---
_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
das_tickbox: '0'
date_created: 2024-06-16T22:01:06Z
date_published: 2024-06-01T00:00:00Z
date_updated: 2026-10-02T11:47:18Z
day: '01'
ddc:
- '510'
department:
- _id: HeEd
doi: 10.4230/LIPIcs.SoCG.2024.76
external_id:
arxiv:
- '2402.15787'
file:
- access_level: open_access
checksum: fbad1de06383a6b7e8a1cb3e8c7205ce
content_type: application/pdf
creator: dernst
date_created: 2024-06-17T08:40:04Z
date_updated: 2024-06-17T08:40:04Z
file_id: '17151'
file_name: 2024_LIPICS_Rote.pdf
file_size: 1430896
relation: main_file
success: 1
file_date_updated: 2024-06-17T08:40:04Z
fulldoi: https://doi.org/10.4230/LIPIcs.SoCG.2024.76
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'
researchdata_availability: no
scopus_import: '1'
status: public
supplementarymaterial: no
title: Grid peeling of parabolas
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: 317138e5-6ab7-11ef-aa6d-ffef3953e345
volume: 293
year: '2024'
...
---
_id: '17891'
abstract:
- lang: eng
text: "Abstract\r\nMethods used in topological data analysis naturally capture higher-order
interactions in point cloud data embedded in a metric space. This methodology
was recently extended to data living in an information space, by which we mean
a space measured with an information theoretical distance. One such setting is
a finite collection of discrete probability distributions embedded in the probability
simplex measured with the relative entropy (Kullback–Leibler divergence). More
generally, one can work with a Bregman divergence parameterized by a different
notion of entropy. While theoretical algorithms exist for this setup, there is
a paucity of implementations for exploring and comparing geometric-topological
properties of various information spaces. The interest of this work is therefore
twofold. First, we propose the first robust algorithms and software for geometric
and topological data analysis in information space. Perhaps surprisingly, despite
working with Bregman divergences, our design reuses robust libraries for the Euclidean
case. Second, using the new software, we take the first steps towards understanding
the geometric-topological structure of these spaces. In particular, we compare
them with the more familiar spaces equipped with the Euclidean and Fisher metrics."
acknowledgement: We thank Anton Nikitenko for first observing that the Wrap complex
can be characterized as stated in Claim (ii) of the Wrap Complex Lemma, and Ondrej
Draganov for correcting a critical mistake in one of our formulas in Section 2.
article_number: '637'
article_processing_charge: Yes
article_type: original
author:
- first_name: Herbert
full_name: Edelsbrunner, Herbert
id: 3FB178DA-F248-11E8-B48F-1D18A9856A87
last_name: Edelsbrunner
orcid: 0000-0002-9823-6833
- first_name: Katharina
full_name: Ölsböck, Katharina
id: 4D4AA390-F248-11E8-B48F-1D18A9856A87
last_name: Ölsböck
orcid: 0000-0002-4672-8297
- first_name: Hubert
full_name: Wagner, Hubert
id: 379CA8B8-F248-11E8-B48F-1D18A9856A87
last_name: Wagner
orcid: 0009-0009-9111-8429
citation:
ama: Edelsbrunner H, Ölsböck K, Wagner H. Understanding higher-order interactions
in information space. Entropy. 2024;26(8). doi:10.3390/e26080637
apa: Edelsbrunner, H., Ölsböck, K., & Wagner, H. (2024). Understanding higher-order
interactions in information space. Entropy. MDPI. https://doi.org/10.3390/e26080637
chicago: Edelsbrunner, Herbert, Katharina Ölsböck, and Hubert Wagner. “Understanding
Higher-Order Interactions in Information Space.” Entropy. MDPI, 2024. https://doi.org/10.3390/e26080637.
ieee: H. Edelsbrunner, K. Ölsböck, and H. Wagner, “Understanding higher-order interactions
in information space,” Entropy, vol. 26, no. 8. MDPI, 2024.
ista: Edelsbrunner H, Ölsböck K, Wagner H. 2024. Understanding higher-order interactions
in information space. Entropy. 26(8), 637.
mla: Edelsbrunner, Herbert, et al. “Understanding Higher-Order Interactions in Information
Space.” Entropy, vol. 26, no. 8, 637, MDPI, 2024, doi:10.3390/e26080637.
short: H. Edelsbrunner, K. Ölsböck, H. Wagner, Entropy 26 (2024).
das_tickbox: '1'
dataavailabilitystatement: 'The software and data used for this study is available
at: https://git.ista.ac.at/katharina.oelsboeck/wrap_2_3-public/ (accessed on 19
July 2024).'
date_created: 2024-09-08T22:01:11Z
date_published: 2024-08-01T00:00:00Z
date_updated: 2026-10-02T11:50:23Z
day: '01'
ddc:
- '510'
department:
- _id: HeEd
doi: 10.3390/e26080637
external_id:
isi:
- '001305543500001'
pmid:
- '39202107'
file:
- access_level: open_access
checksum: 624a9e2c5b49d6c38b88b0f675467ba3
content_type: application/pdf
creator: dernst
date_created: 2024-09-09T09:01:12Z
date_updated: 2024-09-09T09:01:12Z
file_id: '17948'
file_name: 2024_Entropy_Edelsbrunner.pdf
file_size: 8025139
relation: main_file
success: 1
file_date_updated: 2024-09-09T09:01:12Z
fulldoi: https://doi.org/10.3390/e26080637
has_accepted_license: '1'
intvolume: ' 26'
isi: 1
issue: '8'
language:
- iso: eng
month: '08'
oa: 1
oa_version: Published Version
pmid: 1
publication: Entropy
publication_identifier:
eissn:
- 1099-4300
publication_status: published
publisher: MDPI
quality_controlled: '1'
related_material:
link:
- relation: software
url: https://git.ista.ac.at/katharina.oelsboeck/wrap_2_3-public/
researchdata_availability: yes
scopus_import: '1'
status: public
supplementarymaterial: yes
title: Understanding higher-order interactions in information space
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: 26
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
das_tickbox: '0'
date_created: 2024-02-18T23:01:03Z
date_published: 2024-01-01T00:00:00Z
date_updated: 2026-10-02T11:50:59Z
day: '01'
department:
- _id: HeEd
doi: 10.1007/978-3-031-49272-3_23
ec_funded: 1
external_id:
arxiv:
- '2306.13201'
isi:
- '001207939600023'
fulldoi: https://doi.org/10.1007/978-3-031-49272-3_23
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
researchdata_availability: no
scopus_import: '1'
status: public
supplementarymaterial: no
title: Decomposition of geometric graphs into star-forests
type: conference
user_id: 317138e5-6ab7-11ef-aa6d-ffef3953e345
volume: 14465
year: '2024'
...
---
_id: '17190'
abstract:
- lang: eng
text: "For a locally finite set, \U0001D434⊆ℝ\U0001D451\r\n, the \U0001D458\r\nth
Brillouin zone of \U0001D44E∈\U0001D434\r\n is the region of points \U0001D465∈ℝ\U0001D451\r\n
for which ‖\U0001D465−\U0001D44E‖\r\n is the \U0001D458\r\nth smallest among the
Euclidean distances between \U0001D465\r\n and the points in \U0001D434\r\n. If
\U0001D434\r\n is a lattice, the \U0001D458\r\nth Brillouin zones of the points
in \U0001D434\r\n are translates of each other, and together they tile space.
