---
OA_place: publisher
OA_type: hybrid
_id: '15380'
abstract:
- lang: eng
text: The depth of a cell in an arrangement of n (non-vertical) great-spheres in
Sd is the number of great-spheres that pass above the cell. We prove Euler-type
relations, which imply extensions of the classic Dehn–Sommerville relations for
convex polytopes to sublevel sets of the depth function, and we use the relations
to extend the expressions for the number of faces of neighborly polytopes to the
number of cells of levels in neighborly arrangements.
acknowledgement: "The authors thank Uli Wagner and Emo Welzl for comments on an earlier
version of this paper, and for pointing out related work in the prior literature.\r\nOpen
access funding provided by Institute of Science and Technology (IST Austria). This
project has received funding from the European Research Council (ERC) under the
European Union’s Horizon 2020 research and innovation programme, Grant No. 788183,
from the Wittgenstein Prize, Austrian Science Fund (FWF), Grant No. Z 342-N31, and
from the DFG Collaborative Research Center TRR 109, ‘Discretization in Geometry
and Dynamics’, Austrian Science Fund (FWF), Grant No. I 02979-N35."
article_processing_charge: Yes (via OA deal)
article_type: original
author:
- first_name: Ranita
full_name: Biswas, Ranita
id: 3C2B033E-F248-11E8-B48F-1D18A9856A87
last_name: Biswas
orcid: 0000-0002-5372-7890
- first_name: Sebastiano
full_name: Cultrera Di Montesano, Sebastiano
id: 34D2A09C-F248-11E8-B48F-1D18A9856A87
last_name: Cultrera Di Montesano
orcid: 0000-0001-6249-0832
- first_name: Herbert
full_name: Edelsbrunner, Herbert
id: 3FB178DA-F248-11E8-B48F-1D18A9856A87
last_name: Edelsbrunner
orcid: 0000-0002-9823-6833
- first_name: Morteza
full_name: Saghafian, Morteza
id: f86f7148-b140-11ec-9577-95435b8df824
last_name: Saghafian
citation:
ama: 'Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. Depth in arrangements:
Dehn–Sommerville–Euler relations with applications. Journal of Applied and
Computational Topology. 2024;8:557-578. doi:10.1007/s41468-024-00173-w'
apa: 'Biswas, R., Cultrera di Montesano, S., Edelsbrunner, H., & Saghafian,
M. (2024). Depth in arrangements: Dehn–Sommerville–Euler relations with applications.
Journal of Applied and Computational Topology. Springer Nature. https://doi.org/10.1007/s41468-024-00173-w'
chicago: 'Biswas, Ranita, Sebastiano Cultrera di Montesano, Herbert Edelsbrunner,
and Morteza Saghafian. “Depth in Arrangements: Dehn–Sommerville–Euler Relations
with Applications.” Journal of Applied and Computational Topology. Springer
Nature, 2024. https://doi.org/10.1007/s41468-024-00173-w.'
ieee: 'R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, and M. Saghafian, “Depth
in arrangements: Dehn–Sommerville–Euler relations with applications,” Journal
of Applied and Computational Topology, vol. 8. Springer Nature, pp. 557–578,
2024.'
ista: 'Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. 2024. Depth
in arrangements: Dehn–Sommerville–Euler relations with applications. Journal of
Applied and Computational Topology. 8, 557–578.'
mla: 'Biswas, Ranita, et al. “Depth in Arrangements: Dehn–Sommerville–Euler Relations
with Applications.” Journal of Applied and Computational Topology, vol.
8, Springer Nature, 2024, pp. 557–78, doi:10.1007/s41468-024-00173-w.'
short: R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, M. Saghafian, Journal
of Applied and Computational Topology 8 (2024) 557–578.
corr_author: '1'
date_created: 2024-05-12T22:01:03Z
date_published: 2024-09-01T00:00:00Z
date_updated: 2025-05-14T09:27:57Z
day: '01'
ddc:
- '510'
department:
- _id: HeEd
doi: 10.1007/s41468-024-00173-w
ec_funded: 1
external_id:
pmid:
- '39308789'
file:
- access_level: open_access
checksum: 0ee15c1493a6413cf356ab2f32c81a9e
content_type: application/pdf
creator: dernst
date_created: 2025-04-23T08:01:36Z
date_updated: 2025-04-23T08:01:36Z
file_id: '19612'
file_name: 2024_JourApplCompTopo_BiswasRa.pdf
file_size: 522831
relation: main_file
success: 1
file_date_updated: 2025-04-23T08:01:36Z
has_accepted_license: '1'
intvolume: ' 8'
language:
- iso: eng
month: '09'
oa: 1
oa_version: Published Version
page: 557-578
pmid: 1
project:
- _id: 266A2E9E-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '788183'
name: Alpha Shape Theory Extended
- _id: 268116B8-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: Z00342
name: Mathematics, Computer Science
- _id: 2561EBF4-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: I02979-N35
name: Persistence and stability of geometric complexes
publication: Journal of Applied and Computational Topology
publication_identifier:
eissn:
- 2367-1734
issn:
- 2367-1726
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
related_material:
record:
- id: '11658'
relation: earlier_version
status: public
scopus_import: '1'
status: public
title: 'Depth in arrangements: Dehn–Sommerville–Euler relations with applications'
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
short: CC BY (4.0)
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 8
year: '2024'
...
---
_id: '17144'
abstract:
- lang: eng
text: "We prove that the medial axis of closed sets is Hausdorff stable in the following
sense: Let \U0001D4AE ⊆ ℝ^d be a fixed closed set that contains a bounding sphere.
That is, the bounding sphere is part of the set \U0001D4AE. Consider the space
of C^{1,1} diffeomorphisms of ℝ^d to itself, which keep the bounding sphere invariant.
The map from this space of diffeomorphisms (endowed with a Banach norm) to the
space of closed subsets of ℝ^d (endowed with the Hausdorff distance), mapping
a diffeomorphism F to the closure of the medial axis of F(\U0001D4AE), is Lipschitz.
This extends a previous stability result of Chazal and Soufflet on the stability
of the medial axis of C² manifolds under C² ambient diffeomorphisms."
acknowledgement: "This research has been supported by the European Research Council
(ERC), grant No. 788183, by the Wittgenstein Prize, Austrian Science Fund (FWF),
grant No. Z 342-N31, and by the DFG Collaborative Research Center TRR 109, Austrian
Science Fund (FWF), grant No. I 02979-N35.\r\nSupported by the European Union’s
Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie
grant agreement No. 754411, the Austrian science fund (FWF) grant No. M-3073, and
the welcome package from IDEX of the Université Cô d'Azur.\r\nWe are greatly indebted
to Fred Chazal for sharing his insights. We further thank Erin Chambers, Christopher
Fillmore, and Elizabeth Stephenson for early discussions and all members of the
Edelsbrunner group (Institute of Science and Technology Austria) and the Datashape
team (Inria) for the atmosphere in which this research was conducted."
alternative_title:
- LIPIcs
article_number: '69'
article_processing_charge: No
arxiv: 1
author:
- first_name: Hana
full_name: Kourimska, Hana
id: D9B8E14C-3C26-11EA-98F5-1F833DDC885E
last_name: Kourimska
orcid: 0000-0001-7841-0091
- first_name: André
full_name: Lieutier, André
last_name: Lieutier
- first_name: Mathijs
full_name: Wintraecken, Mathijs
id: 307CFBC8-F248-11E8-B48F-1D18A9856A87
last_name: Wintraecken
orcid: 0000-0002-7472-2220
citation:
ama: 'Kourimska H, Lieutier A, Wintraecken M. The medial axis of any closed bounded
set Is Lipschitz stable with respect to the Hausdorff distance Under ambient diffeomorphisms.
In: 40th International Symposium on Computational Geometry. Vol 293. Schloss
Dagstuhl - Leibniz-Zentrum für Informatik; 2024. doi:10.4230/LIPIcs.SoCG.2024.69'
apa: 'Kourimska, H., Lieutier, A., & Wintraecken, M. (2024). The medial axis
of any closed bounded set Is Lipschitz stable with respect to the Hausdorff distance
Under ambient diffeomorphisms. In 40th International Symposium on Computational
Geometry (Vol. 293). Athens, Greece: Schloss Dagstuhl - Leibniz-Zentrum für
Informatik. https://doi.org/10.4230/LIPIcs.SoCG.2024.69'
chicago: Kourimska, Hana, André Lieutier, and Mathijs Wintraecken. “The Medial Axis
of Any Closed Bounded Set Is Lipschitz Stable with Respect to the Hausdorff Distance
Under Ambient Diffeomorphisms.” In 40th International Symposium on Computational
Geometry, Vol. 293. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024.
https://doi.org/10.4230/LIPIcs.SoCG.2024.69.
ieee: H. Kourimska, A. Lieutier, and M. Wintraecken, “The medial axis of any closed
bounded set Is Lipschitz stable with respect to the Hausdorff distance Under ambient
diffeomorphisms,” in 40th International Symposium on Computational Geometry,
Athens, Greece, 2024, vol. 293.
ista: 'Kourimska H, Lieutier A, Wintraecken M. 2024. The medial axis of any closed
bounded set Is Lipschitz stable with respect to the Hausdorff distance Under ambient
diffeomorphisms. 40th International Symposium on Computational Geometry. SoCG:
Symposium on Computational Geometry, LIPIcs, vol. 293, 69.'
mla: Kourimska, Hana, et al. “The Medial Axis of Any Closed Bounded Set Is Lipschitz
Stable with Respect to the Hausdorff Distance Under Ambient Diffeomorphisms.”
