: 'SODA: Symposium on Discrete Algorithms'
start_date: 2024-01-07
corr_author: '1'
date_created: 2024-03-08T10:27:39Z
date_published: 2024-01-04T00:00:00Z
date_updated: 2026-04-07T12:58:47Z
day: '04'
department:
- _id: HeEd
- _id: MoHe
doi: 10.1137/1.9781611977912.11
ec_funded: 1
editor:
- first_name: David P.
full_name: Woodruff, David P.
last_name: Woodruff
external_id:
arxiv:
- '2311.01115'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/2311.01115
month: '01'
oa: 1
oa_version: Preprint
page: 243 - 295
project:
- _id: 266A2E9E-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '788183'
name: Alpha Shape Theory Extended
- _id: 268116B8-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: Z00342
name: Mathematics, Computer Science
- _id: bd9ca328-d553-11ed-ba76-dc4f890cfe62
call_identifier: H2020
grant_number: '101019564'
name: The design and evaluation of modern fully dynamic data structures
- _id: 34def286-11ca-11ed-8bc3-da5948e1613c
grant_number: Z00422
name: Efficient algorithms
- _id: bd9e3a2e-d553-11ed-ba76-8aa684ce17fe
grant_number: P33775
name: Fast Algorithms for a Reactive Network Layer
publication: Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms
(SODA)
publication_identifier:
eisbn:
- '9781611977912'
publication_status: published
publisher: Society for Industrial and Applied Mathematics
quality_controlled: '1'
related_material:
record:
- id: '15094'
relation: dissertation_contains
status: public
scopus_import: '1'
status: public
title: Dynamically maintaining the persistent homology of time series
type: conference
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2024'
...
---
OA_place: publisher
_id: '15094'
abstract:
- lang: eng
text: "Point sets, geometric networks, and arrangements of hyperplanes are fundamental
objects in\r\ndiscrete geometry that have captivated mathematicians for centuries,
if not millennia. This\r\nthesis seeks to cast new light on these structures by
illustrating specific instances where a\r\ntopological perspective, specifically
through discrete Morse theory and persistent homology,\r\nprovides valuable insights.\r\n\r\nAt
first glance, the topology of these geometric objects might seem uneventful: point
sets\r\nessentially lack of topology, arrangements of hyperplanes are a decomposition
of Rd, which\r\nis a contractible space, and the topology of a network primarily
involves the enumeration\r\nof connected components and cycles within the network.
However, beneath this apparent\r\nsimplicity, there lies an array of intriguing
structures, a small subset of which will be uncovered\r\nin this thesis.\r\n\r\nFocused
on three case studies, each addressing one of the mentioned objects, this work\r\nwill
showcase connections that intertwine topology with diverse fields such as combinatorial\r\ngeometry,
algorithms and data structures, and emerging applications like spatial biology.\r\n\r\n"
alternative_title:
- ISTA Thesis
article_processing_charge: No
author:
- first_name: Sebastiano
full_name: Cultrera di Montesano, Sebastiano
id: 34D2A09C-F248-11E8-B48F-1D18A9856A87
last_name: Cultrera di Montesano
orcid: 0000-0001-6249-0832
citation:
ama: Cultrera di Montesano S. Persistence and Morse theory for discrete geometric
structures. 2024. doi:10.15479/at:ista:15094
apa: Cultrera di Montesano, S. (2024). Persistence and Morse theory for discrete
geometric structures. Institute of Science and Technology Austria. https://doi.org/10.15479/at:ista:15094
chicago: Cultrera di Montesano, Sebastiano. “Persistence and Morse Theory for Discrete
Geometric Structures.” Institute of Science and Technology Austria, 2024. https://doi.org/10.15479/at:ista:15094.
ieee: S. Cultrera di Montesano, “Persistence and Morse theory for discrete geometric
structures,” Institute of Science and Technology Austria, 2024.
ista: Cultrera di Montesano S. 2024. Persistence and Morse theory for discrete geometric
structures. Institute of Science and Technology Austria.
