---
OA_place: publisher
OA_type: hybrid
PlanS_conform: '1'
_id: '21407'
abstract:
- lang: eng
text: "This note proves that only a linear number of holes in a Cech complex of
n points in R^d\r\ncan persist over an interval of constant length. Specifically,
for any fixed dimension p <\r\nd and fixed ε > 0, the number of p-dimensional
holes in the ˇ Cech complex at radius 1\r\nthat persist to radius 1+ε is bounded
above by a constant times n,where n is the number\r\nof points. The proof uses
a packing argument supported by relating theCˇ ech complexes\r\nwith corresponding
snap complexes over the cells in a partition of space. The argument\r\nis self-contained
and elementary, relying on geometric and combinatorial constructions\r\nrather
than on the existing theory of sparse approximations or interleavings. The bound\r\nalso
applies to Alpha complexes and Vietoris–Rips complexes. While our result can be\r\ninferred
from prior work on sparse filtrations, to our knowledge, no explicit statement\r\nor
direct proof of this bound appears in the literature."
acknowledgement: The authors would like to thank Michael Lesnick and Primoz Skraba
for their helpful comments regarding sparse approximations of filtrations. We are
also grateful to the anonymous referees for their careful reading and constructive
suggestions. The three authors are supported by the Wittgenstein Prize, Austrian
Science Fund (FWF), grant no. Z 342-N31, by the DFG Collaborative Research Center
TRR 109, Austrian Science Fund (FWF), grant no. I 02979-N35, the U.S. National Science
Foundation (NSF-DMS), grant no. 2005630, and a JSPS Grant-in-Aid for Transformative
Research Areas (A) (22H05107, Y.H.), EPSRC Research Grant EP/Y008642/1.
article_number: '5'
article_processing_charge: Yes (in subscription journal)
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: Matthew
full_name: Kahle, Matthew
last_name: Kahle
- first_name: Shu
full_name: Kanazawa, Shu
last_name: Kanazawa
citation:
ama: Edelsbrunner H, Kahle M, Kanazawa S. Maximum persistent Betti numbers of Čech
complexes. Journal of Applied and Computational Topology. 2026;10. doi:10.1007/s41468-026-00233-3
apa: Edelsbrunner, H., Kahle, M., & Kanazawa, S. (2026). Maximum persistent
Betti numbers of Čech complexes. Journal of Applied and Computational Topology.
Springer Nature. https://doi.org/10.1007/s41468-026-00233-3
chicago: Edelsbrunner, Herbert, Matthew Kahle, and Shu Kanazawa. “Maximum Persistent
Betti Numbers of Čech Complexes.” Journal of Applied and Computational Topology.
Springer Nature, 2026. https://doi.org/10.1007/s41468-026-00233-3.
ieee: H. Edelsbrunner, M. Kahle, and S. Kanazawa, “Maximum persistent Betti numbers
of Čech complexes,” Journal of Applied and Computational Topology, vol.
10. Springer Nature, 2026.
ista: Edelsbrunner H, Kahle M, Kanazawa S. 2026. Maximum persistent Betti numbers
of Čech complexes. Journal of Applied and Computational Topology. 10, 5.
mla: Edelsbrunner, Herbert, et al. “Maximum Persistent Betti Numbers of Čech Complexes.”
Journal of Applied and Computational Topology, vol. 10, 5, Springer Nature,
2026, doi:10.1007/s41468-026-00233-3.
short: H. Edelsbrunner, M. Kahle, S. Kanazawa, Journal of Applied and Computational
Topology 10 (2026).
date_created: 2026-03-08T23:01:45Z
date_published: 2026-03-01T00:00:00Z
date_updated: 2026-03-09T11:31:29Z
day: '01'
ddc:
- '500'
department:
- _id: HeEd
doi: 10.1007/s41468-026-00233-3
external_id:
arxiv:
- '2409.05241'
file:
- access_level: open_access
checksum: 0bf6dc430cafa40c08f260fe17d54595
content_type: application/pdf
creator: dernst
date_created: 2026-03-09T11:29:30Z
date_updated: 2026-03-09T11:29:30Z
file_id: '21416'
file_name: 2026_JourAppliedCompTopology_Edelsbrunner.pdf
file_size: 323111
relation: main_file
success: 1
file_date_updated: 2026-03-09T11:29:30Z
has_accepted_license: '1'
intvolume: ' 10'
language:
- iso: eng
month: '03'
oa: 1
oa_version: Published Version
project:
- _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'
scopus_import: '1'
status: public
title: Maximum persistent Betti numbers of Čech complexes
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
short: CC BY (4.0)
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 10
year: '2026'
...