40th International Symposium on Computational Geometry, vol. 293, 69, Schloss
Dagstuhl - Leibniz-Zentrum für Informatik, 2024, doi:10.4230/LIPIcs.SoCG.2024.69.
short: H. Kourimska, A. Lieutier, M. Wintraecken, in:, 40th International Symposium
on Computational Geometry, Schloss Dagstuhl - Leibniz-Zentrum für Informatik,
2024.
conference:
end_date: 2024-06-14
location: Athens, Greece
name: 'SoCG: Symposium on Computational Geometry'
date_created: 2024-06-16T22:01:06Z
date_published: 2024-06-01T00:00:00Z
date_updated: 2025-04-15T07:16:58Z
day: '01'
ddc:
- '510'
department:
- _id: HeEd
doi: 10.4230/LIPIcs.SoCG.2024.69
ec_funded: 1
external_id:
arxiv:
- '2212.01118'
file:
- access_level: open_access
checksum: b40ff456c19294adb5d9613fcfd751c6
content_type: application/pdf
creator: dernst
date_created: 2024-06-17T08:33:40Z
date_updated: 2024-06-17T08:33:40Z
file_id: '17150'
file_name: 2024_LIPICS_Kourimska.pdf
file_size: 1612558
relation: main_file
success: 1
file_date_updated: 2024-06-17T08:33:40Z
has_accepted_license: '1'
intvolume: ' 293'
language:
- iso: eng
month: '06'
oa: 1
oa_version: Published Version
project:
- _id: 266A2E9E-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '788183'
name: Alpha Shape Theory Extended
- _id: 268116B8-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: Z00342
name: Mathematics, Computer Science
- _id: 2561EBF4-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: I02979-N35
name: Persistence and stability of geometric complexes
- _id: 260C2330-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '754411'
name: ISTplus - Postdoctoral Fellowships
- _id: fc390959-9c52-11eb-aca3-afa58bd282b2
grant_number: M03073
name: Learning and triangulating manifolds via collapses
publication: 40th International Symposium on Computational Geometry
publication_identifier:
isbn:
- '9783959773164'
issn:
- 1868-8969
publication_status: published
publisher: Schloss Dagstuhl - Leibniz-Zentrum für Informatik
quality_controlled: '1'
scopus_import: '1'
status: public
title: The medial axis of any closed bounded set Is Lipschitz stable with respect
to the Hausdorff distance Under ambient diffeomorphisms
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
short: CC BY (4.0)
type: conference
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 293
year: '2024'
...
---
_id: '17145'
abstract:
- lang: eng
text: Grid peeling is the process of repeatedly removing the convex hull vertices
of the grid points that lie inside a given convex curve. It has been conjectured
that, for a more and more refined grid, grid peeling converges to a continuous
process, the affine curve-shortening flow, which deforms the curve based on the
curvature. We prove this conjecture for one class of curves, parabolas with a
vertical axis, and we determine the value of the constant factor in the formula
that relates the two processes.
acknowledgement: Part of this work was done while G.R. enjoyed the hospitality of
the Institute of Science and Technology Austria (ISTA) as a visiting professor during
his sabbatical in the winter semester 2022/23.
alternative_title:
- LIPIcs
article_number: '76'
article_processing_charge: No
arxiv: 1
author:
- first_name: Günter
full_name: Rote, Günter
last_name: Rote
- first_name: Moritz
full_name: Rüber, Moritz
last_name: Rüber
- first_name: Morteza
full_name: Saghafian, Morteza
id: f86f7148-b140-11ec-9577-95435b8df824
last_name: Saghafian
citation:
ama: 'Rote G, Rüber M, Saghafian M. Grid peeling of parabolas. In: 40th International
Symposium on Computational Geometry. Vol 293. Schloss Dagstuhl - Leibniz-Zentrum
für Informatik; 2024. doi:10.4230/LIPIcs.SoCG.2024.76'
apa: 'Rote, G., Rüber, M., & Saghafian, M. (2024). Grid peeling of parabolas.
In 40th International Symposium on Computational Geometry (Vol. 293). Athens,
Greece: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. https://doi.org/10.4230/LIPIcs.SoCG.2024.76'
chicago: Rote, Günter, Moritz Rüber, and Morteza Saghafian. “Grid Peeling of Parabolas.”
In 40th International Symposium on Computational Geometry, Vol. 293. Schloss
Dagstuhl - Leibniz-Zentrum für Informatik, 2024. https://doi.org/10.4230/LIPIcs.SoCG.2024.76.
ieee: G. Rote, M. Rüber, and M. Saghafian, “Grid peeling of parabolas,” in 40th
International Symposium on Computational Geometry, Athens, Greece, 2024, vol.
293.
ista: 'Rote G, Rüber M, Saghafian M. 2024. Grid peeling of parabolas. 40th International
Symposium on Computational Geometry. SoCG: Symposium on Computational Geometry,
LIPIcs, vol. 293, 76.'
mla: Rote, Günter, et al. “Grid Peeling of Parabolas.” 40th International Symposium
on Computational Geometry, vol. 293, 76, Schloss Dagstuhl - Leibniz-Zentrum
für Informatik, 2024, doi:10.4230/LIPIcs.SoCG.2024.76.
short: G. Rote, M. Rüber, M. Saghafian, in:, 40th International Symposium on Computational
Geometry, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024.
conference:
end_date: 2024-06-14
location: Athens, Greece
name: 'SoCG: Symposium on Computational Geometry'
start_date: 2024-06-11
date_created: 2024-06-16T22:01:06Z
date_published: 2024-06-01T00:00:00Z
date_updated: 2024-06-17T08:41:56Z
day: '01'
ddc:
- '510'
department:
- _id: HeEd
doi: 10.4230/LIPIcs.SoCG.2024.76
external_id:
arxiv:
- '2402.15787'
file:
- 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
has_accepted_license: '1'
intvolume: ' 293'
language:
- iso: eng
month: '06'
oa: 1
oa_version: Published Version
publication: 40th International Symposium on Computational Geometry
publication_identifier:
isbn:
- '9783959773164'
issn:
- 1868-8969
publication_status: published
publisher: Schloss Dagstuhl - Leibniz-Zentrum für Informatik
quality_controlled: '1'
scopus_import: '1'
status: public
title: Grid peeling of parabolas
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
short: CC BY (4.0)
type: conference
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 293
year: '2024'
...
---
_id: '17146'
abstract:
- lang: eng
text: The Upper Bound Theorem for convex polytopes implies that the p-th Betti number
of the Čech complex of any set of N points in ℝ^d and any radius satisfies β_p
= O(N^m), with m = min{p+1, ⌈d/2⌉}. We construct sets in even and odd dimensions,
which prove that this upper bound is asymptotically tight. For example, we describe
a set of N = 2(n+1) points in ℝ³ and two radii such that the first Betti number
of the Čech complex at one radius is (n+1)² - 1, and the second Betti number of
the Čech complex at the other radius is n². In particular, there is an arrangement
of n contruent balls in ℝ³ that enclose a quadratic number of voids, which answers
a long-standing open question in computational geometry.
acknowledgement: "The first author is supported by the European Research Council (ERC),
grant no. 788183, and by the DFG Collaborative Research Center TRR 109, Austrian
Science Fund (FWF), grant no. {I 02979-N35.} The second author is supported by the
European Research Council (ERC), grant \"GeoScape\" and by the Hungarian Science
Foundation (NKFIH), grant K-131529. Both authors are supported by the Wittgenstein
Prize, Austrian Science Fund (FWF), grant no. Z 342-N31.\r\nThe authors thank Matt
Kahle for communicating the question about extremal Čech complexes, Ben Schweinhart
for early discussions on the linked circles construction in three dimensions, and
Gábor Tardos for helpful remarks and suggestions."
alternative_title:
- LIPIcs
article_number: '53'
article_processing_charge: No
arxiv: 1
author:
- first_name: Herbert
full_name: Edelsbrunner, Herbert
id: 3FB178DA-F248-11E8-B48F-1D18A9856A87
last_name: Edelsbrunner
orcid: 0000-0002-9823-6833
- first_name: János
full_name: Pach, János
id: E62E3130-B088-11EA-B919-BF823C25FEA4
last_name: Pach
citation:
ama: 'Edelsbrunner H, Pach J. Maximum Betti numbers of Čech complexes. In: 40th
International Symposium on Computational Geometry. Vol 293. Schloss Dagstuhl
- Leibniz-Zentrum für Informatik; 2024. doi:10.4230/LIPIcs.SoCG.2024.53'
apa: 'Edelsbrunner, H., & Pach, J. (2024). Maximum Betti numbers of Čech complexes.
In 40th International Symposium on Computational Geometry (Vol. 293). Athens,
Greece: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. https://doi.org/10.4230/LIPIcs.SoCG.2024.53'
chicago: Edelsbrunner, Herbert, and János Pach. “Maximum Betti Numbers of Čech Complexes.”
