---
OA_place: publisher
OA_type: hybrid
PlanS_conform: '1'
_id: '20456'
abstract:
- lang: eng
text: Given a locally finite set A⊆Rd and a coloring χ:A→{0,1,…,s}, we introduce
the chromatic Delaunay mosaic of χ, which is a Delaunay mosaic in Rs+d that represents
how points of different colors mingle. Our main results are bounds on the size
of the chromatic Delaunay mosaic, in which we assume that d and s are constants.
For example, if A is finite with n=#A, and the coloring is random, then the chromatic
Delaunay mosaic has O(n⌈d/2⌉) cells in expectation. In contrast, for Delone sets
and Poisson point processes in Rd, the expected number of cells within a closed
ball is only a constant times the number of points in this ball. Furthermore,
in R2 all colorings of a dense set of n points have chromatic Delaunay mosaics
of size O(n). This encourages the use of chromatic Delaunay mosaics in applications.
acknowledgement: The fourth author thanks Boris Aronov for insightful discussions
on the size of the overlay of Voronoi tessellations. Open 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
arxiv: 1
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: 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: Biswas R, Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M.
On the size of chromatic Delaunay mosaics. Discrete and Computational Geometry.
2026;75:24-47. doi:10.1007/s00454-025-00778-7
apa: Biswas, R., Cultrera di Montesano, S., Draganov, O., Edelsbrunner, H., &
Saghafian, M. (2026). On the size of chromatic Delaunay mosaics. Discrete and
Computational Geometry. Springer Nature. https://doi.org/10.1007/s00454-025-00778-7
chicago: Biswas, Ranita, Sebastiano Cultrera di Montesano, Ondrej Draganov, Herbert
Edelsbrunner, and Morteza Saghafian. “On the Size of Chromatic Delaunay Mosaics.”
Discrete and Computational Geometry. Springer Nature, 2026. https://doi.org/10.1007/s00454-025-00778-7.
ieee: R. Biswas, S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, and M.
Saghafian, “On the size of chromatic Delaunay mosaics,” Discrete and Computational
Geometry, vol. 75. Springer Nature, pp. 24–47, 2026.
ista: Biswas R, Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M.
2026. On the size of chromatic Delaunay mosaics. Discrete and Computational Geometry.
75, 24–47.
mla: Biswas, Ranita, et al. “On the Size of Chromatic Delaunay Mosaics.” Discrete
and Computational Geometry, vol. 75, Springer Nature, 2026, pp. 24–47, doi:10.1007/s00454-025-00778-7.
short: R. Biswas, S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, M. Saghafian,
Discrete and Computational Geometry 75 (2026) 24–47.
corr_author: '1'
date_created: 2025-10-12T22:01:26Z
date_published: 2026-01-01T00:00:00Z
date_updated: 2026-01-05T13:21:56Z
day: '01'
ddc:
- '510'
department:
- _id: HeEd
doi: 10.1007/s00454-025-00778-7
ec_funded: 1
external_id:
arxiv:
- '2212.03121'
isi:
- '001584166900001'
file:
- access_level: open_access
checksum: 0addb5c1b78142f9fb453bfa04695400
content_type: application/pdf
creator: dernst
date_created: 2026-01-05T13:21:20Z
date_updated: 2026-01-05T13:21:20Z
file_id: '20952'
file_name: 2026_DiscreteCompGeom_Biswas.pdf
file_size: 570922
relation: main_file
success: 1
file_date_updated: 2026-01-05T13:21:20Z
has_accepted_license: '1'
intvolume: ' 75'
isi: 1
language:
- iso: eng
month: '01'
oa: 1
oa_version: Published Version
page: 24-47
project:
- _id: 266A2E9E-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '788183'
name: Alpha Shape Theory Extended
- _id: 268116B8-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: Z00342
name: Mathematics, Computer Science
- _id: 2561EBF4-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: I02979-N35
name: Persistence and stability of geometric complexes
publication: Discrete and Computational Geometry
publication_identifier:
eissn:
- 1432-0444
issn:
- 0179-5376
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
related_material:
record:
- id: '15090'
relation: earlier_version
status: public
scopus_import: '1'
status: public
title: On the size of chromatic Delaunay mosaics
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: 75
year: '2026'
...