Depending on the value of \U0001D458\r\n, they express medium- or long-range order
in the set. We study fundamental geometric and combinatorial properties of Brillouin
zones, focusing on the integer lattice and its perturbations. Our results include
the stability of a Brillouin zone under perturbations, a linear upper bound on
the number of chambers in a zone for lattices in ℝ2\r\n, and the convergence of
the maximum volume of a chamber to zero for the integer lattice."
acknowledgement: The second author is partially supported by the Alexander von Humboldt
Foundation. The sixth author is supported by the European Union's Horizon 2020 research
and innovation programme under Marie Sklodowska-Curie grant agreement 754411, and
by Austrian Science Fund(FWF) grant M-3073. All other authors are supported by European
Research Council (ERC) grant 788183, by the Wittgenstein Prize, by Austrian Science
Fund (FWF) grant Z 342-N31, and by the DFG Collaborative Research Center TRR 109,
Austrian Science Fund (FWF) grant I 02979-N35.
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: Ghafaris, Mohadese
last_name: Ghafaris
- 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: Saghafiant, Morteza
last_name: Saghafiant
- first_name: Mathijs
full_name: Wintraecken, Mathijs
id: 307CFBC8-F248-11E8-B48F-1D18A9856A87
last_name: Wintraecken
orcid: 0000-0002-7472-2220
citation:
ama: Edelsbrunner H, Garber A, Ghafaris M, Heiss T, Saghafiant M, Wintraecken M.
Brillouin zones of integer lattices and their perturbations. SIAM Journal on
Discrete Mathematics. 2024;38(2):1784-1807. doi:10.1137/22M1489071
apa: Edelsbrunner, H., Garber, A., Ghafaris, M., Heiss, T., Saghafiant, M., &
Wintraecken, M. (2024). Brillouin zones of integer lattices and their perturbations.
SIAM Journal on Discrete Mathematics. Society for Industrial and Applied
Mathematics. https://doi.org/10.1137/22M1489071
chicago: Edelsbrunner, Herbert, Alexey Garber, Mohadese Ghafaris, Teresa Heiss,
Morteza Saghafiant, and Mathijs Wintraecken. “Brillouin Zones of Integer Lattices
and Their Perturbations.” SIAM Journal on Discrete Mathematics. Society
for Industrial and Applied Mathematics, 2024. https://doi.org/10.1137/22M1489071.
ieee: H. Edelsbrunner, A. Garber, M. Ghafaris, T. Heiss, M. Saghafiant, and M. Wintraecken,
“Brillouin zones of integer lattices and their perturbations,” SIAM Journal
on Discrete Mathematics, vol. 38, no. 2. Society for Industrial and Applied
Mathematics, pp. 1784–1807, 2024.
ista: Edelsbrunner H, Garber A, Ghafaris M, Heiss T, Saghafiant M, Wintraecken M.
2024. Brillouin zones of integer lattices and their perturbations. SIAM Journal
on Discrete Mathematics. 38(2), 1784–1807.
mla: Edelsbrunner, Herbert, et al. “Brillouin Zones of Integer Lattices and Their
Perturbations.” SIAM Journal on Discrete Mathematics, vol. 38, no. 2, Society
for Industrial and Applied Mathematics, 2024, pp. 1784–807, doi:10.1137/22M1489071.
short: H. Edelsbrunner, A. Garber, M. Ghafaris, T. Heiss, M. Saghafiant, M. Wintraecken,
SIAM Journal on Discrete Mathematics 38 (2024) 1784–1807.
corr_author: '1'
das_tickbox: '0'
date_created: 2024-06-30T22:01:05Z
date_published: 2024-06-07T00:00:00Z
date_updated: 2026-10-02T11:49:10Z
day: '07'
department:
- _id: HeEd
doi: 10.1137/22M1489071
ec_funded: 1
external_id:
arxiv:
- '2204.01077'
isi:
- '001292728600001'
fulldoi: https://doi.org/10.1137/22M1489071
intvolume: ' 38'
isi: 1
issue: '2'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://doi.org/10.48550/arXiv.2204.01077
month: '06'
oa: 1
oa_version: Preprint
page: 1784-1807
project:
- _id: 260C2330-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '754411'
name: ISTplus - Postdoctoral Fellowships
- _id: 266A2E9E-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '788183'
name: Alpha Shape Theory Extended
- _id: fc390959-9c52-11eb-aca3-afa58bd282b2
grant_number: M03073
name: Learning and triangulating manifolds via collapses
- _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: SIAM Journal on Discrete Mathematics
publication_identifier:
issn:
- 0895-4801
publication_status: published
publisher: Society for Industrial and Applied Mathematics
quality_controlled: '1'
researchdata_availability: no
scopus_import: '1'
status: public
supplementarymaterial: no
title: Brillouin zones of integer lattices and their perturbations
type: journal_article
user_id: 317138e5-6ab7-11ef-aa6d-ffef3953e345
volume: 38
year: '2024'
...
---
OA_place: publisher
OA_type: gold
_id: '18556'
abstract:
- lang: eng
text: Given a finite set, A ⊆ ℝ², and a subset, B ⊆ A, the MST-ratio is the combined
length of the minimum spanning trees of B and A⧵B divided by the length of the
minimum spanning tree of A. The question of the supremum, over all sets A, of
the maximum, over all subsets B, is related to the Steiner ratio, and we prove
this sup-max is between 2.154 and 2.427. Restricting ourselves to 2-dimensional
lattices, we prove that the sup-max is 2, while the inf-max is 1.25. By some margin
the most difficult of these results is the upper bound for the inf-max, which
we prove by showing that the hexagonal lattice cannot have MST-ratio larger than
1.25.
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. 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.
alternative_title:
- LIPIcs
article_number: '3'
article_processing_charge: Yes
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. The Euclidean
MST-ratio for bi-colored lattices. In: 32nd International Symposium on Graph
Drawing and Network Visualization. Vol 320. Schloss Dagstuhl - Leibniz-Zentrum
für Informatik; 2024. doi:10.4230/LIPIcs.GD.2024.3'
apa: 'Cultrera di Montesano, S., Draganov, O., Edelsbrunner, H., & Saghafian,
M. (2024). The Euclidean MST-ratio for bi-colored lattices. In 32nd International
Symposium on Graph Drawing and Network Visualization (Vol. 320). Vienna, Austria:
Schloss Dagstuhl - Leibniz-Zentrum für Informatik. https://doi.org/10.4230/LIPIcs.GD.2024.3'
chicago: Cultrera di Montesano, Sebastiano, Ondrej Draganov, Herbert Edelsbrunner,
and Morteza Saghafian. “The Euclidean MST-Ratio for Bi-Colored Lattices.” In 32nd
International Symposium on Graph Drawing and Network Visualization, Vol. 320.
Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. https://doi.org/10.4230/LIPIcs.GD.2024.3.
ieee: S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, and M. Saghafian,
“The Euclidean MST-ratio for bi-colored lattices,” in 32nd International Symposium
on Graph Drawing and Network Visualization, Vienna, Austria, 2024, vol. 320.
ista: 'Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M. 2024. The
Euclidean MST-ratio for bi-colored lattices. 32nd International Symposium on Graph
Drawing and Network Visualization. GD: Graph Drawing and Network Visualization,
LIPIcs, vol. 320, 3.'
mla: Cultrera di Montesano, Sebastiano, et al. “The Euclidean MST-Ratio for Bi-Colored
Lattices.” 32nd International Symposium on Graph Drawing and Network Visualization,
vol. 320, 3, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024, doi:10.4230/LIPIcs.GD.2024.3.
short: S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, M. Saghafian, in:,
32nd International Symposium on Graph Drawing and Network Visualization, Schloss
Dagstuhl - Leibniz-Zentrum für Informatik, 2024.
conference:
end_date: 2024-09-20
location: Vienna, Austria
name: 'GD: Graph Drawing and Network Visualization'
start_date: 2024-09-18
corr_author: '1'
das_tickbox: '0'
date_created: 2024-11-17T23:01:47Z
date_published: 2024-10-28T00:00:00Z
date_updated: 2026-10-02T11:52:23Z
day: '28'
ddc:
- '510'
department:
- _id: HeEd
doi: 10.4230/LIPIcs.GD.2024.3
ec_funded: 1
external_id:
arxiv:
- '2403.10204'
isi:
- '001540278400001'
file:
- access_level: open_access
checksum: 5f9b35e115c3d375e99be78da9054cb4
content_type: application/pdf
creator: dernst
date_created: 2024-11-18T07:49:25Z
date_updated: 2024-11-18T07:49:25Z
file_id: '18560'
file_name: 2024_LIPIcs_CultreradiMontesano.pdf
file_size: 908541
relation: main_file
success: 1
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fulldoi: https://doi.org/10.4230/LIPIcs.GD.2024.3
has_accepted_license: '1'
intvolume: ' 320'
isi: 1
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: 32nd International Symposium on Graph Drawing and Network Visualization
publication_identifier:
isbn:
- '9783959773430'
issn:
- 1868-8969
publication_status: published
publisher: Schloss Dagstuhl - Leibniz-Zentrum für Informatik
quality_controlled: '1'
researchdata_availability: no
scopus_import: '1'
status: public
supplementarymaterial: no
title: The Euclidean MST-ratio for bi-colored lattices
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: 317138e5-6ab7-11ef-aa6d-ffef3953e345
volume: 320
year: '2024'
...
---
_id: '18097'
abstract:
- lang: eng
text: "In our companion paper \"Tight bounds for the learning of homotopy à la Niyogi,
Smale, and Weinberger for subsets of Euclidean spaces and of Riemannian manifolds\"
we gave optimal bounds (in terms of the two one-sided Hausdorff distances) on
a sample P of an input shape \U0001D4AE (either manifold or general set with positive
reach) such that one can infer the homotopy of \U0001D4AE from the union of balls
with some radius centred at P, both in Euclidean space and in a Riemannian manifold
of bounded curvature. The construction showing the optimality of the bounds is
not straightforward. The purpose of this video is to visualize and thus elucidate
said construction in the Euclidean setting."
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. I02979-N35. Mathijs Wintraecken: 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) grant
No. M-3073, and the welcome package from IDEX of the Université Côte d’Azur.\r\nWe
thank Jean-Daniel Boissonnat, Herbert Edelsbrunner, and Mariette Yvinec for discussion."
alternative_title:
- LIPIcs
article_number: '87'
article_processing_charge: Yes
author:
- first_name: Dominique
full_name: Attali, Dominique
last_name: Attali
- first_name: Hana
full_name: Kourimska, Hana
id: D9B8E14C-3C26-11EA-98F5-1F833DDC885E
last_name: Kourimska
orcid: 0000-0001-7841-0091
- first_name: Christopher D
full_name: Fillmore, Christopher D
id: 35638A5C-AAC7-11E9-B0BF-5503E6697425
last_name: Fillmore
- first_name: Ishika
full_name: Ghosh, Ishika
id: ee449b28-344d-11ef-a6d5-9ca430e9e9ff
last_name: Ghosh
- first_name: Andre
full_name: Lieutier, Andre
last_name: Lieutier
- 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: 'Attali D, Kourimska H, Fillmore CD, et al. The ultimate frontier: An optimality
construction for homotopy inference (media exposition). In: 40th International
Symposium on Computational Geometry. Vol 293. Schloss Dagstuhl - Leibniz-Zentrum
für Informatik; 2024. doi:10.4230/LIPIcs.SoCG.2024.87'
apa: 'Attali, D., Kourimska, H., Fillmore, C. D., Ghosh, I., Lieutier, A., Stephenson,
E. R., & Wintraecken, M. (2024). The ultimate frontier: An optimality construction
for homotopy inference (media exposition). In 40th International Symposium
on Computational Geometry (Vol. 293). Athens, Greece: Schloss Dagstuhl - Leibniz-Zentrum
für Informatik. https://doi.org/10.4230/LIPIcs.SoCG.2024.87'
chicago: 'Attali, Dominique, Hana Kourimska, Christopher D Fillmore, Ishika Ghosh,
Andre Lieutier, Elizabeth R Stephenson, and Mathijs Wintraecken. “The Ultimate
Frontier: An Optimality Construction for Homotopy Inference (Media Exposition).”
In 40th International Symposium on Computational Geometry, Vol. 293. Schloss
Dagstuhl - Leibniz-Zentrum für Informatik, 2024. https://doi.org/10.4230/LIPIcs.SoCG.2024.87.'
ieee: 'D. Attali et al., “The ultimate frontier: An optimality construction
for homotopy inference (media exposition),” in 40th International Symposium
on Computational Geometry, Athens, Greece, 2024, vol. 293.'
ista: 'Attali D, Kourimska H, Fillmore CD, Ghosh I, Lieutier A, Stephenson ER, Wintraecken
M. 2024. The ultimate frontier: An optimality construction for homotopy inference
(media exposition). 40th International Symposium on Computational Geometry. SoCG:
Symposium on Computational Geometry, LIPIcs, vol. 293, 87.'
mla: 'Attali, Dominique, et al. “The Ultimate Frontier: An Optimality Construction
for Homotopy Inference (Media Exposition).” 40th International Symposium on
Computational Geometry, vol. 293, 87, Schloss Dagstuhl - Leibniz-Zentrum für
Informatik, 2024, doi:10.4230/LIPIcs.SoCG.2024.87.'
short: D. Attali, H. Kourimska, C.D. Fillmore, I. Ghosh, A. Lieutier, E.R. Stephenson,
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'
start_date: 2024-06-11
corr_author: '1'
das_tickbox: '0'
date_created: 2024-09-19T10:29:48Z
date_published: 2024-06-06T00:00:00Z
date_updated: 2026-10-02T11:51:50Z
day: '06'
ddc:
- '000'
department:
- _id: HeEd
doi: 10.4230/LIPIcs.SoCG.2024.87
ec_funded: 1
file:
- access_level: open_access
checksum: 9355c2e60b8ec285e1b22719c5b73f1a
content_type: application/pdf
creator: dernst
date_created: 2024-09-19T10:30:37Z
date_updated: 2024-09-19T10:30:37Z
file_id: '18098'
file_name: 2024_LIPICs_Attali.pdf
file_size: 3507177
relation: main_file
success: 1
file_date_updated: 2024-09-19T10:30:37Z
fulldoi: https://doi.org/10.4230/LIPIcs.SoCG.2024.87
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'
publication_status: published
publisher: Schloss Dagstuhl - Leibniz-Zentrum für Informatik
quality_controlled: '1'
researchdata_availability: no
status: public
supplementarymaterial: yes
title: 'The ultimate frontier: An optimality construction for homotopy inference (media
exposition)'
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: 317138e5-6ab7-11ef-aa6d-ffef3953e345
volume: 293
year: '2024'
...