In 40th International Symposium on Computational Geometry, Vol. 293. Schloss
Dagstuhl - Leibniz-Zentrum für Informatik, 2024. https://doi.org/10.4230/LIPIcs.SoCG.2024.53.
ieee: H. Edelsbrunner and J. Pach, “Maximum Betti numbers of Čech complexes,” in
40th International Symposium on Computational Geometry, Athens, Greece,
2024, vol. 293.
ista: 'Edelsbrunner H, Pach J. 2024. Maximum Betti numbers of Čech complexes. 40th
International Symposium on Computational Geometry. SoCG: Symposium on Computational
Geometry, LIPIcs, vol. 293, 53.'
mla: Edelsbrunner, Herbert, and János Pach. “Maximum Betti Numbers of Čech Complexes.”
40th International Symposium on Computational Geometry, vol. 293, 53, Schloss
Dagstuhl - Leibniz-Zentrum für Informatik, 2024, doi:10.4230/LIPIcs.SoCG.2024.53.
short: H. Edelsbrunner, J. Pach, in:, 40th International Symposium on Computational
Geometry, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024.
conference:
end_date: 2024-06-14
location: Athens, Greece
name: 'SoCG: Symposium on Computational Geometry'
start_date: 2024-06-11
date_created: 2024-06-16T22:01:06Z
date_published: 2024-06-01T00:00:00Z
date_updated: 2025-12-01T15:19:20Z
day: '01'
ddc:
- '510'
department:
- _id: HeEd
doi: 10.4230/LIPIcs.SoCG.2024.53
ec_funded: 1
external_id:
arxiv:
- '2310.14801'
file:
- access_level: open_access
checksum: 5442d44fb89d77477a87668d6e61aac9
content_type: application/pdf
creator: dernst
date_created: 2024-06-17T08:46:33Z
date_updated: 2024-06-17T08:46:33Z
file_id: '17152'
file_name: 2024_LIPICS_Edelsbrunner.pdf
file_size: 766562
relation: main_file
success: 1
file_date_updated: 2024-06-17T08:46:33Z
has_accepted_license: '1'
intvolume: ' 293'
language:
- iso: eng
month: '06'
oa: 1
oa_version: Published Version
project:
- _id: 266A2E9E-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '788183'
name: Alpha Shape Theory Extended
- _id: 2561EBF4-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: I02979-N35
name: Persistence and stability of geometric complexes
- _id: 268116B8-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: Z00342
name: Mathematics, Computer Science
publication: 40th International Symposium on Computational Geometry
publication_identifier:
isbn:
- '9783959773164'
issn:
- 1868-8969
publication_status: published
publisher: Schloss Dagstuhl - Leibniz-Zentrum für Informatik
quality_controlled: '1'
related_material:
record:
- id: '20657'
relation: later_version
status: public
scopus_import: '1'
status: public
title: Maximum Betti numbers of Čech complexes
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
short: CC BY (4.0)
type: conference
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 293
year: '2024'
...
---
_id: '17170'
abstract:
- lang: eng
text: "In this article we extend and strengthen the seminal work by Niyogi, Smale,
and Weinberger on the learning of the homotopy type from a sample of an underlying
space. In their work, Niyogi, Smale, and Weinberger studied samples of C² manifolds
with positive reach embedded in ℝ^d. We extend their results in the following
ways: - As the ambient space we consider both ℝ^d and Riemannian manifolds with
lower bounded sectional curvature. - In both types of ambient spaces, we study
sets of positive reach - a significantly more general setting than C² manifolds
- as well as general manifolds of positive reach. - The sample P of a set (or
a manifold) \U0001D4AE of positive reach may be noisy. We work with two one-sided
Hausdorff distances - ε and δ - between P and \U0001D4AE. We provide tight bounds
in terms of ε and δ, that guarantee that there exists a parameter r such that
the union of balls of radius r centred at the sample P deformation-retracts to
\U0001D4AE. We exhibit their tightness by an explicit construction. We carefully
distinguish the roles of δ and ε. This is not only essential to achieve tight
bounds, but also sensible in practical situations, since it allows one to adapt
the bound according to sample density and the amount of noise present in the sample
separately."
acknowledgement: "This research has been supported by the European Research Council
(ERC), grant No. 788183, by the Wittgenstein Prize, Austrian Science Fund (FWF),
grant No. Z 342-N31, and by the DFG Collaborative Research Center TRR 109, Austrian
Science Fund (FWF), grant No. I 02979-N35.\r\nWintraecken, Mathijs: 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."
alternative_title:
- LIPIcs
article_processing_charge: No
arxiv: 1
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: André
full_name: Lieutier, André
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. Tight bounds for the learning of
homotopy à la Niyogi, Smale, and Weinberger for subsets of euclidean spaces and
of Riemannian manifolds. In: 40th International Symposium on Computational
Geometry. Vol 293. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2024:11:1-11:19.
doi:10.4230/LIPIcs.SoCG.2024.11'
apa: 'Attali, D., Kourimska, H., Fillmore, C. D., Ghosh, I., Lieutier, A., Stephenson,
E. R., & Wintraecken, M. (2024). Tight bounds for the learning of homotopy
à la Niyogi, Smale, and Weinberger for subsets of euclidean spaces and of Riemannian
manifolds. In 40th International Symposium on Computational Geometry (Vol.
293, p. 11:1-11:19). Athens, Greece: Schloss Dagstuhl - Leibniz-Zentrum für Informatik.
https://doi.org/10.4230/LIPIcs.SoCG.2024.11'
chicago: Attali, Dominique, Hana Kourimska, Christopher D Fillmore, Ishika Ghosh,
André Lieutier, Elizabeth R Stephenson, and Mathijs Wintraecken. “Tight Bounds
for the Learning of Homotopy à La Niyogi, Smale, and Weinberger for Subsets of
Euclidean Spaces and of Riemannian Manifolds.” In 40th International Symposium
on Computational Geometry, 293:11:1-11:19. Schloss Dagstuhl - Leibniz-Zentrum
für Informatik, 2024. https://doi.org/10.4230/LIPIcs.SoCG.2024.11.
ieee: D. Attali et al., “Tight bounds for the learning of homotopy à la Niyogi,
Smale, and Weinberger for subsets of euclidean spaces and of Riemannian manifolds,”
in 40th International Symposium on Computational Geometry, Athens, Greece,
2024, vol. 293, p. 11:1-11:19.
ista: 'Attali D, Kourimska H, Fillmore CD, Ghosh I, Lieutier A, Stephenson ER, Wintraecken
M. 2024. Tight bounds for the learning of homotopy à la Niyogi, Smale, and Weinberger
for subsets of euclidean spaces and of Riemannian manifolds. 40th International
Symposium on Computational Geometry. SoCG: Symposium on Computational Geometry,
LIPIcs, vol. 293, 11:1-11:19.'
mla: Attali, Dominique, et al. “Tight Bounds for the Learning of Homotopy à La Niyogi,
Smale, and Weinberger for Subsets of Euclidean Spaces and of Riemannian Manifolds.”
40th International Symposium on Computational Geometry, vol. 293, Schloss
Dagstuhl - Leibniz-Zentrum für Informatik, 2024, p. 11:1-11:19, doi:10.4230/LIPIcs.SoCG.2024.11.
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, p. 11:1-11:19.
conference:
end_date: 2024-06-14
location: Athens, Greece
name: 'SoCG: Symposium on Computational Geometry'
start_date: 2024-06-11
date_created: 2024-06-25T11:45:58Z
date_published: 2024-06-06T00:00:00Z
date_updated: 2025-04-15T07:16:57Z
day: '06'
ddc:
- '516'
department:
- _id: GradSch
- _id: HeEd
doi: 10.4230/LIPIcs.SoCG.2024.11
ec_funded: 1
external_id:
arxiv:
- '2206.10485'
file:
- access_level: open_access
checksum: 6a2ddc8b51aa58f197a8b294750f1f8d
content_type: application/pdf
creator: cfillmor
date_created: 2024-06-25T11:47:26Z
date_updated: 2024-06-25T11:47:26Z
file_id: '17171'
file_name: LIPIcs.SoCG.2024.11.pdf
file_size: 20886142
relation: main_file
success: 1
file_date_updated: 2024-06-25T11:47:26Z
has_accepted_license: '1'
intvolume: ' 293'
language:
- iso: eng
month: '06'
oa: 1
oa_version: Published Version
page: 11:1-11:19
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: 260C2330-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '754411'
name: ISTplus - Postdoctoral Fellowships
- _id: 2561EBF4-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: I02979-N35
name: Persistence and stability of geometric complexes
- _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:
eissn:
- 1868-8969
isbn:
- '9783959773164'
publication_status: published
publisher: Schloss Dagstuhl - Leibniz-Zentrum für Informatik
quality_controlled: '1'
scopus_import: '1'
status: public
title: Tight bounds for the learning of homotopy à la Niyogi, Smale, and Weinberger
for subsets of euclidean spaces and of Riemannian 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: conference
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 293
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'
date_created: 2024-06-30T22:01:05Z
date_published: 2024-06-07T00:00:00Z
date_updated: 2025-09-08T08:06:04Z
day: '07'
department:
- _id: HeEd
doi: 10.1137/22M1489071
ec_funded: 1
external_id:
arxiv:
- '2204.01077'
isi:
- '001292728600001'
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'
scopus_import: '1'
status: public
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: 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'
date_created: 2025-01-31T17:03:04Z
date_published: 2024-06-09T00:00:00Z
date_updated: 2026-04-07T11:47:29Z
day: '09'
department:
- _id: HeEd
doi: 10.48550/arXiv.2209.14993
ec_funded: 1
external_id:
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
status: public
title: Discrete microlocal Morse theory
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
short: CC BY (4.0)
type: preprint
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2024'
...