---
OA_place: repository
OA_type: green
_id: '20980'
abstract:
- lang: eng
text: 'Morse decompositions partition the flows in a vector field into equivalent
structures. Given such a decomposition, one can define a further summary of its
flow structure by what is called a connection matrix. These matrices, a generalization
of Morse boundary operators from classical Morse theory, capture the connections
made by the flows among the critical structures—such as attractors, repellers,
and orbits—in a vector field. Recently, in the context of combinatorial dynamics,
an efficient persistence-like algorithm to compute connection matrices has been
proposed in Dey, Lipiński, Mrozek, and Slechta [SIAM J. Appl. Dyn. Syst., 23 (2024),
pp. 81–97]. We show that, actually, the classical persistence algorithm with exhaustive
reduction retrieves connection matrices, both simplifying the algorithm of Dey
et al. and bringing the theory of persistence closer to combinatorial dynamical
systems. We supplement this main result with an observation: the concept of persistence
as defined for scalar fields naturally adapts to Morse decompositions whose Morse
sets are filtered with a Lyapunov function. We conclude by presenting preliminary
experimental results.'
acknowledgement: "This research was supported by NSF grants DMS-2301360 and CCF-2437030
as well as from the European Union's Horizon 2020 research and innovation programme
under Marie Sk\\lodowska-Curie grant 101034413.\r\n"
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Tamal K.
full_name: Dey, Tamal K.
last_name: Dey
- first_name: Andrew
full_name: Haas, Andrew
last_name: Haas
- first_name: Michał
full_name: Lipiński, Michał
id: dfffb474-4317-11ee-8f5c-fe3fc95a425e
last_name: Lipiński
orcid: 0000-0001-9789-9750
citation:
ama: Dey TK, Haas A, Lipiński M. Computing a connection matrix and persistence efficiently
from a morse decomposition. SIAM Journal on Applied Dynamical Systems.
2026;25(1):108-130. doi:10.1137/25m1739406
apa: Dey, T. K., Haas, A., & Lipiński, M. (2026). Computing a connection matrix
and persistence efficiently from a morse decomposition. SIAM Journal on Applied
Dynamical Systems. Society for Industrial & Applied Mathematics. https://doi.org/10.1137/25m1739406
chicago: Dey, Tamal K., Andrew Haas, and Michał Lipiński. “Computing a Connection
Matrix and Persistence Efficiently from a Morse Decomposition.” SIAM Journal
on Applied Dynamical Systems. Society for Industrial & Applied Mathematics,
2026. https://doi.org/10.1137/25m1739406.
ieee: T. K. Dey, A. Haas, and M. Lipiński, “Computing a connection matrix and persistence
efficiently from a morse decomposition,” SIAM Journal on Applied Dynamical
Systems, vol. 25, no. 1. Society for Industrial & Applied Mathematics,
pp. 108–130, 2026.
ista: Dey TK, Haas A, Lipiński M. 2026. Computing a connection matrix and persistence
efficiently from a morse decomposition. SIAM Journal on Applied Dynamical Systems.
25(1), 108–130.
mla: Dey, Tamal K., et al. “Computing a Connection Matrix and Persistence Efficiently
from a Morse Decomposition.” SIAM Journal on Applied Dynamical Systems,
vol. 25, no. 1, Society for Industrial & Applied Mathematics, 2026, pp. 108–30,
doi:10.1137/25m1739406.
short: T.K. Dey, A. Haas, M. Lipiński, SIAM Journal on Applied Dynamical Systems
25 (2026) 108–130.
date_created: 2026-01-12T11:17:06Z
date_published: 2026-01-01T00:00:00Z
date_updated: 2026-01-20T07:40:39Z
day: '01'
ddc:
- '510'
department:
- _id: HeEd
doi: 10.1137/25m1739406
ec_funded: 1
external_id:
arxiv:
- '2502.19369'
intvolume: ' 25'
issue: '1'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://doi.org/10.48550/arXiv.2502.19369
month: '01'
oa: 1
oa_version: Preprint
page: 108-130
project:
- _id: fc2ed2f7-9c52-11eb-aca3-c01059dda49c
call_identifier: H2020
grant_number: '101034413'
name: 'IST-BRIDGE: International postdoctoral program'
publication: SIAM Journal on Applied Dynamical Systems
publication_identifier:
issn:
- 1536-0040
publication_status: published
publisher: Society for Industrial & Applied Mathematics
quality_controlled: '1'
scopus_import: '1'
status: public
title: Computing a connection matrix and persistence efficiently from a morse decomposition
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 25
year: '2026'
...
---
OA_place: repository
OA_type: green
_id: '21056'
abstract:
- lang: eng
text: "In this work, we introduce and study what we believe is an intriguing, and,
to the best of our knowledge, previously unknown connection between two fundamental
areas in computational topology, namely topological data analysis (TDA) and knot
theory. Given a function from a topological space to ℝ, TDA provides tools to
simplify and study the importance of topological features: in particular, the
\U0001D459^\U0001D461ℎ-dimensional persistence diagram encodes the topological
changes (or \U0001D459-homology) in the sublevel set as the function value increases
into a set of points in the plane. Given a continuous one parameter family of
such functions, we can combine the persistence diagrams into an object known as
a vineyard, which tracks the evolution of points in the persistence diagram as
the function changes. If we further restrict that family of functions to be periodic,
we identify the two ends of the vineyard, yielding a closed vineyard. This allows
the study of monodromy, which in this context means that following the family
of functions for a period permutes the set of points in a non-trivial way. Recent
work has studied monodromy in the directional persistent homology transform, demonstrating
some interesting connections between an input shape and monodromy in the persistent
homology transform for 0-dimensional homology embedded in ℝ^2.\r\nIn this work,
given a link and a value \U0001D459, we construct a topological space (based on
the given link) and periodic family of functions on this space (based on the Euclidean
distance function), such that the closed \U0001D459-vineyard contains this link.