---
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'
das_tickbox: '0'
date_created: 2023-07-02T22:00:44Z
date_published: 2024-10-01T00:00:00Z
date_updated: 2026-10-02T11:52:56Z
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
fulldoi: https://doi.org/10.1007/s41468-023-00126-9
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: '11660'
relation: earlier_version
status: public
- id: '15094'
relation: dissertation_contains
status: public
researchdata_availability: no
scopus_import: '1'
status: public
supplementarymaterial: no
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: 317138e5-6ab7-11ef-aa6d-ffef3953e345
volume: 8
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'
das_tickbox: '0'
date_created: 2024-05-12T22:01:03Z
date_published: 2024-09-01T00:00:00Z
date_updated: 2026-10-02T11:55:12Z
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
fulldoi: https://doi.org/10.1007/s41468-024-00173-w
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
researchdata_availability: no
scopus_import: '1'
status: public
supplementarymaterial: no
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: 317138e5-6ab7-11ef-aa6d-ffef3953e345
volume: 8
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'
das_tickbox: '1'
dataavailabilitystatement: No data was used for the research described in the article.
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department:
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doi: 10.1016/j.jcta.2024.105889
external_id:
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OA_place: publisher
_id: '18667'
abstract:
- lang: eng
text: "Many chemical and physical properties of materials are determined by the
material’s shape,\r\nfor example the size of its pores and the width of its tunnels.
This makes materials science\r\na prime application area for geometrical and topological
methods. Nevertheless many\r\nmethods in topological data analysis have not been
satisfyingly extended to the needs of\r\nmaterials science. This thesis provides
new methods and new mathematical theorems\r\ntargeted at those specific needs
by answering four different research questions. While the\r\nmotivation for each
of the research questions arises from materials science, the methods\r\nare versatile
and can be applied in different areas as well. \r\n\r\nThe first research question
is concerned with image data, for example a three-dimensional\r\ncomputed tomography
(CT) scan of a material, like sand or stone. There are two commonly\r\nused topologies
for digital images and depending on the application either of them might be\r\nrequired.
However, software for computing the topological data analysis method persistence\r\nhomology,
usually supports only one of the two topologies. We answer the question how to\r\ncompute
persistent homology of an image with respect to one of the two topologies using\r\nsoftware
that is intended for the other topology. \r\n\r\nThe second research question
is concerned with image data as well, and asks how much\r\nof the topological
information of an image is lost when the resolution is coarsened. As\r\ncomputer
tomography scanners are more expensive the higher the resolution, it is an\r\nimportant
question in materials science to know which resolution is enough to get satisfying\r\npersistent
homology. We give theoretical bounds on the information loss based on different\r\ngeometrical
properties of the object to be scanned. In addition, we conduct experiments on\r\nsand
and stone CT image data. \r\n\r\nThe third research question is motivated by comparing
crystalline materials efficiently. As\r\nthe atoms within a crystal repeat periodically,
crystalline materials are either modeled by\r\nunmanageable infinite periodic
point sets, or by one of their fundamental domains, which is\r\nunstable under
perturbation. Therefore a fingerprint of crystalline materials is needed, with\r\nappropriate
properties such that comparing the crystals can be eased by comparing the\r\nfingerprints
instead. We define the density fingerprint and prove the necessary properties.
\r\n\r\nThe fourth research question is motivated by studying the hole-structure
or connectedness,\r\ni.e. persistent homology or merge trees, of crystalline materials.
A common way to deal\r\nwith periodicity is to take a fundamental domain and identify
opposite boundaries to form a\r\ntorus. However, computing persistent homology
or merge trees on that torus loses some\r\nof the information materials scientists
are interested in and is additionally not stable under\r\ncertain noise. We therefore
decorate the merge tree stemming from the torus with additional\r\ninformation
describing the density and growth rate of the periodic copies of a component\r\nwithin
a growing spherical window. We prove all desired properties, like stability and
efficient\r\ncomputability."
acknowledgement: "I was supported by the European Research Council (ERC) Horizon 2020
project\r\n“Alpha Shape Theory Extended” No. 788183 and by the Pöttinger Scholarship.
In addition,\r\nI am very thankful for having been able to attend the second Workshop
for Women in\r\nComputational Topology in July 2019, funded by the Mathematical
Sciences Institute at\r\nANU, the US National Science Foundation through the award
CCF-1841455, the Australian\r\nMathematical Sciences Institute and the Association
for Women in Mathematics. Two of the\r\nprojects presented in this thesis started
there. One of them reached completion thanks to\r\nfunding from the MSRI Summer
Research in Mathematics program awarded to me and my\r\ncollaborators in 2020."
alternative_title:
- ISTA Thesis
article_processing_charge: No
author:
- first_name: Teresa
full_name: Heiss, Teresa
id: 4879BB4E-F248-11E8-B48F-1D18A9856A87
last_name: Heiss
orcid: 0000-0002-1780-2689
citation:
ama: Heiss T. New methods for applying topological data analysis to materials science.
2024. doi:10.15479/at:ista:18667
apa: Heiss, T. (2024). New methods for applying topological data analysis to
materials science. Institute of Science and Technology Austria. https://doi.org/10.15479/at:ista:18667
chicago: Heiss, Teresa. “New Methods for Applying Topological Data Analysis to Materials
Science.” Institute of Science and Technology Austria, 2024. https://doi.org/10.15479/at:ista:18667.
ieee: T. Heiss, “New methods for applying topological data analysis to materials
science,” Institute of Science and Technology Austria, 2024.
ista: Heiss T. 2024. New methods for applying topological data analysis to materials
science. Institute of Science and Technology Austria.
mla: Heiss, Teresa. New Methods for Applying Topological Data Analysis to Materials
Science. Institute of Science and Technology Austria, 2024, doi:10.15479/at:ista:18667.
short: T. Heiss, New Methods for Applying Topological Data Analysis to Materials
Science, Institute of Science and Technology Austria, 2024.
corr_author: '1'
das_tickbox: '0'
date_created: 2024-12-17T16:17:55Z
date_published: 2024-12-17T00:00:00Z
date_updated: 2026-10-02T11:55:42Z
day: '17'
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- '514'
- '516'
- '004'
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relation: source_file
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fulldoi: https://doi.org/10.15479/at:ista:18667
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keyword:
- persistent homology
- topological data analysis
- periodic
- crystalline materials
- images
- fingerprint
language:
- iso: eng
month: '12'
oa: 1
oa_version: Published Version
page: '111'
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call_identifier: H2020
grant_number: '788183'
name: Alpha Shape Theory Extended
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issn:
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publisher: Institute of Science and Technology Austria
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full_name: Edelsbrunner, Herbert
id: 3FB178DA-F248-11E8-B48F-1D18A9856A87
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...