---
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'
...
---
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: '12763'
abstract:
- lang: eng
text: 'Kleinjohann (Archiv der Mathematik 35(1):574–582, 1980; Mathematische Zeitschrift
176(3), 327–344, 1981) and Bangert (Archiv der Mathematik 38(1):54–57, 1982) extended
the reach rch(S) from subsets S of Euclidean space to the reach rchM(S) of subsets
S of Riemannian manifolds M, where M is smooth (we’ll assume at least C3). Bangert
showed that sets of positive reach in Euclidean space and Riemannian manifolds
are very similar. In this paper we introduce a slight variant of Kleinjohann’s
and Bangert’s extension and quantify the similarity between sets of positive reach
in Euclidean space and Riemannian manifolds in a new way: Given p∈M and q∈S, we
bound the local feature size (a local version of the reach) of its lifting to
the tangent space via the inverse exponential map (exp−1p(S)) at q, assuming that
rchM(S) and the geodesic distance dM(p,q) are bounded. These bounds are motivated
by the importance of the reach and local feature size to manifold learning, topological
inference, and triangulating manifolds and the fact that intrinsic approaches
circumvent the curse of dimensionality.'
acknowledgement: "We thank Eddie Aamari, David Cohen-Steiner, Isa Costantini, Fred
Chazal, Ramsay Dyer, André Lieutier, and Alef Sterk for discussion and Pierre Pansu
for encouragement. We further acknowledge the anonymous reviewers whose comments
helped improve the exposition.\r\nThe research leading to these results has received
funding from the European Research Council (ERC) under the European Union’s Seventh
Framework Programme (FP/2007-2013) / ERC Grant Agreement No. 339025 GUDHI (Algorithmic
Foundations of Geometry Understanding in Higher Dimensions). The first author is
further supported by the French government, through the 3IA Côte d’Azur Investments
in the Future project managed by the National Research Agency (ANR) with the reference
number ANR-19-P3IA-0002. The second author is supported by the European Union’s
Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie
Grant Agreement No. 754411 and the Austrian science fund (FWF) M-3073."
article_processing_charge: No
article_type: original
author:
- first_name: Jean Daniel
full_name: Boissonnat, Jean Daniel
last_name: Boissonnat
- first_name: Mathijs
full_name: Wintraecken, Mathijs
id: 307CFBC8-F248-11E8-B48F-1D18A9856A87
last_name: Wintraecken
orcid: 0000-0002-7472-2220
citation:
ama: Boissonnat JD, Wintraecken M. The reach of subsets of manifolds. Journal
of Applied and Computational Topology. 2023;7:619-641. doi:10.1007/s41468-023-00116-x
apa: Boissonnat, J. D., & Wintraecken, M. (2023). The reach of subsets of manifolds.
Journal of Applied and Computational Topology. Springer Nature. https://doi.org/10.1007/s41468-023-00116-x
chicago: Boissonnat, Jean Daniel, and Mathijs Wintraecken. “The Reach of Subsets
of Manifolds.” Journal of Applied and Computational Topology. Springer
Nature, 2023. https://doi.org/10.1007/s41468-023-00116-x.
ieee: J. D. Boissonnat and M. Wintraecken, “The reach of subsets of manifolds,”
Journal of Applied and Computational Topology, vol. 7. Springer Nature,
pp. 619–641, 2023.
ista: Boissonnat JD, Wintraecken M. 2023. The reach of subsets of manifolds. Journal
of Applied and Computational Topology. 7, 619–641.
mla: Boissonnat, Jean Daniel, and Mathijs Wintraecken. “The Reach of Subsets of
Manifolds.” Journal of Applied and Computational Topology, vol. 7, Springer
Nature, 2023, pp. 619–41, doi:10.1007/s41468-023-00116-x.
short: J.D. Boissonnat, M. Wintraecken, Journal of Applied and Computational Topology
7 (2023) 619–641.