mla: Cultrera di Montesano, Sebastiano. Persistence and Morse Theory for Discrete
Geometric Structures. Institute of Science and Technology Austria, 2024, doi:10.15479/at:ista:15094.
short: S. Cultrera di Montesano, Persistence and Morse Theory for Discrete Geometric
Structures, Institute of Science and Technology Austria, 2024.
corr_author: '1'
date_created: 2024-03-08T15:28:10Z
date_published: 2024-03-08T00:00:00Z
date_updated: 2026-04-07T12:58:48Z
day: '08'
ddc:
- '514'
- '500'
- '516'
degree_awarded: PhD
department:
- _id: GradSch
- _id: HeEd
doi: 10.15479/at:ista:15094
ec_funded: 1
file:
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date_created: 2024-03-14T08:55:07Z
date_updated: 2024-03-14T08:55:07Z
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file_name: Thesis Sebastiano.pdf
file_size: 4106872
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success: 1
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checksum: bcbd213490f5a7e68855a092bbce93f1
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date_created: 2024-03-14T08:56:24Z
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language:
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month: '03'
oa: 1
oa_version: Published Version
page: '108'
project:
- _id: 266A2E9E-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '788183'
name: Alpha Shape Theory Extended
- _id: 268116B8-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: Z00342
name: Mathematics, Computer Science
- _id: 0aa4bc98-070f-11eb-9043-e6fff9c6a316
grant_number: I4887
name: Persistent Homology, Algorithms and Stochastic Geometry
- _id: 2561EBF4-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: I02979-N35
name: Persistence and stability of geometric complexes
publication_identifier:
issn:
- 2663-337X
publication_status: published
publisher: Institute of Science and Technology Austria
related_material:
record:
- id: '15091'
relation: part_of_dissertation
status: public
- id: '11660'
relation: part_of_dissertation
status: public
- id: '15090'
relation: part_of_dissertation
status: public
- id: '15093'
relation: part_of_dissertation
status: public
- id: '13182'
relation: part_of_dissertation
status: public
- id: '11658'
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: Persistence and Morse theory for discrete geometric structures
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legal_code_url: https://creativecommons.org/licenses/by-nc-sa/4.0/legalcode
name: Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International (CC
BY-NC-SA 4.0)
short: CC BY-NC-SA (4.0)
type: dissertation
user_id: ba8df636-2132-11f1-aed0-ed93e2281fdd
year: '2024'
...
---
OA_place: publisher
OA_type: hybrid
_id: '15247'
abstract:
- lang: eng
text: Extending the notion of sunflowers, we call a family of at least two sets
an odd-sunflower if every element of the underlying set is contained in an odd
number of sets or in none of them. It follows from the Erdős–Szemerédi conjecture,
recently proved by Naslund and Sawin, that there is a constant <2 such that every
family of subsets of an n-element set that contains no odd-sunflower consists
of at most n sets. We construct such families of size at least 1.5021n. We also
characterize minimal odd-sunflowers of triples.
acknowledgement: We are grateful to Balázs Keszegh, and to the members of the Miklós
Schweitzer Competition committee of 2022 for valuable discussions, and Shira Zerbib
for pointing out several important mathematical typos.
article_number: '105889'
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Peter
full_name: Frankl, Peter
last_name: Frankl
- first_name: János
full_name: Pach, János
id: E62E3130-B088-11EA-B919-BF823C25FEA4
last_name: Pach
- first_name: Dömötör
full_name: Pálvölgyi, Dömötör
last_name: Pálvölgyi
citation:
ama: Frankl P, Pach J, Pálvölgyi D. Odd-sunflowers. Journal of Combinatorial
Theory, Series A. 2024;206(8). doi:10.1016/j.jcta.2024.105889
apa: Frankl, P., Pach, J., & Pálvölgyi, D. (2024). Odd-sunflowers. Journal
of Combinatorial Theory, Series A. Elsevier. https://doi.org/10.1016/j.jcta.2024.105889
chicago: Frankl, Peter, János Pach, and Dömötör Pálvölgyi. “Odd-Sunflowers.” Journal
of Combinatorial Theory, Series A. Elsevier, 2024. https://doi.org/10.1016/j.jcta.2024.105889.