---
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'
das_tickbox: '0'
date_created: 2024-03-08T15:28:10Z
date_published: 2024-03-08T00:00:00Z
date_updated: 2026-10-02T11:57:31Z
day: '08'
ddc:
- '514'
- '500'
- '516'
degree_awarded: PhD
department:
- _id: GradSch
- _id: HeEd
doi: 10.15479/at:ista:15094
doi_confirm: '1'
ec_funded: 1
file:
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creator: scultrer
date_created: 2024-03-14T08:55:07Z
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language:
- iso: eng
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:
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- id: '13182'
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relation: part_of_dissertation
status: public
- id: '15093'
relation: part_of_dissertation
status: public
researchdata_availability: no
status: public
supervisor:
- first_name: Herbert
full_name: Edelsbrunner, Herbert
id: 3FB178DA-F248-11E8-B48F-1D18A9856A87
last_name: Edelsbrunner
orcid: 0000-0002-9823-6833
supplementarymaterial: no
title: Persistence and Morse theory for discrete geometric structures
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short: CC BY-NC-SA (4.0)
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user_id: 317138e5-6ab7-11ef-aa6d-ffef3953e345
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'
das_tickbox: '0'
date_created: 2024-03-08T10:27:39Z
date_published: 2024-01-04T00:00:00Z
date_updated: 2026-10-02T11:57:32Z
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'
fulldoi: https://doi.org/10.1137/1.9781611977912.11
language:
- iso: eng
main_file_link:
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url: https://arxiv.org/abs/2311.01115
month: '01'
oa: 1
oa_version: Preprint
page: 243 - 295
project:
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call_identifier: H2020
grant_number: '788183'
name: Alpha Shape Theory Extended
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call_identifier: FWF
grant_number: Z00342
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call_identifier: H2020
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grant_number: Z00422
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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'
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supplementarymaterial: no
title: Dynamically maintaining the persistent homology of time series
type: conference
user_id: 317138e5-6ab7-11ef-aa6d-ffef3953e345
year: '2024'
...
---
OA_place: repository
_id: '18981'
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\r\nUnion’s Horizon 2020 research and innovation programme,
grant no. 788183, from the Wittgenstein Prize,\r\nAustrian Science Fund (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), grant no. I 02979-N35."
article_processing_charge: No
arxiv: 1
author:
- first_name: Adam
full_name: Brown, Adam
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. arXiv. doi:10.48550/arXiv.2209.14993
apa: Brown, A., & Draganov, O. (n.d.). Discrete microlocal Morse theory. arXiv.
https://doi.org/10.48550/arXiv.2209.14993
chicago: Brown, Adam, and Ondrej Draganov. “Discrete Microlocal Morse Theory.” ArXiv,
n.d. https://doi.org/10.48550/arXiv.2209.14993.
ieee: A. Brown and O. Draganov, “Discrete microlocal Morse theory,” arXiv.
.
ista: Brown A, Draganov O. Discrete microlocal Morse theory. arXiv, 10.48550/arXiv.2209.14993.
mla: Brown, Adam, and Ondrej Draganov. “Discrete Microlocal Morse Theory.” ArXiv,
doi:10.48550/arXiv.2209.14993.
short: A. Brown, O. Draganov, ArXiv (n.d.).
corr_author: '1'
das_tickbox: '0'
date_created: 2025-01-31T17:03:04Z
date_published: 2024-06-09T00:00:00Z
date_updated: 2026-10-02T11:57:00Z
day: '09'
department:
- _id: HeEd
doi: 10.48550/arXiv.2209.14993
ec_funded: 1
external_id:
arxiv:
- '2209.14993'
fulldoi: https://doi.org/10.48550/arXiv.2209.14993
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://doi.org/10.48550/arXiv.2209.14993
month: '06'
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: arXiv
publication_status: draft
related_material:
record:
- id: '20323'
relation: later_version
status: public
- id: '18979'
relation: dissertation_contains
status: public
researchdata_availability: no
status: public
supplementarymaterial: no
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: preprint
user_id: 317138e5-6ab7-11ef-aa6d-ffef3953e345
year: '2024'
...
---
OA_place: repository
_id: '18673'
abstract:
- lang: eng
text: "Motivated by applications to crystalline materials, we generalize the merge
tree and the related barcode of a filtered complex to the periodic setting in
Euclidean space. They are invariant under isometries, changing bases, and indeed
changing lattices. In addition, we prove stability under perturbations and provide
an algorithm that under mild geometric conditions typically satisfied by crystalline
materials takes O((n+m)logn) time, in which n and m are the numbers of vertices
and edges in the quotient complex, respectively.\r\n"
acknowledgement: "Both authors are partially supported by the European Research Council
(ERC) Horizon 2020 project\r\n‘Alpha Shape Theory Extended’, grant no. 788183. The
first author is also partially supported by the DFG\r\nCollaborative Research Center
TRR 109, ‘Discretization in Geometry and Dynamics’, Austrian Science Fund\r\n(FWF),
grant no. I 02979-N35."
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: Teresa
full_name: Heiss, Teresa
id: 4879BB4E-F248-11E8-B48F-1D18A9856A87
last_name: Heiss
orcid: 0000-0002-1780-2689
citation:
ama: Edelsbrunner H, Heiss T. Merge trees of periodic filtrations. arXiv.
doi:10.48550/arXiv.2408.16575
apa: Edelsbrunner, H., & Heiss, T. (n.d.). Merge trees of periodic filtrations.
arXiv. https://doi.org/10.48550/arXiv.2408.16575
chicago: Edelsbrunner, Herbert, and Teresa Heiss. “Merge Trees of Periodic Filtrations.”
ArXiv, n.d. https://doi.org/10.48550/arXiv.2408.16575.
ieee: H. Edelsbrunner and T. Heiss, “Merge trees of periodic filtrations,” arXiv.
.
ista: Edelsbrunner H, Heiss T. Merge trees of periodic filtrations. arXiv, 10.48550/arXiv.2408.16575.
mla: Edelsbrunner, Herbert, and Teresa Heiss. “Merge Trees of Periodic Filtrations.”
ArXiv, doi:10.48550/arXiv.2408.16575.
short: H. Edelsbrunner, T. Heiss, ArXiv (n.d.).
corr_author: '1'
das_tickbox: '0'
date_created: 2024-12-18T14:06:57Z
date_published: 2024-08-29T00:00:00Z
date_updated: 2026-10-02T11:55:43Z
day: '29'
department:
- _id: HeEd
doi: 10.48550/arXiv.2408.16575
ec_funded: 1
external_id:
arxiv:
- '2408.16575'
fulldoi: https://doi.org/10.48550/arXiv.2408.16575
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://doi.org/10.48550/arXiv.2408.16575
month: '08'
oa: 1
oa_version: Preprint
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
publication: arXiv
publication_status: draft
related_material:
record:
- id: '18667'
relation: dissertation_contains
status: public
researchdata_availability: no
status: public
supplementarymaterial: no
title: Merge trees of periodic filtrations
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: 317138e5-6ab7-11ef-aa6d-ffef3953e345
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'
das_tickbox: '0'
date_created: 2024-03-08T10:13:59Z
date_published: 2024-02-07T00:00:00Z
date_updated: 2026-10-02T11:56:13Z
day: '07'
department:
- _id: HeEd
doi: 10.48550/arXiv.2212.03128
external_id:
arxiv:
- '2212.03128'
fulldoi: https://doi.org/10.48550/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: '18979'
relation: dissertation_contains
status: public
- id: '20585'
relation: later_version
status: public
- id: '15094'
relation: dissertation_contains
status: public
researchdata_availability: no
status: public
supplementarymaterial: no
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: 317138e5-6ab7-11ef-aa6d-ffef3953e345
year: '2024'
...