corr_author: '1'
date_created: 2023-03-26T22:01:08Z
date_published: 2023-09-01T00:00:00Z
date_updated: 2025-04-14T07:44:01Z
day: '01'
department:
- _id: HeEd
doi: 10.1007/s41468-023-00116-x
ec_funded: 1
intvolume: ' 7'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://inserm.hal.science/INRIA-SACLAY/hal-04083524v1
month: '09'
oa: 1
oa_version: Submitted Version
page: 619-641
project:
- _id: 260C2330-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '754411'
name: ISTplus - Postdoctoral Fellowships
- _id: fc390959-9c52-11eb-aca3-afa58bd282b2
grant_number: M03073
name: Learning and triangulating manifolds via collapses
publication: Journal of Applied and Computational Topology
publication_identifier:
eissn:
- 2367-1734
issn:
- 2367-1726
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: The reach of subsets of manifolds
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 7
year: '2023'
...
---
_id: '9111'
abstract:
- lang: eng
text: 'We study the probabilistic convergence between the mapper graph and the Reeb
graph of a topological space X equipped with a continuous function f:X→R. We first
give a categorification of the mapper graph and the Reeb graph by interpreting
them in terms of cosheaves and stratified covers of the real line R. We then introduce
a variant of the classic mapper graph of Singh et al. (in: Eurographics symposium
on point-based graphics, 2007), referred to as the enhanced mapper graph, and
demonstrate that such a construction approximates the Reeb graph of (X,f) when
it is applied to points randomly sampled from a probability density function concentrated
on (X,f). Our techniques are based on the interleaving distance of constructible
cosheaves and topological estimation via kernel density estimates. Following Munch
and Wang (In: 32nd international symposium on computational geometry, volume 51
of Leibniz international proceedings in informatics (LIPIcs), Dagstuhl, Germany,
pp 53:1–53:16, 2016), we first show that the mapper graph of (X,f), a constructible
R-space (with a fixed open cover), approximates the Reeb graph of the same space.
We then construct an isomorphism between the mapper of (X,f) to the mapper of
a super-level set of a probability density function concentrated on (X,f). Finally,
building on the approach of Bobrowski et al. (Bernoulli 23(1):288–328, 2017b),
we show that, with high probability, we can recover the mapper of the super-level
set given a sufficiently large sample. Our work is the first to consider the mapper
construction using the theory of cosheaves in a probabilistic setting. It is part
of an ongoing effort to combine sheaf theory, probability, and statistics, to
support topological data analysis with random data.'
acknowledgement: "AB was supported in part by the European Union’s Horizon 2020 research
and innovation\r\nprogramme under the Marie Sklodowska-Curie GrantAgreement No.
754411 and NSF IIS-1513616. OB was supported in part by the Israel Science Foundation,
Grant 1965/19. BW was supported in part by NSF IIS-1513616 and DBI-1661375. EM was
supported in part by NSF CMMI-1800466, DMS-1800446, and CCF-1907591.We would like
to thank the Institute for Mathematics and its Applications for hosting a workshop
titled Bridging Statistics and Sheaves in May 2018, where this work was conceived.\r\nOpen
Access funding provided by Institute of Science and Technology (IST Austria)."
article_processing_charge: Yes (via OA deal)
article_type: original
arxiv: 1
author:
- first_name: Adam
full_name: Brown, Adam
id: 70B7FDF6-608D-11E9-9333-8535E6697425
last_name: Brown
- first_name: Omer
full_name: Bobrowski, Omer
last_name: Bobrowski
- first_name: Elizabeth
full_name: Munch, Elizabeth
last_name: Munch
- first_name: Bei
full_name: Wang, Bei
last_name: Wang
citation:
ama: Brown A, Bobrowski O, Munch E, Wang B. Probabilistic convergence and stability
of random mapper graphs. Journal of Applied and Computational Topology.