ieee: P. Frankl, J. Pach, and D. Pálvölgyi, “Odd-sunflowers,” Journal of Combinatorial
Theory, Series A, vol. 206, no. 8. Elsevier, 2024.
ista: Frankl P, Pach J, Pálvölgyi D. 2024. Odd-sunflowers. Journal of Combinatorial
Theory, Series A. 206(8), 105889.
mla: Frankl, Peter, et al. “Odd-Sunflowers.” Journal of Combinatorial Theory,
Series A, vol. 206, no. 8, 105889, Elsevier, 2024, doi:10.1016/j.jcta.2024.105889.
short: P. Frankl, J. Pach, D. Pálvölgyi, Journal of Combinatorial Theory, Series
A 206 (2024).
corr_author: '1'
date_created: 2024-03-31T22:01:11Z
date_published: 2024-08-01T00:00:00Z
date_updated: 2025-09-04T13:20:39Z
day: '01'
ddc:
- '510'
department:
- _id: HeEd
doi: 10.1016/j.jcta.2024.105889
external_id:
arxiv:
- '2310.16701'
isi:
- '001217739200001'
file:
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checksum: ffc29d65e712849f0d31009271e06a63
content_type: application/pdf
creator: dernst
date_created: 2025-01-09T08:37:20Z
date_updated: 2025-01-09T08:37:20Z
file_id: '18791'
file_name: 2024_JourCombiTheoryA_Frankl.pdf
file_size: 366029
relation: main_file
success: 1
file_date_updated: 2025-01-09T08:37:20Z
has_accepted_license: '1'
intvolume: ' 206'
isi: 1
issue: '8'
language:
- iso: eng
month: '08'
oa: 1
oa_version: Published Version
publication: Journal of Combinatorial Theory, Series A
publication_identifier:
eissn:
- 1096-0899
issn:
- 0097-3165
publication_status: published
publisher: Elsevier
quality_controlled: '1'
scopus_import: '1'
status: public
title: Odd-sunflowers
tmp:
image: /images/cc_by_nc.png
legal_code_url: https://creativecommons.org/licenses/by-nc/4.0/legalcode
name: Creative Commons Attribution-NonCommercial 4.0 International (CC BY-NC 4.0)
short: CC BY-NC (4.0)
type: journal_article
user_id: 317138e5-6ab7-11ef-aa6d-ffef3953e345
volume: 206
year: '2024'
...
---
OA_place: publisher
OA_type: hybrid
_id: '15380'
abstract:
- lang: eng
text: The depth of a cell in an arrangement of n (non-vertical) great-spheres in
Sd is the number of great-spheres that pass above the cell. We prove Euler-type
relations, which imply extensions of the classic Dehn–Sommerville relations for
convex polytopes to sublevel sets of the depth function, and we use the relations
to extend the expressions for the number of faces of neighborly polytopes to the
number of cells of levels in neighborly arrangements.
acknowledgement: "The authors thank Uli Wagner and Emo Welzl for comments on an earlier
version of this paper, and for pointing out related work in the prior literature.\r\nOpen
access funding provided by Institute of Science and Technology (IST Austria). This
project has received funding from the European Research Council (ERC) under the
European Union’s Horizon 2020 research and innovation programme, Grant No. 788183,
from the Wittgenstein Prize, Austrian Science Fund (FWF), Grant No. Z 342-N31, and
from the DFG Collaborative Research Center TRR 109, ‘Discretization in Geometry
and Dynamics’, Austrian Science Fund (FWF), Grant No. I 02979-N35."