This shows that vineyards are topologically as rich as one could possibly hope,
suggesting many future directions of work. Importantly, it has at least two immediate
consequences we explicitly point out:\r\n1.\tMonodromy of any periodicity can
occur in a \U0001D459-vineyard for any \U0001D459. This answers a variant of a
question by Arya and collaborators. To exhibit this as a consequence of our first
main result we also reformulate monodromy in a more geometric way, which may be
of interest in itself.\r\n2.\tTopologically distinguishing closed vineyards is
likely to be difficult (from a complexity theory as well as from a practical perspective)
because of the difficulty of knot and link recognition, which have strong connections
to many NP-hard problems."
acknowledgement: We thank the reviewers for both SODA and ATMCS for their comments,
whichimproved the exposition. We thank Kate Turner for discussion and Clément Maria
for pointing out thatAlexander’s theorem was already (well) known. Mathijs Wintraecken
would like to express his gratitude tothe administrative support he received from
University of Notre Dame during his visit and from Sophie Honnoratand Stephanie
Verdonck at Inria in general.This work has been supported by the ANR grant StratMesh,
ANR-24-CE48-1899, by NSF award 2444309, andthe welcome package from IDEX of the
Université Côte d’Azur, ANR-15-IDEX-01.
article_processing_charge: No
arxiv: 1
author:
- first_name: Erin W.
full_name: Chambers, Erin W.
last_name: Chambers
- first_name: Christopher D
full_name: Fillmore, Christopher D
id: 35638A5C-AAC7-11E9-B0BF-5503E6697425
last_name: Fillmore
- first_name: Elizabeth R
full_name: Stephenson, Elizabeth R
id: 2D04F932-F248-11E8-B48F-1D18A9856A87
last_name: Stephenson
orcid: 0000-0002-6862-208X
- first_name: Mathijs
full_name: Wintraecken, Mathijs
id: 307CFBC8-F248-11E8-B48F-1D18A9856A87
last_name: Wintraecken
orcid: 0000-0002-7472-2220
citation:
ama: 'Chambers EW, Fillmore CD, Stephenson ER, Wintraecken M. Braiding Vineyards.
In: Green Larsen K, Saha B, eds. Proceedings of the 2026 Annual ACM-SIAM Symposium
on Discrete Algorithms. Philadelphia, PA, United States: Society for Industrial
and Applied Mathematics; 2026:6240-6263. doi:10.1137/1.9781611978971.225'
apa: 'Chambers, E. W., Fillmore, C. D., Stephenson, E. R., & Wintraecken, M.
(2026). Braiding Vineyards. In K. Green Larsen & B. Saha (Eds.), Proceedings
of the 2026 Annual ACM-SIAM Symposium on Discrete Algorithms (pp. 6240–6263).
Philadelphia, PA, United States: Society for Industrial and Applied Mathematics.
https://doi.org/10.1137/1.9781611978971.225'
chicago: 'Chambers, Erin W., Christopher D Fillmore, Elizabeth R Stephenson, and
Mathijs Wintraecken. “Braiding Vineyards.” In Proceedings of the 2026 Annual
ACM-SIAM Symposium on Discrete Algorithms, edited by Kasper Green Larsen and
Barna Saha, 6240–63. Philadelphia, PA, United States: Society for Industrial and
Applied Mathematics, 2026. https://doi.org/10.1137/1.9781611978971.225.'
ieee: 'E. W. Chambers, C. D. Fillmore, E. R. Stephenson, and M. Wintraecken, “Braiding
Vineyards,” in Proceedings of the 2026 Annual ACM-SIAM Symposium on Discrete
Algorithms, K. Green Larsen and B. Saha, Eds. Philadelphia, PA, United States:
Society for Industrial and Applied Mathematics, 2026, pp. 6240–6263.'
ista: 'Chambers EW, Fillmore CD, Stephenson ER, Wintraecken M. 2026.Braiding Vineyards.
In: Proceedings of the 2026 Annual ACM-SIAM Symposium on Discrete Algorithms.