---
OA_place: publisher
_id: '18667'
abstract:
- lang: eng
text: "Many chemical and physical properties of materials are determined by the
material’s shape,\r\nfor example the size of its pores and the width of its tunnels.
This makes materials science\r\na prime application area for geometrical and topological
methods. Nevertheless many\r\nmethods in topological data analysis have not been
satisfyingly extended to the needs of\r\nmaterials science. This thesis provides
new methods and new mathematical theorems\r\ntargeted at those specific needs
by answering four different research questions. While the\r\nmotivation for each
of the research questions arises from materials science, the methods\r\nare versatile
and can be applied in different areas as well. \r\n\r\nThe first research question
is concerned with image data, for example a three-dimensional\r\ncomputed tomography
(CT) scan of a material, like sand or stone. There are two commonly\r\nused topologies
for digital images and depending on the application either of them might be\r\nrequired.
However, software for computing the topological data analysis method persistence\r\nhomology,
usually supports only one of the two topologies. We answer the question how to\r\ncompute
persistent homology of an image with respect to one of the two topologies using\r\nsoftware
that is intended for the other topology. \r\n\r\nThe second research question
is concerned with image data as well, and asks how much\r\nof the topological
information of an image is lost when the resolution is coarsened. As\r\ncomputer
tomography scanners are more expensive the higher the resolution, it is an\r\nimportant
question in materials science to know which resolution is enough to get satisfying\r\npersistent
homology. We give theoretical bounds on the information loss based on different\r\ngeometrical
properties of the object to be scanned. In addition, we conduct experiments on\r\nsand
and stone CT image data. \r\n\r\nThe third research question is motivated by comparing
crystalline materials efficiently. As\r\nthe atoms within a crystal repeat periodically,
crystalline materials are either modeled by\r\nunmanageable infinite periodic
point sets, or by one of their fundamental domains, which is\r\nunstable under
perturbation. Therefore a fingerprint of crystalline materials is needed, with\r\nappropriate
properties such that comparing the crystals can be eased by comparing the\r\nfingerprints
instead. We define the density fingerprint and prove the necessary properties.
\r\n\r\nThe fourth research question is motivated by studying the hole-structure
or connectedness,\r\ni.e. persistent homology or merge trees, of crystalline materials.
A common way to deal\r\nwith periodicity is to take a fundamental domain and identify
opposite boundaries to form a\r\ntorus. However, computing persistent homology
or merge trees on that torus loses some\r\nof the information materials scientists
are interested in and is additionally not stable under\r\ncertain noise. We therefore
decorate the merge tree stemming from the torus with additional\r\ninformation
describing the density and growth rate of the periodic copies of a component\r\nwithin
a growing spherical window. We prove all desired properties, like stability and
efficient\r\ncomputability."
acknowledgement: "I was supported by the European Research Council (ERC) Horizon 2020
project\r\n“Alpha Shape Theory Extended” No. 788183 and by the Pöttinger Scholarship.
In addition,\r\nI am very thankful for having been able to attend the second Workshop
for Women in\r\nComputational Topology in July 2019, funded by the Mathematical
Sciences Institute at\r\nANU, the US National Science Foundation through the award
CCF-1841455, the Australian\r\nMathematical Sciences Institute and the Association
for Women in Mathematics. Two of the\r\nprojects presented in this thesis started
there. One of them reached completion thanks to\r\nfunding from the MSRI Summer
Research in Mathematics program awarded to me and my\r\ncollaborators in 2020."
alternative_title:
- ISTA Thesis
article_processing_charge: No
author:
- first_name: Teresa
full_name: Heiss, Teresa
id: 4879BB4E-F248-11E8-B48F-1D18A9856A87
last_name: Heiss
orcid: 0000-0002-1780-2689
citation:
ama: Heiss T. New methods for applying topological data analysis to materials science.
2024. doi:10.15479/at:ista:18667
apa: Heiss, T. (2024). New methods for applying topological data analysis to
materials science. Institute of Science and Technology Austria. https://doi.org/10.15479/at:ista:18667
chicago: Heiss, Teresa. “New Methods for Applying Topological Data Analysis to Materials
Science.” Institute of Science and Technology Austria, 2024. https://doi.org/10.15479/at:ista:18667.
ieee: T. Heiss, “New methods for applying topological data analysis to materials
science,” Institute of Science and Technology Austria, 2024.
ista: Heiss T. 2024. New methods for applying topological data analysis to materials
science. Institute of Science and Technology Austria.
mla: Heiss, Teresa. New Methods for Applying Topological Data Analysis to Materials
Science. Institute of Science and Technology Austria, 2024, doi:10.15479/at:ista:18667.
short: T. Heiss, New Methods for Applying Topological Data Analysis to Materials
Science, Institute of Science and Technology Austria, 2024.
corr_author: '1'
date_created: 2024-12-17T16:17:55Z
date_published: 2024-12-17T00:00:00Z
date_updated: 2026-04-07T12:54:10Z
day: '17'
ddc:
- '514'
- '516'
- '004'
degree_awarded: PhD
department:
- _id: GradSch
- _id: HeEd
doi: 10.15479/at:ista:18667
ec_funded: 1
file:
- access_level: open_access
checksum: 247bb057aed2fba1cd4711917aaa2d77
content_type: application/pdf
creator: theiss
date_created: 2024-12-19T10:24:46Z
date_updated: 2024-12-19T10:24:46Z
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checksum: 9648b45c07a008ee11a07f99856a139d
content_type: application/zip
creator: theiss
date_created: 2024-12-19T10:24:50Z
date_updated: 2024-12-19T10:24:50Z
file_id: '18687'
file_name: PhD_Thesis.zip
file_size: 17197731
relation: source_file
file_date_updated: 2024-12-19T10:24:50Z
has_accepted_license: '1'
keyword:
- persistent homology
- topological data analysis
- periodic
- crystalline materials
- images
- fingerprint
language:
- iso: eng
month: '12'
oa: 1
oa_version: Published Version
page: '111'
project:
- _id: 266A2E9E-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '788183'
name: Alpha Shape Theory Extended
publication_identifier:
isbn:
- 978-3-99078-052-7
issn:
- 2663-337X
publication_status: published
publisher: Institute of Science and Technology Austria
related_material:
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- id: '18673'
relation: part_of_dissertation
status: public
- id: '9345'
relation: part_of_dissertation
status: public
status: public
supervisor:
- first_name: Herbert
full_name: Edelsbrunner, Herbert
id: 3FB178DA-F248-11E8-B48F-1D18A9856A87
last_name: Edelsbrunner
orcid: 0000-0002-9823-6833
title: New methods for applying topological data analysis to materials science
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
short: CC BY (4.0)
type: dissertation
user_id: ba8df636-2132-11f1-aed0-ed93e2281fdd
year: '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'
date_created: 2024-12-18T14:06:57Z
date_published: 2024-08-29T00:00:00Z
date_updated: 2026-04-07T12:54:09Z
day: '29'
department:
- _id: HeEd
doi: 10.48550/arXiv.2408.16575
ec_funded: 1
external_id:
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:
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- id: '18667'
relation: dissertation_contains
status: public
status: public
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: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2024'
...
---
OA_place: publisher
_id: '15094'
abstract:
- lang: eng
text: "Point sets, geometric networks, and arrangements of hyperplanes are fundamental
objects in\r\ndiscrete geometry that have captivated mathematicians for centuries,
if not millennia. This\r\nthesis seeks to cast new light on these structures by
illustrating specific instances where a\r\ntopological perspective, specifically
through discrete Morse theory and persistent homology,\r\nprovides valuable insights.\r\n\r\nAt
first glance, the topology of these geometric objects might seem uneventful: point
sets\r\nessentially lack of topology, arrangements of hyperplanes are a decomposition
of Rd, which\r\nis a contractible space, and the topology of a network primarily
involves the enumeration\r\nof connected components and cycles within the network.