article_processing_charge: Yes (via OA deal)
article_type: original
author:
- first_name: Ranita
full_name: Biswas, Ranita
id: 3C2B033E-F248-11E8-B48F-1D18A9856A87
last_name: Biswas
orcid: 0000-0002-5372-7890
- first_name: Sebastiano
full_name: Cultrera Di Montesano, Sebastiano
id: 34D2A09C-F248-11E8-B48F-1D18A9856A87
last_name: Cultrera Di Montesano
orcid: 0000-0001-6249-0832
- first_name: Herbert
full_name: Edelsbrunner, Herbert
id: 3FB178DA-F248-11E8-B48F-1D18A9856A87
last_name: Edelsbrunner
orcid: 0000-0002-9823-6833
- first_name: Morteza
full_name: Saghafian, Morteza
id: f86f7148-b140-11ec-9577-95435b8df824
last_name: Saghafian
citation:
ama: 'Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. Depth in arrangements:
Dehn–Sommerville–Euler relations with applications. Journal of Applied and
Computational Topology. 2024;8:557-578. doi:10.1007/s41468-024-00173-w'
apa: 'Biswas, R., Cultrera di Montesano, S., Edelsbrunner, H., & Saghafian,
M. (2024). Depth in arrangements: Dehn–Sommerville–Euler relations with applications.
Journal of Applied and Computational Topology. Springer Nature. https://doi.org/10.1007/s41468-024-00173-w'
chicago: 'Biswas, Ranita, Sebastiano Cultrera di Montesano, Herbert Edelsbrunner,
and Morteza Saghafian. “Depth in Arrangements: Dehn–Sommerville–Euler Relations
with Applications.” Journal of Applied and Computational Topology. Springer
Nature, 2024. https://doi.org/10.1007/s41468-024-00173-w.'
ieee: 'R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, and M. Saghafian, “Depth
in arrangements: Dehn–Sommerville–Euler relations with applications,” Journal
of Applied and Computational Topology, vol. 8. Springer Nature, pp. 557–578,
2024.'
ista: 'Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. 2024. Depth
in arrangements: Dehn–Sommerville–Euler relations with applications. Journal of
Applied and Computational Topology. 8, 557–578.'
mla: 'Biswas, Ranita, et al. “Depth in Arrangements: Dehn–Sommerville–Euler Relations
with Applications.” Journal of Applied and Computational Topology, vol.
8, Springer Nature, 2024, pp. 557–78, doi:10.1007/s41468-024-00173-w.'
short: R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, M. Saghafian, Journal
of Applied and Computational Topology 8 (2024) 557–578.
corr_author: '1'
date_created: 2024-05-12T22:01:03Z
date_published: 2024-09-01T00:00:00Z
date_updated: 2025-05-14T09:27:57Z
day: '01'
ddc:
- '510'
department:
- _id: HeEd
doi: 10.1007/s41468-024-00173-w
ec_funded: 1
external_id:
pmid:
- '39308789'
file:
- access_level: open_access
checksum: 0ee15c1493a6413cf356ab2f32c81a9e
content_type: application/pdf
creator: dernst
date_created: 2025-04-23T08:01:36Z
date_updated: 2025-04-23T08:01:36Z
file_id: '19612'
file_name: 2024_JourApplCompTopo_BiswasRa.pdf
file_size: 522831
relation: main_file
success: 1
file_date_updated: 2025-04-23T08:01:36Z
has_accepted_license: '1'
intvolume: ' 8'
language:
- iso: eng
month: '09'
oa: 1
oa_version: Published Version
page: 557-578
pmid: 1
project:
- _id: 266A2E9E-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '788183'
name: Alpha Shape Theory Extended
- _id: 268116B8-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: Z00342
name: Mathematics, Computer Science
- _id: 2561EBF4-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: I02979-N35
name: Persistence and stability of geometric complexes
publication: Journal of Applied and Computational Topology
publication_identifier:
eissn:
- 2367-1734
issn:
- 2367-1726
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
related_material:
record:
- id: '11658'
relation: earlier_version
status: public
scopus_import: '1'
status: public
title: 'Depth in arrangements: Dehn–Sommerville–Euler relations with applications'
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: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 8
year: '2024'
...
---
_id: '17144'
abstract:
- lang: eng
text: "We prove that the medial axis of closed sets is Hausdorff stable in the following
sense: Let \U0001D4AE ⊆ ℝ^d be a fixed closed set that contains a bounding sphere.