, 6240–6263.'
mla: Chambers, Erin W., et al. “Braiding Vineyards.” Proceedings of the 2026
Annual ACM-SIAM Symposium on Discrete Algorithms, edited by Kasper Green Larsen
and Barna Saha, Society for Industrial and Applied Mathematics, 2026, pp. 6240–63,
doi:10.1137/1.9781611978971.225.
short: E.W. Chambers, C.D. Fillmore, E.R. Stephenson, M. Wintraecken, in:, K. Green
Larsen, B. Saha (Eds.), Proceedings of the 2026 Annual ACM-SIAM Symposium on Discrete
Algorithms, Society for Industrial and Applied Mathematics, Philadelphia, PA,
United States, 2026, pp. 6240–6263.
date_created: 2026-01-28T12:58:16Z
date_published: 2026-01-07T00:00:00Z
date_updated: 2026-02-16T08:06:23Z
day: '07'
department:
- _id: HeEd
doi: 10.1137/1.9781611978971.225
editor:
- first_name: Kasper
full_name: Green Larsen, Kasper
last_name: Green Larsen
- first_name: Barna
full_name: Saha, Barna
last_name: Saha
external_id:
arxiv:
- '2504.11203'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://doi.org/10.48550/arXiv.2504.11203
month: '01'
oa: 1
oa_version: Preprint
page: 6240-6263
place: Philadelphia, PA, United States
publication: Proceedings of the 2026 Annual ACM-SIAM Symposium on Discrete Algorithms
publication_identifier:
eisbn:
- '9781611978971'
publication_status: published
publisher: Society for Industrial and Applied Mathematics
quality_controlled: '1'
related_material:
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- id: '21051'
relation: earlier_version
status: public
status: public
title: Braiding Vineyards
type: book_chapter
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2026'
...
---
DOAJ_listed: '1'
OA_place: publisher
OA_type: gold
PlanS_conform: '1'
_id: '21115'
abstract:
- lang: eng
text: Quantifying cell morphology is central to understanding cellular regulation,
fate, and heterogeneity, yet conventional image-based analyses often struggle
with diverse or irregular shapes. We present a computational framework that uses
topological data analysis to characterise and compare single-cell morphologies
from fluorescence microscopy. Each cell is represented by its contour together
with the position of its nucleus, from which we construct a filtration based on
a radial distance function and derive a persistence diagram encoding the shape’s
topological evolution. The similarity between two cells is quantified using the
2-Wasserstein distance between their diagrams, yielding a shape distance we call
the PH distance. We apply this method to two representative experimental systems—primary
human mesenchymal stem cells (hMSCs) and HeLa cells—and show that PH distances
enable the detection of outliers in those systems, the identification of sub-populations,
and the quantification of shape heterogeneity. We benchmark PH against three established
contour-based distances (aspect ratio, Fourier descriptors, and elastic shape
analysis) and show that PH offers better separation between cell types and greater
robustness when clustering heterogeneous populations. Together, these results
demonstrate that persistent-homology-based signatures provide a principled and
sensitive approach for analysing cell morphology in settings where traditional
geometric or image-based descriptors are insufficient.
acknowledgement: We thank Stephan Huckemann, Katharine Turner, Benjamin Eltzner, Stephan
Tillmann, Fariza Rashid, Vanessa Robins, and Lamiae Azizi for many useful discussions
at various stages of this project. FR and PY gratefully acknowledge Matthias Weiss
(Experimental Physics I, University of Bayreuth, Germany) for granting access to
cell culture and laboratories, as well as funding consumables and the fruitful discussion
that contributed to this work. For open access purposes, the author has applied
a CC BY public copyright license to any author-accepted manuscript version arising
from this submission.
article_number: e1013890
article_processing_charge: Yes
article_type: original
author:
- first_name: Yossi
full_name: Bleile, Yossi
id: 920a7385-7995-11ef-9bfd-8c434cd8f3c2
last_name: Bleile
orcid: 0000-0002-4861-9174
- first_name: Pooja
full_name: Yadav, Pooja
last_name: Yadav
- first_name: Patrice
full_name: Koehl, Patrice
last_name: Koehl
- first_name: Florian
full_name: Rehfeldt, Florian
last_name: Rehfeldt
citation:
ama: 'Bleile Y, Yadav P, Koehl P, Rehfeldt F. Persistence diagrams as morphological
signatures of cells: A method to measure and compare cells within a population.
PLoS Computational Biology. 2026;22. doi:10.1371/journal.pcbi.1013890'
apa: 'Bleile, Y., Yadav, P., Koehl, P., & Rehfeldt, F. (2026). Persistence diagrams
as morphological signatures of cells: A method to measure and compare cells within
a population. PLoS Computational Biology. Public Library of Science. https://doi.org/10.1371/journal.pcbi.1013890'
chicago: 'Bleile, Yossi, Pooja Yadav, Patrice Koehl, and Florian Rehfeldt. “Persistence
Diagrams as Morphological Signatures of Cells: A Method to Measure and Compare
Cells within a Population.” PLoS Computational Biology. Public Library
of Science, 2026. https://doi.org/10.1371/journal.pcbi.1013890.'
ieee: 'Y. Bleile, P. Yadav, P. Koehl, and F. Rehfeldt, “Persistence diagrams as
morphological signatures of cells: A method to measure and compare cells within
a population,” PLoS Computational Biology, vol. 22. Public Library of Science,
2026.'
ista: 'Bleile Y, Yadav P, Koehl P, Rehfeldt F. 2026. Persistence diagrams as morphological
signatures of cells: A method to measure and compare cells within a population.