---
_id: '12086'
abstract:
- lang: eng
text: We present a simple algorithm for computing higher-order Delaunay mosaics
that works in Euclidean spaces of any finite dimensions. The algorithm selects
the vertices of the order-k mosaic from incrementally constructed lower-order
mosaics and uses an algorithm for weighted first-order Delaunay mosaics as a black-box
to construct the order-k mosaic from its vertices. Beyond this black-box, the
algorithm uses only combinatorial operations, thus facilitating easy implementation.
We extend this algorithm to compute higher-order α-shapes and provide open-source
implementations. We present experimental results for properties of higher-order
Delaunay mosaics of random point sets.
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.
article_processing_charge: Yes (via OA deal)
article_type: original
author:
- first_name: Herbert
full_name: Edelsbrunner, Herbert
id: 3FB178DA-F248-11E8-B48F-1D18A9856A87
last_name: Edelsbrunner
orcid: 0000-0002-9823-6833
- first_name: Georg F
full_name: Osang, Georg F
id: 464B40D6-F248-11E8-B48F-1D18A9856A87
last_name: Osang
orcid: 0000-0002-8882-5116
citation:
ama: Edelsbrunner H, Osang GF. A simple algorithm for higher-order Delaunay mosaics
and alpha shapes. Algorithmica. 2023;85:277-295. doi:10.1007/s00453-022-01027-6
apa: Edelsbrunner, H., & Osang, G. F. (2023). A simple algorithm for higher-order
Delaunay mosaics and alpha shapes. Algorithmica. Springer Nature. https://doi.org/10.1007/s00453-022-01027-6
chicago: Edelsbrunner, Herbert, and Georg F Osang. “A Simple Algorithm for Higher-Order
Delaunay Mosaics and Alpha Shapes.” Algorithmica. Springer Nature, 2023.
https://doi.org/10.1007/s00453-022-01027-6.
ieee: H. Edelsbrunner and G. F. Osang, “A simple algorithm for higher-order Delaunay
mosaics and alpha shapes,” Algorithmica, vol. 85. Springer Nature, pp.
277–295, 2023.
ista: Edelsbrunner H, Osang GF. 2023. A simple algorithm for higher-order Delaunay
mosaics and alpha shapes. Algorithmica. 85, 277–295.
mla: Edelsbrunner, Herbert, and Georg F. Osang. “A Simple Algorithm for Higher-Order
Delaunay Mosaics and Alpha Shapes.” Algorithmica, vol. 85, Springer Nature,
2023, pp. 277–95, doi:10.1007/s00453-022-01027-6.
short: H. Edelsbrunner, G.F. Osang, Algorithmica 85 (2023) 277–295.
corr_author: '1'
date_created: 2022-09-11T22:01:57Z
date_published: 2023-01-01T00:00:00Z
date_updated: 2025-04-23T08:46:48Z
day: '01'
ddc:
- '510'
department:
- _id: HeEd
doi: 10.1007/s00453-022-01027-6
ec_funded: 1
external_id:
isi:
- '000846967100001'
pmid:
- '36687803'
file:
- access_level: open_access
checksum: 71685ca5121f4c837f40c3f8eb50c915
content_type: application/pdf
creator: dernst
date_created: 2023-01-20T10:02:48Z
date_updated: 2023-01-20T10:02:48Z
file_id: '12322'
file_name: 2023_Algorithmica_Edelsbrunner.pdf
file_size: 911017
relation: main_file
success: 1
file_date_updated: 2023-01-20T10:02:48Z
fulldoi: https://doi.org/10.1007/s00453-022-01027-6
has_accepted_license: '1'
intvolume: ' 85'
isi: 1
language:
- iso: eng
month: '01'
oa: 1
oa_version: Published Version
page: 277-295
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: Algorithmica
publication_identifier:
eissn:
- 1432-0541
issn:
- 0178-4617
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: A simple algorithm for higher-order Delaunay mosaics and alpha shapes
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: 85
year: '2023'
...
---
_id: '12287'
abstract:
- lang: eng
text: We present criteria for establishing a triangulation of a manifold. Given
a manifold M, a simplicial complex A, and a map H from the underlying space of
A to M, our criteria are presented in local coordinate charts for M, and ensure
that H is a homeomorphism. These criteria do not require a differentiable structure,
or even an explicit metric on M. No Delaunay property of A is assumed. The result
provides a triangulation guarantee for algorithms that construct a simplicial
complex by working in local coordinate patches. Because the criteria are easily
verified in such a setting, they are expected to be of general use.
acknowledgement: "This work has been funded by the European Research Council under
the European Union’s ERC Grant Agreement number 339025 GUDHI (Algorithmic Foundations
of Geometric Understanding in Higher Dimensions). Arijit Ghosh is supported by Ramanujan
Fellowship (No. SB/S2/RJN-064/2015). Part of this work was done when Arijit Ghosh
was a Researcher at Max-Planck-Institute for Informatics, Germany, supported by
the IndoGerman Max Planck Center for Computer Science (IMPECS). Mathijs Wintraecken
also received funding from the European Union’s Horizon 2020 research and innovation
programme under the Marie Skłodowska-Curie grant agreement No. 754411 and the Austrian
Science Fund (FWF): M-3073. A part of the results described in this paper were presented
at SoCG 2018 and in [3]. \r\nOpen access funding provided by the Austrian Science
Fund (FWF)."
article_processing_charge: No
article_type: original
author:
- first_name: Jean-Daniel
full_name: Boissonnat, Jean-Daniel
last_name: Boissonnat
- first_name: Ramsay
full_name: Dyer, Ramsay
last_name: Dyer
- first_name: Arijit
full_name: Ghosh, Arijit
last_name: Ghosh
- first_name: Mathijs
full_name: Wintraecken, Mathijs
id: 307CFBC8-F248-11E8-B48F-1D18A9856A87
last_name: Wintraecken
orcid: 0000-0002-7472-2220
citation:
ama: Boissonnat J-D, Dyer R, Ghosh A, Wintraecken M. Local criteria for triangulating
general manifolds. Discrete & Computational Geometry. 2023;69:156-191.
doi:10.1007/s00454-022-00431-7
apa: Boissonnat, J.-D., Dyer, R., Ghosh, A., & Wintraecken, M. (2023). Local
criteria for triangulating general manifolds. Discrete & Computational
Geometry. Springer Nature. https://doi.org/10.1007/s00454-022-00431-7
chicago: Boissonnat, Jean-Daniel, Ramsay Dyer, Arijit Ghosh, and Mathijs Wintraecken.
“Local Criteria for Triangulating General Manifolds.” Discrete & Computational
Geometry. Springer Nature, 2023. https://doi.org/10.1007/s00454-022-00431-7.
ieee: J.-D. Boissonnat, R. Dyer, A. Ghosh, and M. Wintraecken, “Local criteria for
triangulating general manifolds,” Discrete & Computational Geometry,
vol. 69. Springer Nature, pp. 156–191, 2023.
ista: Boissonnat J-D, Dyer R, Ghosh A, Wintraecken M. 2023. Local criteria for triangulating
general manifolds. Discrete & Computational Geometry. 69, 156–191.
mla: Boissonnat, Jean-Daniel, et al. “Local Criteria for Triangulating General Manifolds.”