That is, the bounding sphere is part of the set \U0001D4AE. Consider the space
of C^{1,1} diffeomorphisms of ℝ^d to itself, which keep the bounding sphere invariant.
The map from this space of diffeomorphisms (endowed with a Banach norm) to the
space of closed subsets of ℝ^d (endowed with the Hausdorff distance), mapping
a diffeomorphism F to the closure of the medial axis of F(\U0001D4AE), is Lipschitz.
This extends a previous stability result of Chazal and Soufflet on the stability
of the medial axis of C² manifolds under C² ambient diffeomorphisms."
acknowledgement: "This research has been supported by the European Research Council
(ERC), grant No. 788183, by the Wittgenstein Prize, Austrian Science Fund (FWF),
grant No. Z 342-N31, and by the DFG Collaborative Research Center TRR 109, Austrian
Science Fund (FWF), grant No. I 02979-N35.\r\nSupported by the European Union’s
Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie
grant agreement No. 754411, the Austrian science fund (FWF) grant No. M-3073, and
the welcome package from IDEX of the Université Cô d'Azur.\r\nWe are greatly indebted
to Fred Chazal for sharing his insights. We further thank Erin Chambers, Christopher
Fillmore, and Elizabeth Stephenson for early discussions and all members of the
Edelsbrunner group (Institute of Science and Technology Austria) and the Datashape
team (Inria) for the atmosphere in which this research was conducted."
alternative_title:
- LIPIcs
article_number: '69'
article_processing_charge: No
arxiv: 1
author:
- first_name: Hana
full_name: Kourimska, Hana
id: D9B8E14C-3C26-11EA-98F5-1F833DDC885E
last_name: Kourimska
orcid: 0000-0001-7841-0091
- first_name: André
full_name: Lieutier, André
last_name: Lieutier
- first_name: Mathijs
full_name: Wintraecken, Mathijs
id: 307CFBC8-F248-11E8-B48F-1D18A9856A87
last_name: Wintraecken
orcid: 0000-0002-7472-2220
citation:
ama: 'Kourimska H, Lieutier A, Wintraecken M. The medial axis of any closed bounded
set Is Lipschitz stable with respect to the Hausdorff distance Under ambient diffeomorphisms.
In: 40th International Symposium on Computational Geometry. Vol 293. Schloss
Dagstuhl - Leibniz-Zentrum für Informatik; 2024. doi:10.4230/LIPIcs.SoCG.2024.69'
apa: 'Kourimska, H., Lieutier, A., & Wintraecken, M. (2024). The medial axis
of any closed bounded set Is Lipschitz stable with respect to the Hausdorff distance
Under ambient diffeomorphisms. In 40th International Symposium on Computational
Geometry (Vol. 293). Athens, Greece: Schloss Dagstuhl - Leibniz-Zentrum für
Informatik. https://doi.org/10.4230/LIPIcs.SoCG.2024.69'
chicago: Kourimska, Hana, André Lieutier, and Mathijs Wintraecken. “The Medial Axis
of Any Closed Bounded Set Is Lipschitz Stable with Respect to the Hausdorff Distance
Under Ambient Diffeomorphisms.” In 40th International Symposium on Computational
Geometry, Vol. 293. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024.
https://doi.org/10.4230/LIPIcs.SoCG.2024.69.
ieee: H. Kourimska, A. Lieutier, and M. Wintraecken, “The medial axis of any closed
bounded set Is Lipschitz stable with respect to the Hausdorff distance Under ambient
diffeomorphisms,” in 40th International Symposium on Computational Geometry,
Athens, Greece, 2024, vol. 293.
ista: 'Kourimska H, Lieutier A, Wintraecken M. 2024. The medial axis of any closed
bounded set Is Lipschitz stable with respect to the Hausdorff distance Under ambient
diffeomorphisms. 40th International Symposium on Computational Geometry. SoCG:
Symposium on Computational Geometry, LIPIcs, vol. 293, 69.'
mla: Kourimska, Hana, et al. “The Medial Axis of Any Closed Bounded Set Is Lipschitz
Stable with Respect to the Hausdorff Distance Under Ambient Diffeomorphisms.”