PLoS Computational Biology. 22, e1013890.'
mla: 'Bleile, Yossi, et al. “Persistence Diagrams as Morphological Signatures of
Cells: A Method to Measure and Compare Cells within a Population.” PLoS Computational
Biology, vol. 22, e1013890, Public Library of Science, 2026, doi:10.1371/journal.pcbi.1013890.'
short: Y. Bleile, P. Yadav, P. Koehl, F. Rehfeldt, PLoS Computational Biology 22
(2026).
corr_author: '1'
date_created: 2026-01-30T10:36:32Z
date_published: 2026-01-28T00:00:00Z
date_updated: 2026-02-12T14:23:54Z
day: '28'
ddc:
- '000'
department:
- _id: HeEd
doi: 10.1371/journal.pcbi.1013890
external_id:
pmid:
- '41604421'
file:
- access_level: open_access
checksum: 3899d929ee9be0453c95524e49992d72
content_type: application/pdf
creator: dernst
date_created: 2026-02-10T07:13:06Z
date_updated: 2026-02-10T07:13:06Z
file_id: '21204'
file_name: 2026_PloSCompBio_Bleile.pdf
file_size: 8908746
relation: main_file
success: 1
file_date_updated: 2026-02-10T07:13:06Z
has_accepted_license: '1'
intvolume: ' 22'
language:
- iso: eng
month: '01'
oa: 1
oa_version: Published Version
pmid: 1
publication: PLoS Computational Biology
publication_identifier:
issn:
- 1553-7358
publication_status: published
publisher: Public Library of Science
quality_controlled: '1'
related_material:
link:
- relation: software
url: https://github.com/yossibokorbleile/correa
scopus_import: '1'
status: public
title: 'Persistence diagrams as morphological signatures of cells: A method to measure
and compare cells within a population'
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: 22
year: '2026'
...
---
OA_place: publisher
OA_type: hybrid
PlanS_conform: '1'
_id: '21232'
abstract:
- lang: eng
text: "Abstract\r\n In this paper,
we consider a simple class of stratified spaces – 2-complexes. We present an algorithm
that learns the abstract structure of an embedded 2-complex from a point cloud
sampled from it. We use tools and inspiration from computational geometry, algebraic
topology, and topological data analysis and prove the correctness of the identified
abstract structure under assumptions on the embedding."
acknowledgement: The author would like to thank Kate Turner, Chris Williams, Jonathan
Spreer, Stephan Tillmann, Vanessa Robins, Vigleik Angeltveit, Martin Helmer, and
James Morgan for very helpful discussions; and thanks Sara Kališnik Hintz and Paul
Bendich for comments on an earlier version. Additonally, the author would like to
thank both reviewers for their very insightful and helpful comments, without which
the paper would be infinitely less coherent than it currently is. Open access funding
provided by Institute of Science and Technology (IST Austria). The work in this
paper was supported by an Australian Federal Government Grant, 2019-2022, Stratified
Space Learning.
article_number: '17'
article_processing_charge: Yes (via OA deal)
article_type: original
arxiv: 1
author:
- first_name: Yossi
full_name: Bleile, Yossi
id: 920a7385-7995-11ef-9bfd-8c434cd8f3c2
last_name: Bleile
orcid: 0000-0002-4861-9174
citation:
ama: 'Bleile Y. Towards stratified space learning: 2-complexes. La Matematica.
2026;5. doi:10.1007/s44007-025-00183-9'
apa: 'Bleile, Y. (2026). Towards stratified space learning: 2-complexes. La Matematica.
Springer Nature. https://doi.org/10.1007/s44007-025-00183-9'
chicago: 'Bleile, Yossi. “Towards Stratified Space Learning: 2-Complexes.” La
Matematica. Springer Nature, 2026. https://doi.org/10.1007/s44007-025-00183-9.'
ieee: 'Y. Bleile, “Towards stratified space learning: 2-complexes,” La Matematica,
vol. 5. Springer Nature, 2026.'
ista: 'Bleile Y. 2026. Towards stratified space learning: 2-complexes. La Matematica.