Discrete & Computational Geometry, vol. 69, Springer Nature, 2023,
pp. 156–91, doi:10.1007/s00454-022-00431-7.
short: J.-D. Boissonnat, R. Dyer, A. Ghosh, M. Wintraecken, Discrete & Computational
Geometry 69 (2023) 156–191.
corr_author: '1'
date_created: 2023-01-16T10:04:06Z
date_published: 2023-01-01T00:00:00Z
date_updated: 2025-04-14T07:44:00Z
day: '01'
ddc:
- '510'
department:
- _id: HeEd
doi: 10.1007/s00454-022-00431-7
ec_funded: 1
external_id:
isi:
- '000862193600001'
file:
- access_level: open_access
checksum: 46352e0ee71e460848f88685ca852681
content_type: application/pdf
creator: dernst
date_created: 2023-02-02T11:01:10Z
date_updated: 2023-02-02T11:01:10Z
file_id: '12488'
file_name: 2023_DiscreteCompGeometry_Boissonnat.pdf
file_size: 582850
relation: main_file
success: 1
file_date_updated: 2023-02-02T11:01:10Z
fulldoi: https://doi.org/10.1007/s00454-022-00431-7
has_accepted_license: '1'
intvolume: ' 69'
isi: 1
keyword:
- Computational Theory and Mathematics
- Discrete Mathematics and Combinatorics
- Geometry and Topology
- Theoretical Computer Science
language:
- iso: eng
month: '01'
oa: 1
oa_version: Published Version
page: 156-191
project:
- _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: Discrete & 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: Local criteria for triangulating general manifolds
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: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 69
year: '2023'
...
---
_id: '12544'
abstract:
- lang: eng
text: Geometry is crucial in our efforts to comprehend the structures and dynamics
of biomolecules. For example, volume, surface area, and integrated mean and Gaussian
curvature of the union of balls representing a molecule are used to quantify its
interactions with the water surrounding it in the morphometric implicit solvent
models. The Alpha Shape theory provides an accurate and reliable method for computing
these geometric measures. In this paper, we derive homogeneous formulas for the
expressions of these measures and their derivatives with respect to the atomic
coordinates, and we provide algorithms that implement them into a new software
package, AlphaMol. The only variables in these formulas are the interatomic distances,
making them insensitive to translations and rotations. AlphaMol includes a sequential
algorithm and a parallel algorithm. In the parallel version, we partition the
atoms of the molecule of interest into 3D rectangular blocks, using a kd-tree
algorithm. We then apply the sequential algorithm of AlphaMol to each block, augmented
by a buffer zone to account for atoms whose ball representations may partially
cover the block. The current parallel version of AlphaMol leads to a 20-fold speed-up
compared to an independent serial implementation when using 32 processors. For
instance, it takes 31 s to compute the geometric measures and derivatives of each
atom in a viral capsid with more than 26 million atoms on 32 Intel processors
running at 2.7 GHz. The presence of the buffer zones, however, leads to redundant
computations, which ultimately limit the impact of using multiple processors.
AlphaMol is available as an OpenSource software.
acknowledgement: "P.K. acknowledges support from the University of California Multicampus
Research Programs and Initiatives (Grant No. M21PR3267) and from the NSF (Grant
No.1760485). H.E. acknowledges support from the European Research Council (ERC)
under the European Union’s Horizon 2020 research and innovation program, 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.\r\nOpen Access
is funded by the Austrian Science Fund (FWF)."
article_processing_charge: No
article_type: original
author:
- first_name: Patrice
full_name: Koehl, Patrice
last_name: Koehl
- 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
citation:
ama: Koehl P, Akopyan A, Edelsbrunner H. Computing the volume, surface area, mean,
and Gaussian curvatures of molecules and their derivatives. Journal of Chemical
Information and Modeling. 2023;63(3):973-985. doi:10.1021/acs.jcim.2c01346
apa: Koehl, P., Akopyan, A., & Edelsbrunner, H. (2023). Computing the volume,
surface area, mean, and Gaussian curvatures of molecules and their derivatives.
Journal of Chemical Information and Modeling. American Chemical Society.
https://doi.org/10.1021/acs.jcim.2c01346
chicago: Koehl, Patrice, Arseniy Akopyan, and Herbert Edelsbrunner. “Computing the
Volume, Surface Area, Mean, and Gaussian Curvatures of Molecules and Their Derivatives.”
Journal of Chemical Information and Modeling. American Chemical Society,
2023. https://doi.org/10.1021/acs.jcim.2c01346.
ieee: P. Koehl, A. Akopyan, and H. Edelsbrunner, “Computing the volume, surface
area, mean, and Gaussian curvatures of molecules and their derivatives,” Journal
of Chemical Information and Modeling, vol. 63, no. 3. American Chemical Society,
pp. 973–985, 2023.
ista: Koehl P, Akopyan A, Edelsbrunner H. 2023. Computing the volume, surface area,
mean, and Gaussian curvatures of molecules and their derivatives. Journal of Chemical
Information and Modeling. 63(3), 973–985.
mla: Koehl, Patrice, et al. “Computing the Volume, Surface Area, Mean, and Gaussian
Curvatures of Molecules and Their Derivatives.” Journal of Chemical Information
and Modeling, vol. 63, no. 3, American Chemical Society, 2023, pp. 973–85,
doi:10.1021/acs.jcim.2c01346.
short: P. Koehl, A. Akopyan, H. Edelsbrunner, Journal of Chemical Information and
Modeling 63 (2023) 973–985.
corr_author: '1'
date_created: 2023-02-12T23:00:59Z
date_published: 2023-02-13T00:00:00Z
date_updated: 2025-04-15T07:16:52Z
day: '13'
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doi: 10.1021/acs.jcim.2c01346
ec_funded: 1
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grant_number: Z00342
name: Mathematics, Computer Science
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call_identifier: FWF
grant_number: I02979-N35
name: Persistence and stability of geometric complexes
publication: Journal of Chemical Information and Modeling
publication_identifier:
eissn:
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issn:
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publication_status: published
publisher: American Chemical Society
quality_controlled: '1'
scopus_import: '1'
status: public
title: Computing the volume, surface area, mean, and Gaussian curvatures of molecules
and their derivatives
tmp:
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name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
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type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 63
year: '2023'
...
---
_id: '12709'
abstract:
- lang: eng
text: Given a finite set A ⊂ ℝ^d, let Cov_{r,k} denote the set of all points within
distance r to at least k points of A. Allowing r and k to vary, we obtain a 2-parameter
family of spaces that grow larger when r increases or k decreases, called the
multicover bifiltration. Motivated by the problem of computing the homology of
this bifiltration, we introduce two closely related combinatorial bifiltrations,
one polyhedral and the other simplicial, which are both topologically equivalent
to the multicover bifiltration and far smaller than a Čech-based model considered
in prior work of Sheehy. Our polyhedral construction is a bifiltration of the
rhomboid tiling of Edelsbrunner and Osang, and can be efficiently computed using
a variant of an algorithm given by these authors as well. Using an implementation
for dimension 2 and 3, we provide experimental results. Our simplicial construction
is useful for understanding the polyhedral construction and proving its correctness.
acknowledgement: We thank the anonymous reviewers for many helpful comments and suggestions,
which led to substantial improvements of the paper. The first two authors were supported
by the Austrian Science Fund (FWF) grant number P 29984-N35 and W1230. The first
author was partly supported by an Austrian Marshall Plan Scholarship, and by the
Brummer & Partners MathDataLab. A conference version of this paper was presented
at the 37th International Symposium on Computational Geometry (SoCG 2021). Open
access funding provided by the Royal Institute of Technology.
article_processing_charge: Yes (via OA deal)
article_type: original
arxiv: 1
author:
- first_name: René
full_name: Corbet, René
last_name: Corbet
- first_name: Michael
full_name: Kerber, Michael
id: 36E4574A-F248-11E8-B48F-1D18A9856A87
last_name: Kerber
orcid: 0000-0002-8030-9299
- first_name: Michael
full_name: Lesnick, Michael
last_name: Lesnick
- first_name: Georg F
full_name: Osang, Georg F
id: 464B40D6-F248-11E8-B48F-1D18A9856A87
last_name: Osang
orcid: 0000-0002-8882-5116
citation:
ama: Corbet R, Kerber M, Lesnick M, Osang GF. Computing the multicover bifiltration.