40th International Symposium on Computational Geometry, vol. 293, 69, Schloss
Dagstuhl - Leibniz-Zentrum für Informatik, 2024, doi:10.4230/LIPIcs.SoCG.2024.69.
short: H. Kourimska, A. Lieutier, M. Wintraecken, in:, 40th International Symposium
on Computational Geometry, Schloss Dagstuhl - Leibniz-Zentrum für Informatik,
2024.
conference:
end_date: 2024-06-14
location: Athens, Greece
name: 'SoCG: Symposium on Computational Geometry'
date_created: 2024-06-16T22:01:06Z
date_published: 2024-06-01T00:00:00Z
date_updated: 2025-04-15T07:16:58Z
day: '01'
ddc:
- '510'
department:
- _id: HeEd
doi: 10.4230/LIPIcs.SoCG.2024.69
ec_funded: 1
external_id:
arxiv:
- '2212.01118'
file:
- access_level: open_access
checksum: b40ff456c19294adb5d9613fcfd751c6
content_type: application/pdf
creator: dernst
date_created: 2024-06-17T08:33:40Z
date_updated: 2024-06-17T08:33:40Z
file_id: '17150'
file_name: 2024_LIPICS_Kourimska.pdf
file_size: 1612558
relation: main_file
success: 1
file_date_updated: 2024-06-17T08:33:40Z
has_accepted_license: '1'
intvolume: ' 293'
language:
- iso: eng
month: '06'
oa: 1
oa_version: Published Version
project:
- _id: 266A2E9E-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '788183'
name: Alpha Shape Theory Extended
- _id: 268116B8-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: Z00342
name: Mathematics, Computer Science
- _id: 2561EBF4-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: I02979-N35
name: Persistence and stability of geometric complexes
- _id: 260C2330-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '754411'
name: ISTplus - Postdoctoral Fellowships
- _id: fc390959-9c52-11eb-aca3-afa58bd282b2
grant_number: M03073
name: Learning and triangulating manifolds via collapses
publication: 40th International Symposium on Computational Geometry
publication_identifier:
isbn:
- '9783959773164'
issn:
- 1868-8969
publication_status: published
publisher: Schloss Dagstuhl - Leibniz-Zentrum für Informatik
quality_controlled: '1'
scopus_import: '1'
status: public
title: The medial axis of any closed bounded set Is Lipschitz stable with respect
to the Hausdorff distance Under ambient diffeomorphisms
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: conference
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 293
year: '2024'
...
---
_id: '17145'
abstract:
- lang: eng
text: Grid peeling is the process of repeatedly removing the convex hull vertices
of the grid points that lie inside a given convex curve. It has been conjectured
that, for a more and more refined grid, grid peeling converges to a continuous
process, the affine curve-shortening flow, which deforms the curve based on the
curvature. We prove this conjecture for one class of curves, parabolas with a
vertical axis, and we determine the value of the constant factor in the formula
that relates the two processes.
acknowledgement: Part of this work was done while G.R. enjoyed the hospitality of
the Institute of Science and Technology Austria (ISTA) as a visiting professor during
his sabbatical in the winter semester 2022/23.
alternative_title:
- LIPIcs
article_number: '76'
article_processing_charge: No
arxiv: 1
author:
- first_name: Günter
full_name: Rote, Günter
last_name: Rote
- first_name: Moritz
full_name: Rüber, Moritz
last_name: Rüber
- first_name: Morteza
full_name: Saghafian, Morteza
id: f86f7148-b140-11ec-9577-95435b8df824
last_name: Saghafian
citation:
ama: 'Rote G, Rüber M, Saghafian M. Grid peeling of parabolas. In: 40th International
Symposium on Computational Geometry. Vol 293. Schloss Dagstuhl - Leibniz-Zentrum
für Informatik; 2024. doi:10.4230/LIPIcs.SoCG.2024.76'
apa: 'Rote, G., Rüber, M., & Saghafian, M. (2024). Grid peeling of parabolas.