However, beneath this apparent\r\nsimplicity, there lies an array of intriguing
structures, a small subset of which will be uncovered\r\nin this thesis.\r\n\r\nFocused
on three case studies, each addressing one of the mentioned objects, this work\r\nwill
showcase connections that intertwine topology with diverse fields such as combinatorial\r\ngeometry,
algorithms and data structures, and emerging applications like spatial biology.\r\n\r\n"
alternative_title:
- ISTA Thesis
article_processing_charge: No
author:
- first_name: Sebastiano
full_name: Cultrera di Montesano, Sebastiano
id: 34D2A09C-F248-11E8-B48F-1D18A9856A87
last_name: Cultrera di Montesano
orcid: 0000-0001-6249-0832
citation:
ama: Cultrera di Montesano S. Persistence and Morse theory for discrete geometric
structures. 2024. doi:10.15479/at:ista:15094
apa: Cultrera di Montesano, S. (2024). Persistence and Morse theory for discrete
geometric structures. Institute of Science and Technology Austria. https://doi.org/10.15479/at:ista:15094
chicago: Cultrera di Montesano, Sebastiano. “Persistence and Morse Theory for Discrete
Geometric Structures.” Institute of Science and Technology Austria, 2024. https://doi.org/10.15479/at:ista:15094.
ieee: S. Cultrera di Montesano, “Persistence and Morse theory for discrete geometric
structures,” Institute of Science and Technology Austria, 2024.
ista: Cultrera di Montesano S. 2024. Persistence and Morse theory for discrete geometric
structures. Institute of Science and Technology Austria.
mla: Cultrera di Montesano, Sebastiano. Persistence and Morse Theory for Discrete
Geometric Structures. Institute of Science and Technology Austria, 2024, doi:10.15479/at:ista:15094.
short: S. Cultrera di Montesano, Persistence and Morse Theory for Discrete Geometric
Structures, Institute of Science and Technology Austria, 2024.
corr_author: '1'
date_created: 2024-03-08T15:28:10Z
date_published: 2024-03-08T00:00:00Z
date_updated: 2026-04-07T12:58:48Z
day: '08'
ddc:
- '514'
- '500'
- '516'
degree_awarded: PhD
department:
- _id: GradSch
- _id: HeEd
doi: 10.15479/at:ista:15094
ec_funded: 1
file:
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creator: scultrer
date_created: 2024-03-14T08:55:07Z
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file_id: '15112'
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language:
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oa: 1
oa_version: Published Version
page: '108'
project:
- _id: 266A2E9E-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '788183'
name: Alpha Shape Theory Extended
- _id: 268116B8-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: Z00342
name: Mathematics, Computer Science
- _id: 0aa4bc98-070f-11eb-9043-e6fff9c6a316
grant_number: I4887
name: Persistent Homology, Algorithms and Stochastic Geometry
- _id: 2561EBF4-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: I02979-N35
name: Persistence and stability of geometric complexes
publication_identifier:
issn:
- 2663-337X
publication_status: published
publisher: Institute of Science and Technology Austria
related_material:
record:
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relation: part_of_dissertation
status: public
- id: '11660'
relation: part_of_dissertation
status: public
- id: '15090'
relation: part_of_dissertation
status: public
- id: '15093'
relation: part_of_dissertation
status: public
- id: '13182'
relation: part_of_dissertation
status: public
- id: '11658'
relation: part_of_dissertation
status: public
status: public
supervisor:
- first_name: Herbert
full_name: Edelsbrunner, Herbert
id: 3FB178DA-F248-11E8-B48F-1D18A9856A87
last_name: Edelsbrunner
orcid: 0000-0002-9823-6833
title: Persistence and Morse theory for discrete geometric structures
tmp:
image: /images/cc_by_nc_sa.png
legal_code_url: https://creativecommons.org/licenses/by-nc-sa/4.0/legalcode
name: Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International (CC
BY-NC-SA 4.0)
short: CC BY-NC-SA (4.0)
type: dissertation
user_id: ba8df636-2132-11f1-aed0-ed93e2281fdd
year: '2024'
...
---
OA_place: publisher
OA_type: hybrid
_id: '13182'
abstract:
- lang: eng
text: "We characterize critical points of 1-dimensional maps paired in persistent
homology\r\ngeometrically and this way get elementary proofs of theorems about
the symmetry\r\nof persistence diagrams and the variation of such maps. In particular,
we identify\r\nbranching points and endpoints of networks as the sole source of
asymmetry and\r\nrelate the cycle basis in persistent homology with a version
of the stable marriage\r\nproblem. Our analysis provides the foundations of fast
algorithms for maintaining a\r\ncollection of sorted lists together with its persistence
diagram."
acknowledgement: Open access funding provided by Austrian Science Fund (FWF). This
project has received funding from the European Research Council (ERC) under the
European Union’s Horizon 2020 research and innovation programme, grant no. 788183,
from the Wittgenstein Prize, Austrian Science Fund (FWF), Grant No. Z 342-N31, and
from the DFG Collaborative Research Center TRR 109, ‘Discretization in Geometry
and Dynamics’, Austrian Science Fund (FWF), Grant No. I 02979-N35. The authors of
this paper thank anonymous reviewers for their constructive criticism and Monika
Henzinger for detailed comments on an earlier version of this paper.
article_processing_charge: Yes (via OA deal)
article_type: original
author:
- first_name: Ranita
full_name: Biswas, Ranita
id: 3C2B033E-F248-11E8-B48F-1D18A9856A87
last_name: Biswas
orcid: 0000-0002-5372-7890
- first_name: Sebastiano
full_name: Cultrera Di Montesano, Sebastiano
id: 34D2A09C-F248-11E8-B48F-1D18A9856A87
last_name: Cultrera Di Montesano
orcid: 0000-0001-6249-0832
- first_name: Herbert
full_name: Edelsbrunner, Herbert
id: 3FB178DA-F248-11E8-B48F-1D18A9856A87
last_name: Edelsbrunner
orcid: 0000-0002-9823-6833
- first_name: Morteza
full_name: Saghafian, Morteza
id: f86f7148-b140-11ec-9577-95435b8df824
last_name: Saghafian
citation:
ama: Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. Geometric characterization
of the persistence of 1D maps. Journal of Applied and Computational Topology.
2024;8:1101-1119. doi:10.1007/s41468-023-00126-9
apa: Biswas, R., Cultrera di Montesano, S., Edelsbrunner, H., & Saghafian, M.
(2024). Geometric characterization of the persistence of 1D maps. Journal of
Applied and Computational Topology. Springer Nature. https://doi.org/10.1007/s41468-023-00126-9
chicago: Biswas, Ranita, Sebastiano Cultrera di Montesano, Herbert Edelsbrunner,
and Morteza Saghafian. “Geometric Characterization of the Persistence of 1D Maps.”
Journal of Applied and Computational Topology. Springer Nature, 2024. https://doi.org/10.1007/s41468-023-00126-9.
ieee: R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, and M. Saghafian, “Geometric
characterization of the persistence of 1D maps,” Journal of Applied and Computational
Topology, vol. 8. Springer Nature, pp. 1101–1119, 2024.
ista: Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. 2024. Geometric
characterization of the persistence of 1D maps. Journal of Applied and Computational
Topology. 8, 1101–1119.
mla: Biswas, Ranita, et al. “Geometric Characterization of the Persistence of 1D
Maps.” Journal of Applied and Computational Topology, vol. 8, Springer
Nature, 2024, pp. 1101–19, doi:10.1007/s41468-023-00126-9.
short: R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, M. Saghafian, Journal
of Applied and Computational Topology 8 (2024) 1101–1119.
corr_author: '1'
date_created: 2023-07-02T22:00:44Z
date_published: 2024-10-01T00:00:00Z
date_updated: 2026-04-07T12:58:47Z
day: '01'
ddc:
- '000'
department:
- _id: HeEd
doi: 10.1007/s41468-023-00126-9
ec_funded: 1
external_id:
pmid:
- '39678706'
file:
- access_level: open_access
checksum: d493df5088c222b88d9ca46b623ad0ee
content_type: application/pdf
creator: dernst
date_created: 2025-01-09T07:39:41Z
date_updated: 2025-01-09T07:39:41Z
file_id: '18783'
file_name: 2024_JourApplCompTopo_Biswas.pdf
file_size: 476896
relation: main_file
success: 1
file_date_updated: 2025-01-09T07:39:41Z
has_accepted_license: '1'
intvolume: ' 8'
language:
- iso: eng
month: '10'
oa: 1
oa_version: Published Version
page: 1101-1119
pmid: 1
project:
- _id: 266A2E9E-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '788183'
name: Alpha Shape Theory Extended
- _id: 0aa4bc98-070f-11eb-9043-e6fff9c6a316
grant_number: I4887
name: Persistent Homology, Algorithms and Stochastic Geometry
- _id: 268116B8-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: Z00342
name: Mathematics, Computer Science
publication: Journal of Applied and Computational Topology
publication_identifier:
eissn:
- 2367-1734
issn:
- 2367-1726
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
related_material:
record:
- id: '15094'
relation: dissertation_contains
status: public
scopus_import: '1'
status: public
title: Geometric characterization of the persistence of 1D maps
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
short: CC BY (4.0)
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 8
year: '2024'
...
---
_id: '15093'
abstract:
- lang: eng
text: We present a dynamic data structure for maintaining the persistent homology
of a time series of real numbers. The data structure supports local operations,
including the insertion and deletion of an item and the cutting and concatenating
of lists, each in time O(log n + k), in which n counts the critical items and
k the changes in the augmented persistence diagram. To achieve this, we design
a tailor-made tree structure with an unconventional representation, referred to
as banana tree, which may be useful in its own right.
acknowledgement: The first and second authors are funded by the European Research Council under the
European Union’s Horizon 2020 research and innovation programme, ERC grant no. 788183,“Alpha
Shape Theory Extended (Alpha)”, by the Wittgenstein Prize, FWF grant no. Z 342-N31,
and by the DFG Collaborative Research Center TRR 109, FWF grant no. I 02979-N35.The
third author received funding by the European Research Council under the European
Union’s Horizon 2020research and innovation programme, ERC grant no. 101019564, “The Design of Modern Fully Dynamic DataStructures
(MoDynStruct)”, and by the Austrian Science Fund through the Wittgenstein Prize
with FWF grant no. Z 422-N, and also by FWF grant no. I 5982-N, and by FWF grant
no. P 33775-N, with additional funding from the netidee SCIENCE Stiftung, 2020–2024. The
fourth author is funded by the Vienna Graduate School on Computational Optimization,
FWF project no. W1260-N35.
article_processing_charge: No
arxiv: 1
author:
- first_name: Sebastiano
full_name: Cultrera di Montesano, Sebastiano
id: 34D2A09C-F248-11E8-B48F-1D18A9856A87
last_name: Cultrera di Montesano
orcid: 0000-0001-6249-0832
- first_name: Herbert
full_name: Edelsbrunner, Herbert
id: 3FB178DA-F248-11E8-B48F-1D18A9856A87
last_name: Edelsbrunner
orcid: 0000-0002-9823-6833
- first_name: Monika H
full_name: Henzinger, Monika H
id: 540c9bbd-f2de-11ec-812d-d04a5be85630
last_name: Henzinger
orcid: 0000-0002-5008-6530
- first_name: Lara
full_name: Ost, Lara
last_name: Ost
citation:
ama: 'Cultrera di Montesano S, Edelsbrunner H, Henzinger M, Ost L. Dynamically maintaining
the persistent homology of time series. In: Woodruff DP, ed. Proceedings of
the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA). Society
for Industrial and Applied Mathematics; 2024:243-295. doi:10.1137/1.9781611977912.11'
apa: 'Cultrera di Montesano, S., Edelsbrunner, H., Henzinger, M., & Ost, L.