5, 17.'
mla: 'Bleile, Yossi. “Towards Stratified Space Learning: 2-Complexes.” La Matematica,
vol. 5, 17, Springer Nature, 2026, doi:10.1007/s44007-025-00183-9.'
short: Y. Bleile, La Matematica 5 (2026).
corr_author: '1'
date_created: 2026-02-16T10:44:44Z
date_published: 2026-02-08T00:00:00Z
date_updated: 2026-02-23T10:20:10Z
day: '08'
ddc:
- '510'
department:
- _id: HeEd
doi: 10.1007/s44007-025-00183-9
external_id:
arxiv:
- '2305.02724'
file:
- access_level: open_access
checksum: 6cae2efb47b025af22a8539c606a4e09
content_type: application/pdf
creator: dernst
date_created: 2026-02-23T10:18:52Z
date_updated: 2026-02-23T10:18:52Z
file_id: '21347'
file_name: 2026_LaMatematica_Bleile.pdf
file_size: 15051582
relation: main_file
success: 1
file_date_updated: 2026-02-23T10:18:52Z
has_accepted_license: '1'
intvolume: ' 5'
language:
- iso: eng
month: '02'
oa: 1
oa_version: Published Version
publication: La Matematica
publication_identifier:
issn:
- 2730-9657
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: 'Towards stratified space learning: 2-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: 5
year: '2026'
...
---
OA_place: repository
OA_type: green
_id: '21374'
abstract:
- lang: eng
text: "Let . S be a set of distinct points in general position in the\r\nEuclidean
plane. A plane Hamiltonian path on . S is a crossing-free geometric path such
that every point of .S is a vertex of the path. It is\r\nknown that, if. S is
sufficiently large, there exist three edge-disjoint plane\r\nHamiltonian paths
on . S. In this paper we study an edge-constrained\r\nversion of the problem of
finding Hamiltonian paths on a point set. We\r\nfirst consider the problem of
finding a single plane Hamiltonian path . π\r\nwith endpoints .s, t ∈ S and constraints
given by a segment . ab, where\r\n.a, b ∈ S. We consider the following scenarios:
(i) .ab ∈ π; (ii) .ab π. We\r\ncharacterize those quintuples . S, a, b, s, t for
which . π exists. Secondly,\r\nwe consider the problem of finding two plane Hamiltonian
paths . π1, π2\r\non a set . S with constraints given by a segment . ab, where
.a, b ∈ S. We\r\nconsider the following scenarios: (i) .π1 and .π2 share no edges
and .ab is\r\nan edge of . π1; (ii) .π1 and .π2 share no edges and none of them
includes\r\n.ab as an edge; (iii) both .π1 and .π2 include .ab as an edge and
share no\r\nother edges. In all cases, we characterize those triples . S, a, b
for which\r\n.π1 and .π2 exist."
acknowledgement: "We thank the organizers of the HOMONOLO 2024 workshop in Nová Louka,
Czech Republic, for the fruitful atmosphere where the research on this project was
initiated.\r\n\r\nT. Antić, A. Džuklevski, J. Kratochvíl and M. Saumell received
funding from GAČR grant 23–04949X, T.A and A.Dž were additionally supported by GAUK
grant SVV–2025–260822. G. Liotta was supported in part by MUR of Italy, PRIN Project
no. 2022TS4Y3N – EXPAND and PON Project ARS01_00540. J. Fiala was in part supported
by GAČR grant 25-16847S."
alternative_title:
- LNCS
article_processing_charge: No
arxiv: 1
author:
- first_name: Todor
full_name: Antić, Todor
last_name: Antić
- first_name: Aleksa
full_name: Džuklevski, Aleksa
last_name: Džuklevski
- first_name: Jiří
full_name: Fiala, Jiří
last_name: Fiala
- first_name: Jan
full_name: Kratochvíl, Jan
last_name: Kratochvíl
- first_name: Giuseppe
full_name: Liotta, Giuseppe
last_name: Liotta
- first_name: Morteza
full_name: Saghafian, Morteza
id: f86f7148-b140-11ec-9577-95435b8df824
last_name: Saghafian
- first_name: Maria
full_name: Saumell, Maria
last_name: Saumell
- first_name: Johannes
full_name: Zink, Johannes
last_name: Zink
citation:
ama: 'Antić T, Džuklevski A, Fiala J, et al. Edge-constrained Hamiltonian paths
on a point set. In: 51st International Conference on Current Trends in Theory
and Practice of Computer Science. Vol 16448. Springer Nature; 2026:532-546.
doi:10.1007/978-3-032-17801-5_39'
apa: 'Antić, T., Džuklevski, A., Fiala, J., Kratochvíl, J., Liotta, G., Saghafian,
M., … Zink, J. (2026). Edge-constrained Hamiltonian paths on a point set. In 51st
International Conference on Current Trends in Theory and Practice of Computer
Science (Vol. 16448, pp. 532–546). Krakow, Poland: Springer Nature. https://doi.org/10.1007/978-3-032-17801-5_39'
chicago: Antić, Todor, Aleksa Džuklevski, Jiří Fiala, Jan Kratochvíl, Giuseppe Liotta,
Morteza Saghafian, Maria Saumell, and Johannes Zink. “Edge-Constrained Hamiltonian
Paths on a Point Set.” In 51st International Conference on Current Trends in
Theory and Practice of Computer Science, 16448:532–46. Springer Nature, 2026.