Discrete and Computational Geometry. 2023;70:376-405. doi:10.1007/s00454-022-00476-8
apa: Corbet, R., Kerber, M., Lesnick, M., & Osang, G. F. (2023). Computing the
multicover bifiltration. Discrete and Computational Geometry. Springer
Nature. https://doi.org/10.1007/s00454-022-00476-8
chicago: Corbet, René, Michael Kerber, Michael Lesnick, and Georg F Osang. “Computing
the Multicover Bifiltration.” Discrete and Computational Geometry. Springer
Nature, 2023. https://doi.org/10.1007/s00454-022-00476-8.
ieee: R. Corbet, M. Kerber, M. Lesnick, and G. F. Osang, “Computing the multicover
bifiltration,” Discrete and Computational Geometry, vol. 70. Springer Nature,
pp. 376–405, 2023.
ista: Corbet R, Kerber M, Lesnick M, Osang GF. 2023. Computing the multicover bifiltration.
Discrete and Computational Geometry. 70, 376–405.
mla: Corbet, René, et al. “Computing the Multicover Bifiltration.” Discrete and
Computational Geometry, vol. 70, Springer Nature, 2023, pp. 376–405, doi:10.1007/s00454-022-00476-8.
short: R. Corbet, M. Kerber, M. Lesnick, G.F. Osang, Discrete and Computational
Geometry 70 (2023) 376–405.
date_created: 2023-03-05T23:01:06Z
date_published: 2023-09-01T00:00:00Z
date_updated: 2025-07-10T12:01:57Z
day: '01'
ddc:
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department:
- _id: HeEd
doi: 10.1007/s00454-022-00476-8
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language:
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page: 376-405
pmid: 1
publication: Discrete and Computational Geometry
publication_identifier:
eissn:
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issn:
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publication_status: published
publisher: Springer Nature
quality_controlled: '1'
related_material:
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relation: earlier_version
status: public
scopus_import: '1'
status: public
title: Computing the multicover bifiltration
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: 70
year: '2023'
...
---
_id: '12763'
abstract:
- lang: eng
text: 'Kleinjohann (Archiv der Mathematik 35(1):574–582, 1980; Mathematische Zeitschrift
176(3), 327–344, 1981) and Bangert (Archiv der Mathematik 38(1):54–57, 1982) extended
the reach rch(S) from subsets S of Euclidean space to the reach rchM(S) of subsets
S of Riemannian manifolds M, where M is smooth (we’ll assume at least C3). Bangert
showed that sets of positive reach in Euclidean space and Riemannian manifolds
are very similar. In this paper we introduce a slight variant of Kleinjohann’s
and Bangert’s extension and quantify the similarity between sets of positive reach
in Euclidean space and Riemannian manifolds in a new way: Given p∈M and q∈S, we
bound the local feature size (a local version of the reach) of its lifting to
the tangent space via the inverse exponential map (exp−1p(S)) at q, assuming that
rchM(S) and the geodesic distance dM(p,q) are bounded. These bounds are motivated
by the importance of the reach and local feature size to manifold learning, topological
inference, and triangulating manifolds and the fact that intrinsic approaches
circumvent the curse of dimensionality.'
acknowledgement: "We thank Eddie Aamari, David Cohen-Steiner, Isa Costantini, Fred
Chazal, Ramsay Dyer, André Lieutier, and Alef Sterk for discussion and Pierre Pansu
for encouragement. We further acknowledge the anonymous reviewers whose comments
helped improve the exposition.\r\nThe research leading to these results has received
funding from the European Research Council (ERC) under the European Union’s Seventh
Framework Programme (FP/2007-2013) / ERC Grant Agreement No. 339025 GUDHI (Algorithmic
Foundations of Geometry Understanding in Higher Dimensions). The first author is
further supported by the French government, through the 3IA Côte d’Azur Investments
in the Future project managed by the National Research Agency (ANR) with the reference
number ANR-19-P3IA-0002. The second author is supported by the European Union’s
Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie
Grant Agreement No. 754411 and the Austrian science fund (FWF) M-3073."
article_processing_charge: No
article_type: original
author:
- first_name: Jean Daniel
full_name: Boissonnat, Jean Daniel
last_name: Boissonnat
- first_name: Mathijs
full_name: Wintraecken, Mathijs
id: 307CFBC8-F248-11E8-B48F-1D18A9856A87
last_name: Wintraecken
orcid: 0000-0002-7472-2220
citation:
ama: Boissonnat JD, Wintraecken M. The reach of subsets of manifolds. Journal
of Applied and Computational Topology. 2023;7:619-641. doi:10.1007/s41468-023-00116-x
apa: Boissonnat, J. D., & Wintraecken, M. (2023). The reach of subsets of manifolds.
Journal of Applied and Computational Topology. Springer Nature. https://doi.org/10.1007/s41468-023-00116-x
chicago: Boissonnat, Jean Daniel, and Mathijs Wintraecken. “The Reach of Subsets
of Manifolds.” Journal of Applied and Computational Topology. Springer
Nature, 2023. https://doi.org/10.1007/s41468-023-00116-x.
ieee: J. D. Boissonnat and M. Wintraecken, “The reach of subsets of manifolds,”
Journal of Applied and Computational Topology, vol. 7. Springer Nature,
pp. 619–641, 2023.
ista: Boissonnat JD, Wintraecken M. 2023. The reach of subsets of manifolds. Journal
of Applied and Computational Topology. 7, 619–641.
mla: Boissonnat, Jean Daniel, and Mathijs Wintraecken. “The Reach of Subsets of
Manifolds.” Journal of Applied and Computational Topology, vol. 7, Springer
Nature, 2023, pp. 619–41, doi:10.1007/s41468-023-00116-x.
short: J.D. Boissonnat, M. Wintraecken, Journal of Applied and Computational Topology
7 (2023) 619–641.
corr_author: '1'
date_created: 2023-03-26T22:01:08Z
date_published: 2023-09-01T00:00:00Z
date_updated: 2025-04-14T07:44:01Z
day: '01'
department:
- _id: HeEd
doi: 10.1007/s41468-023-00116-x
ec_funded: 1
fulldoi: https://doi.org/10.1007/s41468-023-00116-x
intvolume: ' 7'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://inserm.hal.science/INRIA-SACLAY/hal-04083524v1
month: '09'
oa: 1
oa_version: Submitted Version
page: 619-641
project:
- _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: Journal of Applied and Computational Topology
publication_identifier:
eissn:
- 2367-1734
issn:
- 2367-1726
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: The reach of subsets of manifolds
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 7
year: '2023'
...