(2024). Dynamically maintaining the persistent homology of time series. In D.
P. Woodruff (Ed.), Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete
Algorithms (SODA) (pp. 243–295). Alexandria, VA, USA: Society for Industrial
and Applied Mathematics. https://doi.org/10.1137/1.9781611977912.11'
chicago: Cultrera di Montesano, Sebastiano, Herbert Edelsbrunner, Monika Henzinger,
and Lara Ost. “Dynamically Maintaining the Persistent Homology of Time Series.”
In Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms
(SODA), edited by David P. Woodruff, 243–95. Society for Industrial and Applied
Mathematics, 2024. https://doi.org/10.1137/1.9781611977912.11.
ieee: S. Cultrera di Montesano, H. Edelsbrunner, M. Henzinger, and L. Ost, “Dynamically
maintaining the persistent homology of time series,” in Proceedings of the
2024 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), Alexandria,
VA, USA, 2024, pp. 243–295.
ista: 'Cultrera di Montesano S, Edelsbrunner H, Henzinger M, Ost L. 2024. Dynamically
maintaining the persistent homology of time series. Proceedings of the 2024 Annual
ACM-SIAM Symposium on Discrete Algorithms (SODA). SODA: Symposium on Discrete
Algorithms, 243–295.'
mla: Cultrera di Montesano, Sebastiano, et al. “Dynamically Maintaining the Persistent
Homology of Time Series.” Proceedings of the 2024 Annual ACM-SIAM Symposium
on Discrete Algorithms (SODA), edited by David P. Woodruff, Society for Industrial
and Applied Mathematics, 2024, pp. 243–95, doi:10.1137/1.9781611977912.11.
short: S. Cultrera di Montesano, H. Edelsbrunner, M. Henzinger, L. Ost, in:, D.P.
Woodruff (Ed.), Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete
Algorithms (SODA), Society for Industrial and Applied Mathematics, 2024, pp. 243–295.
conference:
end_date: 2024-01-10
location: Alexandria, VA, USA
name: 'SODA: Symposium on Discrete Algorithms'
start_date: 2024-01-07
corr_author: '1'
date_created: 2024-03-08T10:27:39Z
date_published: 2024-01-04T00:00:00Z
date_updated: 2026-04-07T12:58:47Z
day: '04'
department:
- _id: HeEd
- _id: MoHe
doi: 10.1137/1.9781611977912.11
ec_funded: 1
editor:
- first_name: David P.
full_name: Woodruff, David P.
last_name: Woodruff
external_id:
arxiv:
- '2311.01115'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/2311.01115
month: '01'
oa: 1
oa_version: Preprint
page: 243 - 295
project:
- _id: 266A2E9E-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '788183'
name: Alpha Shape Theory Extended
- _id: 268116B8-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: Z00342
name: Mathematics, Computer Science
- _id: bd9ca328-d553-11ed-ba76-dc4f890cfe62
call_identifier: H2020
grant_number: '101019564'
name: The design and evaluation of modern fully dynamic data structures
- _id: 34def286-11ca-11ed-8bc3-da5948e1613c
grant_number: Z00422
name: Efficient algorithms
- _id: bd9e3a2e-d553-11ed-ba76-8aa684ce17fe
grant_number: P33775
name: Fast Algorithms for a Reactive Network Layer
publication: Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms
(SODA)
publication_identifier:
eisbn:
- '9781611977912'
publication_status: published
publisher: Society for Industrial and Applied Mathematics
quality_controlled: '1'
related_material:
record:
- id: '15094'
relation: dissertation_contains
status: public
scopus_import: '1'
status: public
title: Dynamically maintaining the persistent homology of time series
type: conference
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2024'
...
---
OA_place: repository
_id: '15091'
abstract:
- lang: eng
text: "Motivated by applications in the medical sciences, we study finite chromatic\r\nsets
in Euclidean space from a topological perspective. Based on the persistent\r\nhomology
for images, kernels and cokernels, we design provably stable\r\nhomological quantifiers
that describe the geometric micro- and macro-structure\r\nof how the color classes
mingle. These can be efficiently computed using\r\nchromatic variants of Delaunay
and alpha complexes, and code that does these\r\ncomputations is provided."
article_number: '2212.03128'
article_processing_charge: No
arxiv: 1
author:
- first_name: Sebastiano
full_name: Cultrera di Montesano, Sebastiano
id: 34D2A09C-F248-11E8-B48F-1D18A9856A87
last_name: Cultrera di Montesano
orcid: 0000-0001-6249-0832
- first_name: Ondrej
full_name: Draganov, Ondrej
id: 2B23F01E-F248-11E8-B48F-1D18A9856A87
last_name: Draganov
orcid: 0000-0003-0464-3823
- first_name: Herbert
full_name: Edelsbrunner, Herbert
id: 3FB178DA-F248-11E8-B48F-1D18A9856A87
last_name: Edelsbrunner
orcid: 0000-0002-9823-6833
- first_name: Morteza
full_name: Saghafian, Morteza
id: f86f7148-b140-11ec-9577-95435b8df824
last_name: Saghafian
citation:
ama: Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M. Chromatic
alpha complexes. arXiv. doi:10.48550/arXiv.2212.03128
apa: Cultrera di Montesano, S., Draganov, O., Edelsbrunner, H., & Saghafian,
M. (n.d.). Chromatic alpha complexes. arXiv. https://doi.org/10.48550/arXiv.2212.03128
chicago: Cultrera di Montesano, Sebastiano, Ondrej Draganov, Herbert Edelsbrunner,
and Morteza Saghafian. “Chromatic Alpha Complexes.” ArXiv, n.d. https://doi.org/10.48550/arXiv.2212.03128.
ieee: S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, and M. Saghafian,
“Chromatic alpha complexes,” arXiv. .
ista: Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M. Chromatic
alpha complexes. arXiv, 2212.03128.
mla: Cultrera di Montesano, Sebastiano, et al. “Chromatic Alpha Complexes.” ArXiv,
2212.03128, doi:10.48550/arXiv.2212.03128.
short: S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, M. Saghafian, ArXiv
(n.d.).
corr_author: '1'
date_created: 2024-03-08T10:13:59Z
date_published: 2024-02-07T00:00:00Z
date_updated: 2026-04-07T12:58:47Z
day: '07'
department:
- _id: HeEd
doi: 10.48550/arXiv.2212.03128
external_id:
arxiv:
- '2212.03128'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/2212.03128
month: '02'
oa: 1
oa_version: Preprint
publication: arXiv
publication_status: draft
related_material:
record:
- id: '20585'
relation: later_version
status: public
- id: '18979'
relation: dissertation_contains
status: public
- id: '15094'
relation: dissertation_contains
status: public
status: public
title: Chromatic alpha complexes
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
short: CC BY (4.0)
type: preprint
user_id: 8b945eb4-e2f2-11eb-945a-df72226e66a9
year: '2024'
...
---
_id: '15012'
abstract:
- lang: eng
text: We solve a problem of Dujmović and Wood (2007) by showing that a complete
convex geometric graph on n vertices cannot be decomposed into fewer than n-1
star-forests, each consisting of noncrossing edges. This bound is clearly tight.
We also discuss similar questions for abstract graphs.
acknowledgement: János Pach’s Research partially supported by European Research Council
(ERC), grant “GeoScape” No. 882971 and by the Hungarian Science Foundation (NKFIH),
grant K-131529. Work by Morteza Saghafian is partially supported by the European
Research Council (ERC), grant No. 788183, and by the Wittgenstein Prize, Austrian
Science Fund (FWF), grant No. Z 342-N31.
alternative_title:
- LNCS
article_processing_charge: No
arxiv: 1
author:
- first_name: János
full_name: Pach, János
id: E62E3130-B088-11EA-B919-BF823C25FEA4
last_name: Pach
- first_name: Morteza
full_name: Saghafian, Morteza
id: f86f7148-b140-11ec-9577-95435b8df824
last_name: Saghafian
- first_name: Patrick
full_name: Schnider, Patrick
last_name: Schnider
citation:
ama: 'Pach J, Saghafian M, Schnider P. Decomposition of geometric graphs into star-forests.
In: 31st International Symposium on Graph Drawing and Network Visualization.