2021;5(1):99-140. doi:10.1007/s41468-020-00063-x
apa: Brown, A., Bobrowski, O., Munch, E., & Wang, B. (2021). Probabilistic convergence
and stability of random mapper graphs. Journal of Applied and Computational
Topology. Springer Nature. https://doi.org/10.1007/s41468-020-00063-x
chicago: Brown, Adam, Omer Bobrowski, Elizabeth Munch, and Bei Wang. “Probabilistic
Convergence and Stability of Random Mapper Graphs.” Journal of Applied and
Computational Topology. Springer Nature, 2021. https://doi.org/10.1007/s41468-020-00063-x.
ieee: A. Brown, O. Bobrowski, E. Munch, and B. Wang, “Probabilistic convergence
and stability of random mapper graphs,” Journal of Applied and Computational
Topology, vol. 5, no. 1. Springer Nature, pp. 99–140, 2021.
ista: Brown A, Bobrowski O, Munch E, Wang B. 2021. Probabilistic convergence and
stability of random mapper graphs. Journal of Applied and Computational Topology.
5(1), 99–140.
mla: Brown, Adam, et al. “Probabilistic Convergence and Stability of Random Mapper
Graphs.” Journal of Applied and Computational Topology, vol. 5, no. 1,
Springer Nature, 2021, pp. 99–140, doi:10.1007/s41468-020-00063-x.
short: A. Brown, O. Bobrowski, E. Munch, B. Wang, Journal of Applied and Computational
Topology 5 (2021) 99–140.
date_created: 2021-02-11T14:41:02Z
date_published: 2021-03-01T00:00:00Z
date_updated: 2025-04-14T07:43:51Z
day: '01'
ddc:
- '510'
department:
- _id: HeEd
doi: 10.1007/s41468-020-00063-x
ec_funded: 1
external_id:
arxiv:
- '1909.03488'
file:
- access_level: open_access
checksum: 3f02e9d47c428484733da0f588a3c069
content_type: application/pdf
creator: dernst
date_created: 2021-02-11T14:43:59Z
date_updated: 2021-02-11T14:43:59Z
file_id: '9112'
file_name: 2020_JourApplCompTopology_Brown.pdf
file_size: 2090265
relation: main_file
success: 1
file_date_updated: 2021-02-11T14:43:59Z
has_accepted_license: '1'
intvolume: ' 5'
issue: '1'
language:
- iso: eng
month: '03'
oa: 1
oa_version: Published Version
page: 99-140
project:
- _id: 260C2330-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '754411'
name: ISTplus - Postdoctoral Fellowships
publication: Journal of Applied and Computational Topology
publication_identifier:
eissn:
- 2367-1734
issn:
- 2367-1726
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: Probabilistic convergence and stability of random mapper graphs
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: c635000d-4b10-11ee-a964-aac5a93f6ac1
volume: 5
year: '2021'
...
---
_id: '15064'
abstract:
- lang: eng
text: We call a continuous self-map that reveals itself through a discrete set of
point-value pairs a sampled dynamical system. Capturing the available information
with chain maps on Delaunay complexes, we use persistent homology to quantify
the evidence of recurrent behavior. We establish a sampling theorem to recover
the eigenspaces of the endomorphism on homology induced by the self-map. Using
a combinatorial gradient flow arising from the discrete Morse theory for Čech
and Delaunay complexes, we construct a chain map to transform the problem from
the natural but expensive Čech complexes to the computationally efficient Delaunay
triangulations. The fast chain map algorithm has applications beyond dynamical
systems.
acknowledgement: This research has been supported by the DFG Collaborative Research
Center SFB/TRR 109 “Discretization in Geometry and Dynamics”, by Polish MNiSzW Grant
No. 2621/7.PR/12/2013/2, by the Polish National Science Center under Maestro Grant
No. 2014/14/A/ST1/00453 and Grant No. DEC-2013/09/N/ST6/02995. Open Access funding
provided by Projekt DEAL.
article_processing_charge: Yes (via OA deal)
article_type: original
author:
- first_name: U.
full_name: Bauer, U.
last_name: Bauer
- first_name: Herbert
full_name: Edelsbrunner, Herbert
id: 3FB178DA-F248-11E8-B48F-1D18A9856A87
last_name: Edelsbrunner
orcid: 0000-0002-9823-6833
- first_name: Grzegorz
full_name: Jablonski, Grzegorz
id: 4483EF78-F248-11E8-B48F-1D18A9856A87
last_name: Jablonski
orcid: 0000-0002-3536-9866
- first_name: M.
full_name: Mrozek, M.