https://doi.org/10.1007/978-3-032-17801-5_39.
ieee: T. Antić et al., “Edge-constrained Hamiltonian paths on a point set,”
in 51st International Conference on Current Trends in Theory and Practice of
Computer Science, Krakow, Poland, 2026, vol. 16448, pp. 532–546.
ista: 'Antić T, Džuklevski A, Fiala J, Kratochvíl J, Liotta G, Saghafian M, Saumell
M, Zink J. 2026. Edge-constrained Hamiltonian paths on a point set. 51st International
Conference on Current Trends in Theory and Practice of Computer Science. SOFSEM:
Conference on Current Trends in Theory and Practice of Computer Science, LNCS,
vol. 16448, 532–546.'
mla: Antić, Todor, et al. “Edge-Constrained Hamiltonian Paths on a Point Set.” 51st
International Conference on Current Trends in Theory and Practice of Computer
Science, vol. 16448, Springer Nature, 2026, pp. 532–46, doi:10.1007/978-3-032-17801-5_39.
short: T. Antić, A. Džuklevski, J. Fiala, J. Kratochvíl, G. Liotta, M. Saghafian,
M. Saumell, J. Zink, in:, 51st International Conference on Current Trends in Theory
and Practice of Computer Science, Springer Nature, 2026, pp. 532–546.
conference:
end_date: 2026-02-13
location: Krakow, Poland
name: 'SOFSEM: Conference on Current Trends in Theory and Practice of Computer Science'
start_date: 2026-02-09
date_created: 2026-03-01T23:01:40Z
date_published: 2026-02-13T00:00:00Z
date_updated: 2026-03-02T08:49:20Z
day: '13'
department:
- _id: HeEd
doi: 10.1007/978-3-032-17801-5_39
external_id:
arxiv:
- '2511.22526'
intvolume: ' 16448'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://doi.org/10.48550/arXiv.2511.22526
month: '02'
oa: 1
oa_version: Preprint
page: 532-546
publication: 51st International Conference on Current Trends in Theory and Practice
of Computer Science
publication_identifier:
eissn:
- 1611-3349
isbn:
- '9783032178008'
issn:
- 0302-9743
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: Edge-constrained Hamiltonian paths on a point set
type: conference
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 16448
year: '2026'
...
---
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: repository
OA_type: green
_id: '21410'
abstract:
- lang: eng
text: Given a finite set of red and blue points in R^d, the MST-ratio is defined
as the total length of the Euclidean minimum spanning trees of the red points
and the blue points, divided by the length of the Euclidean minimum spanning tree
of their union. The MST-ratio has recently gained attention due to its direct
interpretation in topological models for studying point sets with applications
in spatial biology. The maximum MST-ratio of a point set is the maximum MST-ratio
over all proper colorings of its points by red and blue. We prove that finding
the maximum MST-ratio of a given point set is NP-hard when the dimension is part
of the input. Moreover, we present a quadratic-time 3-approximation algorithm
for this problem. As part of the proof, we show that in any metric space, the
maximum MST-ratio is smaller than 3. Furthermore, we study the average MST-ratio
over all colorings of a set of n points. We show that this average is always at
least n-2/n-1, and for n random points uniformly distributed in a d-dimensional
unit cube, the average tends to (math formular) in expectation as n approaches
infinity.
acknowledgement: "A. J. Ameli—Supported by the project COALESCE (ERC grant no. 853234).\r\nM.
Saghafian—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: Afrouz
full_name: Jabal Ameli, Afrouz
last_name: Jabal Ameli
- first_name: Faezeh
full_name: Motiei, Faezeh
last_name: Motiei
- first_name: Morteza
full_name: Saghafian, Morteza
id: f86f7148-b140-11ec-9577-95435b8df824
last_name: Saghafian
citation:
ama: 'Jabal Ameli A, Motiei F, Saghafian M. On the MST-ratio: Theoretical bounds
and complexity of finding the maximum. In: 20th International Conference and
Workshops on Algorithms and Computation. Vol 16444. Springer Nature; 2026:386-401.
doi:10.1007/978-981-95-7127-7_26'
apa: 'Jabal Ameli, A., Motiei, F., & Saghafian, M. (2026). On the MST-ratio:
Theoretical bounds and complexity of finding the maximum. In 20th International
Conference and Workshops on Algorithms and Computation (Vol. 16444, pp. 386–401).