In 40th International Symposium on Computational Geometry (Vol. 293). Athens,
Greece: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. https://doi.org/10.4230/LIPIcs.SoCG.2024.76'
chicago: Rote, Günter, Moritz Rüber, and Morteza Saghafian. “Grid Peeling of Parabolas.”
In 40th International Symposium on Computational Geometry, Vol. 293. Schloss
Dagstuhl - Leibniz-Zentrum für Informatik, 2024. https://doi.org/10.4230/LIPIcs.SoCG.2024.76.
ieee: G. Rote, M. Rüber, and M. Saghafian, “Grid peeling of parabolas,” in 40th
International Symposium on Computational Geometry, Athens, Greece, 2024, vol.
293.
ista: 'Rote G, Rüber M, Saghafian M. 2024. Grid peeling of parabolas. 40th International
Symposium on Computational Geometry. SoCG: Symposium on Computational Geometry,
LIPIcs, vol. 293, 76.'
mla: Rote, Günter, et al. “Grid Peeling of Parabolas.” 40th International Symposium
on Computational Geometry, vol. 293, 76, Schloss Dagstuhl - Leibniz-Zentrum
für Informatik, 2024, doi:10.4230/LIPIcs.SoCG.2024.76.
short: G. Rote, M. Rüber, M. Saghafian, in:, 40th International Symposium on Computational
Geometry, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024.
conference:
end_date: 2024-06-14
location: Athens, Greece
name: 'SoCG: Symposium on Computational Geometry'
start_date: 2024-06-11
date_created: 2024-06-16T22:01:06Z
date_published: 2024-06-01T00:00:00Z
date_updated: 2024-06-17T08:41:56Z
day: '01'
ddc:
- '510'
department:
- _id: HeEd
doi: 10.4230/LIPIcs.SoCG.2024.76
external_id:
arxiv:
- '2402.15787'
file:
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checksum: fbad1de06383a6b7e8a1cb3e8c7205ce
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creator: dernst
date_created: 2024-06-17T08:40:04Z
date_updated: 2024-06-17T08:40:04Z
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has_accepted_license: '1'
intvolume: ' 293'
language:
- iso: eng
month: '06'
oa: 1
oa_version: Published Version
publication: 40th International Symposium on Computational Geometry
publication_identifier:
isbn:
- '9783959773164'
issn:
- 1868-8969
publication_status: published
publisher: Schloss Dagstuhl - Leibniz-Zentrum für Informatik
quality_controlled: '1'
scopus_import: '1'
status: public
title: Grid peeling of parabolas
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image: /images/cc_by.png
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...
---
_id: '17146'
abstract:
- lang: eng
text: The Upper Bound Theorem for convex polytopes implies that the p-th Betti number
of the Čech complex of any set of N points in ℝ^d and any radius satisfies β_p
= O(N^m), with m = min{p+1, ⌈d/2⌉}. We construct sets in even and odd dimensions,
which prove that this upper bound is asymptotically tight. For example, we describe
a set of N = 2(n+1) points in ℝ³ and two radii such that the first Betti number
of the Čech complex at one radius is (n+1)² - 1, and the second Betti number of
the Čech complex at the other radius is n². In particular, there is an arrangement
of n contruent balls in ℝ³ that enclose a quadratic number of voids, which answers
a long-standing open question in computational geometry.
acknowledgement: "The first author is supported by the European Research Council (ERC),
grant no. 788183, and by the DFG Collaborative Research Center TRR 109, Austrian
Science Fund (FWF), grant no. {I 02979-N35.} The second author is supported by the
European Research Council (ERC), grant \"GeoScape\" and by the Hungarian Science
Foundation (NKFIH), grant K-131529. Both authors are supported by the Wittgenstein
Prize, Austrian Science Fund (FWF), grant no. Z 342-N31.\r\nThe authors thank Matt
Kahle for communicating the question about extremal Čech complexes, Ben Schweinhart
for early discussions on the linked circles construction in three dimensions, and
Gábor Tardos for helpful remarks and suggestions."