last_name: Mrozek
citation:
ama: Bauer U, Edelsbrunner H, Jablonski G, Mrozek M. Čech-Delaunay gradient flow
and homology inference for self-maps. Journal of Applied and Computational
Topology. 2020;4(4):455-480. doi:10.1007/s41468-020-00058-8
apa: Bauer, U., Edelsbrunner, H., Jablonski, G., & Mrozek, M. (2020). Čech-Delaunay
gradient flow and homology inference for self-maps. Journal of Applied and
Computational Topology. Springer Nature. https://doi.org/10.1007/s41468-020-00058-8
chicago: Bauer, U., Herbert Edelsbrunner, Grzegorz Jablonski, and M. Mrozek. “Čech-Delaunay
Gradient Flow and Homology Inference for Self-Maps.” Journal of Applied and
Computational Topology. Springer Nature, 2020. https://doi.org/10.1007/s41468-020-00058-8.
ieee: U. Bauer, H. Edelsbrunner, G. Jablonski, and M. Mrozek, “Čech-Delaunay gradient
flow and homology inference for self-maps,” Journal of Applied and Computational
Topology, vol. 4, no. 4. Springer Nature, pp. 455–480, 2020.
ista: Bauer U, Edelsbrunner H, Jablonski G, Mrozek M. 2020. Čech-Delaunay gradient
flow and homology inference for self-maps. Journal of Applied and Computational
Topology. 4(4), 455–480.
mla: Bauer, U., et al. “Čech-Delaunay Gradient Flow and Homology Inference for Self-Maps.”
Journal of Applied and Computational Topology, vol. 4, no. 4, Springer
Nature, 2020, pp. 455–80, doi:10.1007/s41468-020-00058-8.
short: U. Bauer, H. Edelsbrunner, G. Jablonski, M. Mrozek, Journal of Applied and
Computational Topology 4 (2020) 455–480.
date_created: 2024-03-04T10:47:49Z
date_published: 2020-12-01T00:00:00Z
date_updated: 2024-03-04T10:54:04Z
day: '01'
ddc:
- '500'
department:
- _id: HeEd
doi: 10.1007/s41468-020-00058-8
file:
- access_level: open_access
checksum: eed1168b6e66cd55272c19bb7fca8a1c
content_type: application/pdf
creator: dernst
date_created: 2024-03-04T10:52:42Z
date_updated: 2024-03-04T10:52:42Z
file_id: '15065'
file_name: 2020_JourApplCompTopology_Bauer.pdf
file_size: 851190
relation: main_file
success: 1
file_date_updated: 2024-03-04T10:52:42Z
has_accepted_license: '1'
intvolume: ' 4'
issue: '4'
language:
- iso: eng
month: '12'
oa: 1
oa_version: Published Version
page: 455-480
publication: Journal of Applied and Computational Topology
publication_identifier:
eissn:
- 2367-1734
issn:
- 2367-1726
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: Čech-Delaunay gradient flow and homology inference for self-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: 4
year: '2020'
...
---
_id: '6671'
abstract:
- lang: eng
text: 'In this paper we discuss three results. The first two concern general sets
of positive reach: we first characterize the reach of a closed set by means of
a bound on the metric distortion between the distance measured in the ambient
Euclidean space and the shortest path distance measured in the set. Secondly,
we prove that the intersection of a ball with radius less than the reach with
the set is geodesically convex, meaning that the shortest path between any two
points in the intersection lies itself in the intersection. For our third result
we focus on manifolds with positive reach and give a bound on the angle between
tangent spaces at two different points in terms of the reach and the distance
between the two points.'
article_processing_charge: Yes (via OA deal)
article_type: original
author:
- first_name: Jean-Daniel
full_name: Boissonnat, Jean-Daniel
last_name: Boissonnat
- 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: Boissonnat J-D, Lieutier A, Wintraecken M. The reach, metric distortion, geodesic
convexity and the variation of tangent spaces. Journal of Applied and Computational
Topology. 2019;3(1-2):29–58. doi:10.1007/s41468-019-00029-8
apa: Boissonnat, J.-D., Lieutier, A., & Wintraecken, M. (2019). The reach, metric
distortion, geodesic convexity and the variation of tangent spaces. Journal
of Applied and Computational Topology. Springer Nature. https://doi.org/10.1007/s41468-019-00029-8
chicago: Boissonnat, Jean-Daniel, André Lieutier, and Mathijs Wintraecken. “The
Reach, Metric Distortion, Geodesic Convexity and the Variation of Tangent Spaces.”