Vol 14465. Springer Nature; 2024:339-346. doi:10.1007/978-3-031-49272-3_23'
apa: 'Pach, J., Saghafian, M., & Schnider, P. (2024). Decomposition of geometric
graphs into star-forests. In 31st International Symposium on Graph Drawing
and Network Visualization (Vol. 14465, pp. 339–346). Isola delle Femmine,
Palermo, Italy: Springer Nature. https://doi.org/10.1007/978-3-031-49272-3_23'
chicago: Pach, János, Morteza Saghafian, and Patrick Schnider. “Decomposition of Geometric
Graphs into Star-Forests.” In 31st International Symposium on Graph Drawing
and Network Visualization, 14465:339–46. Springer Nature, 2024. https://doi.org/10.1007/978-3-031-49272-3_23.
ieee: J. Pach, M. Saghafian, and P. Schnider, “Decomposition of geometric graphs
into star-forests,” in 31st International Symposium on Graph Drawing and Network
Visualization, Isola delle Femmine, Palermo, Italy, 2024, vol. 14465, pp.
339–346.
ista: 'Pach J, Saghafian M, Schnider P. 2024. Decomposition of geometric graphs
into star-forests. 31st International Symposium on Graph Drawing and Network Visualization.
GD: Graph Drawing and Network Visualization, LNCS, vol. 14465, 339–346.'
mla: Pach, János, et al. “Decomposition of Geometric Graphs into Star-Forests.”
31st International Symposium on Graph Drawing and Network Visualization,
vol. 14465, Springer Nature, 2024, pp. 339–46, doi:10.1007/978-3-031-49272-3_23.
short: J. Pach, M. Saghafian, P. Schnider, in:, 31st International Symposium on
Graph Drawing and Network Visualization, Springer Nature, 2024, pp. 339–346.
conference:
end_date: 2023-09-22
location: Isola delle Femmine, Palermo, Italy
name: 'GD: Graph Drawing and Network Visualization'
start_date: 2023-09-20
date_created: 2024-02-18T23:01:03Z
date_published: 2024-01-01T00:00:00Z
date_updated: 2026-04-16T09:12:37Z
day: '01'
department:
- _id: HeEd
doi: 10.1007/978-3-031-49272-3_23
ec_funded: 1
external_id:
arxiv:
- '2306.13201'
isi:
- '001207939600023'
intvolume: ' 14465'
isi: 1
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://doi.org/10.48550/arXiv.2306.13201
month: '01'
oa: 1
oa_version: Preprint
page: 339-346
project:
- _id: 266A2E9E-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '788183'
name: Alpha Shape Theory Extended
- _id: 268116B8-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: Z00342
name: Mathematics, Computer Science
publication: 31st International Symposium on Graph Drawing and Network Visualization
publication_identifier:
eisbn:
- '9783031492723'
eissn:
- 1611-3349
isbn:
- '9783031492716'
issn:
- 0302-9743
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
related_material:
record:
- id: '21253'
relation: later_version
status: public
scopus_import: '1'
status: public
title: Decomposition of geometric graphs into star-forests
type: conference
user_id: ba8df636-2132-11f1-aed0-ed93e2281fdd
volume: 14465
year: '2024'
...
---
_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
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: '14362'
abstract:
- lang: eng
text: "Motivated by recent applications to entropy theory in dynamical systems,
we generalise notions introduced by Matthews and define weakly weighted and componentwise
weakly weighted (generalised) quasi-metrics. We then systematise and extend to
full generality the correspondences between these objects and other structures
arising in theoretical computer science and dynamics. In particular, we study
the correspondences with weak partial metrics and, if the underlying space is
a semilattice, with invariant (generalised) quasi-metrics satisfying the descending
path condition, and with strictly monotone semi(-co-)valuations.\r\nWe conclude
discussing, for endomorphisms of generalised quasi-metric semilattices, a generalisation
of both the known intrinsic semilattice entropy and the semigroup entropy."
article_number: '114129'
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Ilaria
full_name: Castellano, Ilaria
last_name: Castellano
- first_name: Anna
full_name: Giordano Bruno, Anna
last_name: Giordano Bruno
- first_name: Nicolò
full_name: Zava, Nicolò
id: c8b3499c-7a77-11eb-b046-aa368cbbf2ad
last_name: Zava
orcid: 0000-0001-8686-1888
citation:
ama: Castellano I, Giordano Bruno A, Zava N. Weakly weighted generalised quasi-metric
spaces and semilattices. Theoretical Computer Science. 2023;977. doi:10.1016/j.tcs.2023.114129
apa: Castellano, I., Giordano Bruno, A., & Zava, N. (2023). Weakly weighted
generalised quasi-metric spaces and semilattices. Theoretical Computer Science.
Elsevier. https://doi.org/10.1016/j.tcs.2023.114129
chicago: Castellano, Ilaria, Anna Giordano Bruno, and Nicolò Zava. “Weakly Weighted
Generalised Quasi-Metric Spaces and Semilattices.” Theoretical Computer Science.
Elsevier, 2023. https://doi.org/10.1016/j.tcs.2023.114129.
ieee: I. Castellano, A. Giordano Bruno, and N. Zava, “Weakly weighted generalised
quasi-metric spaces and semilattices,” Theoretical Computer Science, vol.
977. Elsevier, 2023.
ista: Castellano I, Giordano Bruno A, Zava N. 2023. Weakly weighted generalised
quasi-metric spaces and semilattices. Theoretical Computer Science. 977, 114129.
mla: Castellano, Ilaria, et al. “Weakly Weighted Generalised Quasi-Metric Spaces
and Semilattices.” Theoretical Computer Science, vol. 977, 114129, Elsevier,
2023, doi:10.1016/j.tcs.2023.114129.
short: I. Castellano, A. Giordano Bruno, N. Zava, Theoretical Computer Science 977
(2023).
corr_author: '1'
date_created: 2023-09-24T22:01:11Z
date_published: 2023-10-25T00:00:00Z
date_updated: 2024-10-09T21:07:00Z
day: '25'
department:
- _id: HeEd
doi: 10.1016/j.tcs.2023.114129
external_id:
arxiv:
- '2212.08424'
isi:
- '001076934000001'
intvolume: ' 977'
isi: 1
language:
- iso: eng
main_file_link:
- open_access: '1'
url: 'https://doi.org/10.48550/arXiv.2212.08424 '
month: '10'
oa: 1
oa_version: Preprint
publication: Theoretical Computer Science
publication_identifier:
issn:
- 0304-3975
publication_status: published
publisher: Elsevier
quality_controlled: '1'
scopus_import: '1'
status: public
title: Weakly weighted generalised quasi-metric spaces and semilattices
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 977
year: '2023'
...
---
_id: '14464'
abstract:
- lang: eng
text: 'Given a triangle Δ, we study the problem of determining the smallest enclosing
and largest embedded isosceles triangles of Δ with respect to area and perimeter.
This problem was initially posed by Nandakumar [17, 22] and was first studied
by Kiss, Pach, and Somlai [13], who showed that if Δ′ is the smallest area isosceles
triangle containing Δ, then Δ′ and Δ share a side and an angle. In the present
paper, we prove that for any triangle Δ, every maximum area isosceles triangle
embedded in Δ and every maximum perimeter isosceles triangle embedded in Δ shares
a side and an angle with Δ. Somewhat surprisingly, the case of minimum perimeter
enclosing triangles is different: there are infinite families of triangles Δ whose
minimum perimeter isosceles containers do not share a side and an angle with Δ.'
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Áron
full_name: Ambrus, Áron
last_name: Ambrus
- first_name: Mónika
full_name: Csikós, Mónika
last_name: Csikós
- first_name: Gergely
full_name: Kiss, Gergely
last_name: Kiss
- first_name: János
full_name: Pach, János
id: E62E3130-B088-11EA-B919-BF823C25FEA4
last_name: Pach
- first_name: Gábor
full_name: Somlai, Gábor
last_name: Somlai
citation:
ama: Ambrus Á, Csikós M, Kiss G, Pach J, Somlai G. Optimal embedded and enclosing
isosceles triangles. International Journal of Foundations of Computer Science.
2023;34(7):737-760. doi:10.1142/S012905412342008X
apa: Ambrus, Á., Csikós, M., Kiss, G., Pach, J., & Somlai, G. (2023). Optimal
embedded and enclosing isosceles triangles. International Journal of Foundations
of Computer Science. World Scientific Publishing. https://doi.org/10.1142/S012905412342008X
chicago: Ambrus, Áron, Mónika Csikós, Gergely Kiss, János Pach, and Gábor Somlai.
“Optimal Embedded and Enclosing Isosceles Triangles.” International Journal
of Foundations of Computer Science. World Scientific Publishing, 2023. https://doi.org/10.1142/S012905412342008X.
ieee: Á. Ambrus, M. Csikós, G. Kiss, J. Pach, and G. Somlai, “Optimal embedded and
enclosing isosceles triangles,” International Journal of Foundations of Computer
Science, vol. 34, no. 7. World Scientific Publishing, pp. 737–760, 2023.
ista: Ambrus Á, Csikós M, Kiss G, Pach J, Somlai G. 2023. Optimal embedded and enclosing
isosceles triangles. International Journal of Foundations of Computer Science.
34(7), 737–760.
mla: Ambrus, Áron, et al. “Optimal Embedded and Enclosing Isosceles Triangles.”
International Journal of Foundations of Computer Science, vol. 34, no.