alternative_title:
- LIPIcs
article_number: '53'
article_processing_charge: No
arxiv: 1
author:
- first_name: Herbert
full_name: Edelsbrunner, Herbert
id: 3FB178DA-F248-11E8-B48F-1D18A9856A87
last_name: Edelsbrunner
orcid: 0000-0002-9823-6833
- first_name: János
full_name: Pach, János
id: E62E3130-B088-11EA-B919-BF823C25FEA4
last_name: Pach
citation:
ama: 'Edelsbrunner H, Pach J. Maximum Betti numbers of Čech complexes. In: 40th
International Symposium on Computational Geometry. Vol 293. Schloss Dagstuhl
- Leibniz-Zentrum für Informatik; 2024. doi:10.4230/LIPIcs.SoCG.2024.53'
apa: 'Edelsbrunner, H., & Pach, J. (2024). Maximum Betti numbers of Čech complexes.
In 40th International Symposium on Computational Geometry (Vol. 293). Athens,
Greece: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. https://doi.org/10.4230/LIPIcs.SoCG.2024.53'
chicago: Edelsbrunner, Herbert, and János Pach. “Maximum Betti Numbers of Čech Complexes.”
In 40th International Symposium on Computational Geometry, Vol. 293. Schloss
Dagstuhl - Leibniz-Zentrum für Informatik, 2024. https://doi.org/10.4230/LIPIcs.SoCG.2024.53.
ieee: H. Edelsbrunner and J. Pach, “Maximum Betti numbers of Čech complexes,” in
40th International Symposium on Computational Geometry, Athens, Greece,
2024, vol. 293.
ista: 'Edelsbrunner H, Pach J. 2024. Maximum Betti numbers of Čech complexes. 40th
International Symposium on Computational Geometry. SoCG: Symposium on Computational
Geometry, LIPIcs, vol. 293, 53.'
mla: Edelsbrunner, Herbert, and János Pach. “Maximum Betti Numbers of Čech Complexes.”
40th International Symposium on Computational Geometry, vol. 293, 53, Schloss
Dagstuhl - Leibniz-Zentrum für Informatik, 2024, doi:10.4230/LIPIcs.SoCG.2024.53.
short: H. Edelsbrunner, J. Pach, in:, 40th International Symposium on Computational
Geometry, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024.
conference:
end_date: 2024-06-14
location: Athens, Greece
name: 'SoCG: Symposium on Computational Geometry'
start_date: 2024-06-11
date_created: 2024-06-16T22:01:06Z
date_published: 2024-06-01T00:00:00Z
date_updated: 2025-12-01T15:19:20Z
day: '01'
ddc:
- '510'
department:
- _id: HeEd
doi: 10.4230/LIPIcs.SoCG.2024.53
ec_funded: 1
external_id:
arxiv:
- '2310.14801'
file:
- access_level: open_access
checksum: 5442d44fb89d77477a87668d6e61aac9
content_type: application/pdf
creator: dernst
date_created: 2024-06-17T08:46:33Z
date_updated: 2024-06-17T08:46:33Z
file_id: '17152'
file_name: 2024_LIPICS_Edelsbrunner.pdf
file_size: 766562
relation: main_file
success: 1
file_date_updated: 2024-06-17T08:46:33Z
has_accepted_license: '1'
intvolume: ' 293'
language:
- iso: eng
month: '06'
oa: 1
oa_version: Published Version
project:
- _id: 266A2E9E-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '788183'
name: Alpha Shape Theory Extended
- _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: 40th International Symposium on Computational Geometry
publication_identifier:
isbn:
- '9783959773164'
issn:
- 1868-8969
publication_status: published
publisher: Schloss Dagstuhl - Leibniz-Zentrum für Informatik
quality_controlled: '1'
related_material:
record:
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relation: later_version
status: public
scopus_import: '1'
status: public
title: Maximum Betti numbers of Čech 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: conference
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 293
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...