Journal of Applied and Computational Topology. Springer Nature, 2019. https://doi.org/10.1007/s41468-019-00029-8.
ieee: J.-D. Boissonnat, A. Lieutier, and M. Wintraecken, “The reach, metric distortion,
geodesic convexity and the variation of tangent spaces,” Journal of Applied
and Computational Topology, vol. 3, no. 1–2. Springer Nature, pp. 29–58, 2019.
ista: Boissonnat J-D, Lieutier A, Wintraecken M. 2019. The reach, metric distortion,
geodesic convexity and the variation of tangent spaces. Journal of Applied and
Computational Topology. 3(1–2), 29–58.
mla: Boissonnat, Jean-Daniel, et al. “The Reach, Metric Distortion, Geodesic Convexity
and the Variation of Tangent Spaces.” Journal of Applied and Computational
Topology, vol. 3, no. 1–2, Springer Nature, 2019, pp. 29–58, doi:10.1007/s41468-019-00029-8.
short: J.-D. Boissonnat, A. Lieutier, M. Wintraecken, Journal of Applied and Computational
Topology 3 (2019) 29–58.
corr_author: '1'
date_created: 2019-07-24T08:37:29Z
date_published: 2019-06-01T00:00:00Z
date_updated: 2025-04-14T07:44:06Z
day: '01'
ddc:
- '000'
department:
- _id: HeEd
doi: 10.1007/s41468-019-00029-8
ec_funded: 1
file:
- access_level: open_access
checksum: a5b244db9f751221409cf09c97ee0935
content_type: application/pdf
creator: dernst
date_created: 2019-07-31T08:09:56Z
date_updated: 2020-07-14T12:47:36Z
file_id: '6741'
file_name: 2019_JournAppliedComputTopol_Boissonnat.pdf
file_size: 2215157
relation: main_file
file_date_updated: 2020-07-14T12:47:36Z
has_accepted_license: '1'
intvolume: ' 3'
issue: 1-2
language:
- iso: eng
month: '06'
oa: 1
oa_version: Published Version
page: 29–58
project:
- _id: 260C2330-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '754411'
name: ISTplus - Postdoctoral Fellowships
- _id: B67AFEDC-15C9-11EA-A837-991A96BB2854
name: IST Austria Open Access Fund
publication: Journal of Applied and Computational Topology
publication_identifier:
eissn:
- 2367-1734
issn:
- 2367-1726
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: The reach, metric distortion, geodesic convexity and the variation of tangent
spaces
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: 3
year: '2019'
...
---
_id: '6774'
abstract:
- lang: eng
text: "A central problem of algebraic topology is to understand the homotopy groups
\ \U0001D70B\U0001D451(\U0001D44B) of a topological space X. For the computational
version of the problem, it is well known that there is no algorithm to decide
whether the fundamental group \U0001D70B1(\U0001D44B) of a given finite simplicial
complex X is trivial. On the other hand, there are several algorithms that, given
a finite simplicial complex X that is simply connected (i.e., with \U0001D70B1(\U0001D44B)
\ trivial), compute the higher homotopy group \U0001D70B\U0001D451(\U0001D44B)
\ for any given \U0001D451≥2 . However, these algorithms come with a caveat:
They compute the isomorphism type of \U0001D70B\U0001D451(\U0001D44B) , \U0001D451≥2
\ as an abstract finitely generated abelian group given by generators and relations,
but they work with very implicit representations of the elements of \U0001D70B\U0001D451(\U0001D44B)
. Converting elements of this abstract group into explicit geometric maps from
the d-dimensional sphere \U0001D446\U0001D451 to X has been one of the main
unsolved problems in the emerging field of computational homotopy theory. Here
we present an algorithm that, given a simply connected space X, computes \U0001D70B\U0001D451(\U0001D44B)
\ and represents its elements as simplicial maps from a suitable triangulation
of the d-sphere \U0001D446\U0001D451 to X. For fixed d, the algorithm runs
in time exponential in size(\U0001D44B) , the number of simplices of X. Moreover,
we prove that this is optimal: For every fixed \U0001D451≥2 , we construct a
family of simply connected spaces X such that for any simplicial map representing
a generator of \U0001D70B\U0001D451(\U0001D44B) , the size of the triangulation
of \U0001D446\U0001D451 on which the map is defined, is exponential in size(\U0001D44B)
."