7, World Scientific Publishing, 2023, pp. 737–60, doi:10.1142/S012905412342008X.
short: Á. Ambrus, M. Csikós, G. Kiss, J. Pach, G. Somlai, International Journal
of Foundations of Computer Science 34 (2023) 737–760.
date_created: 2023-10-29T23:01:18Z
date_published: 2023-10-05T00:00:00Z
date_updated: 2023-12-13T13:04:55Z
day: '05'
department:
- _id: HeEd
doi: 10.1142/S012905412342008X
external_id:
arxiv:
- '2205.11637'
isi:
- '001080874400001'
intvolume: ' 34'
isi: 1
issue: '7'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://doi.org/10.48550/arXiv.2205.11637
month: '10'
oa: 1
oa_version: Preprint
page: 737-760
publication: International Journal of Foundations of Computer Science
publication_identifier:
eissn:
- 1793-6373
issn:
- 0129-0541
publication_status: published
publisher: World Scientific Publishing
quality_controlled: '1'
scopus_import: '1'
status: public
title: Optimal embedded and enclosing isosceles triangles
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 34
year: '2023'
...
---
_id: '14557'
abstract:
- lang: eng
text: Motivated by a problem posed in [10], we investigate the closure operators
of the category SLatt of join semilattices and its subcategory SLattO of join
semilattices with bottom element. In particular, we show that there are only finitely
many closure operators of both categories, and provide a complete classification.
We use this result to deduce the known fact that epimorphisms of SLatt and SLattO
are surjective. We complement the paper with two different proofs of this result
using either generators or Isbell’s zigzag theorem.
acknowledgement: "The first and second named authors are members of GNSAGA – INdAM.\r\nThe
third named author was supported by the FWF Grant, Project number I4245–N35"
article_processing_charge: No
article_type: original
author:
- first_name: D.
full_name: Dikranjan, D.
last_name: Dikranjan
- first_name: A.
full_name: Giordano Bruno, A.
last_name: Giordano Bruno
- first_name: Nicolò
full_name: Zava, Nicolò
id: c8b3499c-7a77-11eb-b046-aa368cbbf2ad
last_name: Zava
orcid: 0000-0001-8686-1888
citation:
ama: Dikranjan D, Giordano Bruno A, Zava N. Epimorphisms and closure operators of
categories of semilattices. Quaestiones Mathematicae. 2023;46(S1):191-221.
doi:10.2989/16073606.2023.2247731
apa: Dikranjan, D., Giordano Bruno, A., & Zava, N. (2023). Epimorphisms and
closure operators of categories of semilattices. Quaestiones Mathematicae.
Taylor & Francis. https://doi.org/10.2989/16073606.2023.2247731
chicago: Dikranjan, D., A. Giordano Bruno, and Nicolò Zava. “Epimorphisms and Closure
Operators of Categories of Semilattices.” Quaestiones Mathematicae. Taylor
& Francis, 2023. https://doi.org/10.2989/16073606.2023.2247731.
ieee: D. Dikranjan, A. Giordano Bruno, and N. Zava, “Epimorphisms and closure operators
of categories of semilattices,” Quaestiones Mathematicae, vol. 46, no.
S1. Taylor & Francis, pp. 191–221, 2023.
ista: Dikranjan D, Giordano Bruno A, Zava N. 2023. Epimorphisms and closure operators
of categories of semilattices. Quaestiones Mathematicae. 46(S1), 191–221.
mla: Dikranjan, D., et al. “Epimorphisms and Closure Operators of Categories of
Semilattices.” Quaestiones Mathematicae, vol. 46, no. S1, Taylor &
Francis, 2023, pp. 191–221, doi:10.2989/16073606.2023.2247731.
short: D. Dikranjan, A. Giordano Bruno, N. Zava, Quaestiones Mathematicae 46 (2023)
191–221.
date_created: 2023-11-19T23:00:55Z
date_published: 2023-11-01T00:00:00Z
date_updated: 2025-09-09T13:23:12Z
day: '01'
department:
- _id: HeEd
doi: 10.2989/16073606.2023.2247731
external_id:
isi:
- '001098712000006'
intvolume: ' 46'
isi: 1
issue: S1
language:
- iso: eng
month: '11'
oa_version: None
page: 191-221
project:
- _id: 26AD5D90-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: I04245
name: Algebraic Footprints of Geometric Features in Homology
publication: Quaestiones Mathematicae
publication_identifier:
eissn:
- 1727-933X
issn:
- 1607-3606
publication_status: published
publisher: Taylor & Francis
quality_controlled: '1'
scopus_import: '1'
status: public
title: Epimorphisms and closure operators of categories of semilattices
type: journal_article
user_id: 317138e5-6ab7-11ef-aa6d-ffef3953e345
volume: 46
year: '2023'
...
---
_id: '14739'
abstract:
- lang: eng
text: Attempts to incorporate topological information in supervised learning tasks
have resulted in the creation of several techniques for vectorizing persistent
homology barcodes. In this paper, we study thirteen such methods. Besides describing
an organizational framework for these methods, we comprehensively benchmark them
against three well-known classification tasks. Surprisingly, we discover that
the best-performing method is a simple vectorization, which consists only of a
few elementary summary statistics. Finally, we provide a convenient web application
which has been designed to facilitate exploration and experimentation with various
vectorization methods.
acknowledgement: "The work of Maria-Jose Jimenez, Eduardo Paluzo-Hidalgo and Manuel
Soriano-Trigueros was supported in part by the Spanish grant Ministerio de Ciencia
e Innovacion under Grants TED2021-129438B-I00 and PID2019-107339GB-I00, and in part
by REXASI-PRO H-EU project, call HORIZON-CL4-2021-HUMAN-01-01 under Grant 101070028.
The work of\r\nMaria-Jose Jimenez was supported by a grant of Convocatoria de la
Universidad de Sevilla para la recualificacion del sistema universitario español,
2021-23, funded by the European Union, NextGenerationEU. The work of Vidit Nanda
was supported in part by EPSRC under Grant EP/R018472/1 and in part by US AFOSR
under Grant FA9550-22-1-0462. \r\nWe are grateful to the team of GUDHI and TEASPOON
developers, for their work and their support. We are also grateful to Streamlit
for providing extra resources to deploy the web app\r\nonline on Streamlit community
cloud. We thank the anonymous referees for their helpful suggestions."
article_processing_charge: Yes (in subscription journal)
article_type: original
author:
- first_name: Dashti
full_name: Ali, Dashti
last_name: Ali
- first_name: Aras
full_name: Asaad, Aras
last_name: Asaad
- first_name: Maria-Jose
full_name: Jimenez, Maria-Jose
last_name: Jimenez
- first_name: Vidit
full_name: Nanda, Vidit
last_name: Nanda
- first_name: Eduardo
full_name: Paluzo-Hidalgo, Eduardo
last_name: Paluzo-Hidalgo
- first_name: Manuel
full_name: Soriano Trigueros, Manuel
id: 15ebd7cf-15bf-11ee-aebd-bb4bb5121ea8
last_name: Soriano Trigueros
orcid: 0000-0003-2449-1433
citation:
ama: Ali D, Asaad A, Jimenez M-J, Nanda V, Paluzo-Hidalgo E, Soriano Trigueros M.
A survey of vectorization methods in topological data analysis. IEEE Transactions
on Pattern Analysis and Machine Intelligence. 2023;45(12):14069-14080. doi:10.1109/tpami.2023.3308391
apa: Ali, D., Asaad, A., Jimenez, M.-J., Nanda, V., Paluzo-Hidalgo, E., & Soriano
Trigueros, M. (2023). A survey of vectorization methods in topological data analysis.
IEEE Transactions on Pattern Analysis and Machine Intelligence. IEEE. https://doi.org/10.1109/tpami.2023.3308391
chicago: Ali, Dashti, Aras Asaad, Maria-Jose Jimenez, Vidit Nanda, Eduardo Paluzo-Hidalgo,
and Manuel Soriano Trigueros. “A Survey of Vectorization Methods in Topological
Data Analysis.” IEEE Transactions on Pattern Analysis and Machine Intelligence.
IEEE, 2023. https://doi.org/10.1109/tpami.2023.3308391.
ieee: D. Ali, A. Asaad, M.-J. Jimenez, V. Nanda, E. Paluzo-Hidalgo, and M. Soriano
Trigueros, “A survey of vectorization methods in topological data analysis,” IEEE
Transactions on Pattern Analysis and Machine Intelligence, vol. 45, no. 12.
IEEE, pp. 14069–14080, 2023.
ista: Ali D, Asaad A, Jimenez M-J, Nanda V, Paluzo-Hidalgo E, Soriano Trigueros
M. 2023. A survey of vectorization methods in topological data analysis. IEEE
Transactions on Pattern Analysis and Machine Intelligence. 45(12), 14069–14080.
mla: Ali, Dashti, et al. “A Survey of Vectorization Methods in Topological Data
Analysis.” IEEE Transactions on Pattern Analysis and Machine Intelligence,
vol. 45, no. 12, IEEE, 2023, pp. 14069–80, doi:10.1109/tpami.2023.3308391.
short: D. Ali, A. Asaad, M.-J. Jimenez, V. Nanda, E. Paluzo-Hidalgo, M. Soriano
Trigueros, IEEE Transactions on Pattern Analysis and Machine Intelligence 45 (2023)
14069–14080.
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---
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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)."
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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
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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:
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title: Local criteria for triangulating general manifolds
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...