article_type: original
author:
- first_name: Marek
full_name: Filakovský, Marek
id: 3E8AF77E-F248-11E8-B48F-1D18A9856A87
last_name: Filakovský
- first_name: Peter
full_name: Franek, Peter
id: 473294AE-F248-11E8-B48F-1D18A9856A87
last_name: Franek
orcid: 0000-0001-8878-8397
- first_name: Uli
full_name: Wagner, Uli
id: 36690CA2-F248-11E8-B48F-1D18A9856A87
last_name: Wagner
orcid: 0000-0002-1494-0568
- first_name: Stephan Y
full_name: Zhechev, Stephan Y
id: 3AA52972-F248-11E8-B48F-1D18A9856A87
last_name: Zhechev
citation:
ama: Filakovský M, Franek P, Wagner U, Zhechev SY. Computing simplicial representatives
of homotopy group elements. Journal of Applied and Computational Topology.
2018;2(3-4):177-231. doi:10.1007/s41468-018-0021-5
apa: Filakovský, M., Franek, P., Wagner, U., & Zhechev, S. Y. (2018). Computing
simplicial representatives of homotopy group elements. Journal of Applied and
Computational Topology. Springer. https://doi.org/10.1007/s41468-018-0021-5
chicago: Filakovský, Marek, Peter Franek, Uli Wagner, and Stephan Y Zhechev. “Computing
Simplicial Representatives of Homotopy Group Elements.” Journal of Applied
and Computational Topology. Springer, 2018. https://doi.org/10.1007/s41468-018-0021-5.
ieee: M. Filakovský, P. Franek, U. Wagner, and S. Y. Zhechev, “Computing simplicial
representatives of homotopy group elements,” Journal of Applied and Computational
Topology, vol. 2, no. 3–4. Springer, pp. 177–231, 2018.
ista: Filakovský M, Franek P, Wagner U, Zhechev SY. 2018. Computing simplicial representatives
of homotopy group elements. Journal of Applied and Computational Topology. 2(3–4),
177–231.
mla: Filakovský, Marek, et al. “Computing Simplicial Representatives of Homotopy
Group Elements.” Journal of Applied and Computational Topology, vol. 2,
no. 3–4, Springer, 2018, pp. 177–231, doi:10.1007/s41468-018-0021-5.
short: M. Filakovský, P. Franek, U. Wagner, S.Y. Zhechev, Journal of Applied and
Computational Topology 2 (2018) 177–231.
corr_author: '1'
date_created: 2019-08-08T06:47:40Z
date_published: 2018-12-01T00:00:00Z
date_updated: 2026-04-08T13:56:01Z
day: '01'
ddc:
- '514'
department:
- _id: UlWa
doi: 10.1007/s41468-018-0021-5
file:
- access_level: open_access
checksum: cf9e7fcd2a113dd4828774fc75cdb7e8
content_type: application/pdf
creator: dernst
date_created: 2019-08-08T06:55:21Z
date_updated: 2020-07-14T12:47:40Z
file_id: '6775'
file_name: 2018_JourAppliedComputTopology_Filakovsky.pdf
file_size: 1056278
relation: main_file
file_date_updated: 2020-07-14T12:47:40Z
has_accepted_license: '1'
intvolume: ' 2'
issue: 3-4
language:
- iso: eng
month: '12'
oa: 1
oa_version: Published Version
page: 177-231
project:
- _id: 25F8B9BC-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: M01980
name: Robust Invariants of Nonlinear Systems
- _id: 3AC91DDA-15DF-11EA-824D-93A3E7B544D1
call_identifier: FWF
name: FWF Open Access Fund
publication: Journal of Applied and Computational Topology
publication_identifier:
eissn:
- 2367-1734
issn:
- 2367-1726
publication_status: published
publisher: Springer
quality_controlled: '1'
related_material:
record:
- id: '6681'
relation: dissertation_contains
status: public
scopus_import: '1'
status: public
title: Computing simplicial representatives of homotopy group elements
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: 2
year: '2018'
...