Perugia, Italy: Springer Nature. https://doi.org/10.1007/978-981-95-7127-7_26'
chicago: 'Jabal Ameli, Afrouz, Faezeh Motiei, and Morteza Saghafian. “On the MST-Ratio:
Theoretical Bounds and Complexity of Finding the Maximum.” In 20th International
Conference and Workshops on Algorithms and Computation, 16444:386–401. Springer
Nature, 2026. https://doi.org/10.1007/978-981-95-7127-7_26.'
ieee: 'A. Jabal Ameli, F. Motiei, and M. Saghafian, “On the MST-ratio: Theoretical
bounds and complexity of finding the maximum,” in 20th International Conference
and Workshops on Algorithms and Computation, Perugia, Italy, 2026, vol. 16444,
pp. 386–401.'
ista: 'Jabal Ameli A, Motiei F, Saghafian M. 2026. On the MST-ratio: Theoretical
bounds and complexity of finding the maximum. 20th International Conference and
Workshops on Algorithms and Computation. WALCOM: International Conference and
Workshops on Algorithms and Computation, LNCS, vol. 16444, 386–401.'
mla: 'Jabal Ameli, Afrouz, et al. “On the MST-Ratio: Theoretical Bounds and Complexity
of Finding the Maximum.” 20th International Conference and Workshops on Algorithms
and Computation, vol. 16444, Springer Nature, 2026, pp. 386–401, doi:10.1007/978-981-95-7127-7_26.'
short: A. Jabal Ameli, F. Motiei, M. Saghafian, in:, 20th International Conference
and Workshops on Algorithms and Computation, Springer Nature, 2026, pp. 386–401.
conference:
end_date: 2026-03-06
location: Perugia, Italy
name: 'WALCOM: International Conference and Workshops on Algorithms and Computation'
start_date: 2026-03-04
date_created: 2026-03-08T23:01:45Z
date_published: 2026-02-14T00:00:00Z
date_updated: 2026-03-09T10:25:41Z
day: '14'
department:
- _id: HeEd
doi: 10.1007/978-981-95-7127-7_26
ec_funded: 1
external_id:
arxiv:
- '2409.11079'
intvolume: ' 16444'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://doi.org/10.48550/arXiv.2409.11079
month: '02'
oa: 1
oa_version: Preprint
page: 386-401
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: 20th International Conference and Workshops on Algorithms and Computation
publication_identifier:
eissn:
- 1611-3349
isbn:
- '9789819571260'
issn:
- 0302-9743
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: 'On the MST-ratio: Theoretical bounds and complexity of finding the maximum'
type: conference
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 16444
year: '2026'
...
---
OA_place: publisher
_id: '21021'
abstract:
- lang: eng
text: This thesis examines how geometry and topology intersect in the representation,
transformation, and analysis of complex shapes. It considers how continuous manifolds
relate to their discrete analogues, how topological structures evolve in persistence
vineyards, and how tools from topological data analysis can illuminate problems
in mathematical physics. Central to this exploration is the question of how structure,
both geometric and topological, persists or changes under approximation, sampling,
or deformation. The work develops new approaches to skeletal and grid-based representations
of surfaces, reveals the full expressive capacity of persistence vineyards, and
applies topological methods to the longstanding problem of equilibria in electrostatic
fields. These threads braid together into a broader understanding of how topology
and geometry inform one another across theory, computation, and application.
acknowledged_ssus:
- _id: M-Shop
- _id: ScienComp
acknowledgement: "The research presented in this thesis was funded by the DFG Collaborative
Research\r\nCenter TRR 109, ‘Discretization in Geometry and Dynamics’.\r\n"
alternative_title:
- ISTA Thesis
article_processing_charge: No
author:
- first_name: Christopher D
full_name: Fillmore, Christopher D
id: 35638A5C-AAC7-11E9-B0BF-5503E6697425
last_name: Fillmore
citation:
ama: Fillmore CD. Braiding geometry and topology to study shapes and data. 2026.
doi:10.15479/AT-ISTA-21021
apa: Fillmore, C. D. (2026). Braiding geometry and topology to study shapes and
data. Institute of Science and Technology Austria. https://doi.org/10.15479/AT-ISTA-21021
chicago: Fillmore, Christopher D. “Braiding Geometry and Topology to Study Shapes
and Data.” Institute of Science and Technology Austria, 2026. https://doi.org/10.15479/AT-ISTA-21021.
ieee: C. D. Fillmore, “Braiding geometry and topology to study shapes and data,”
Institute of Science and Technology Austria, 2026.
ista: Fillmore CD. 2026. Braiding geometry and topology to study shapes and data.
Institute of Science and Technology Austria.
mla: Fillmore, Christopher D. Braiding Geometry and Topology to Study Shapes
and Data. Institute of Science and Technology Austria, 2026, doi:10.15479/AT-ISTA-21021.
short: C.D. Fillmore, Braiding Geometry and Topology to Study Shapes and Data, Institute
of Science and Technology Austria, 2026.
corr_author: '1'
date_created: 2026-01-20T21:38:40Z
date_published: 2026-01-21T00:00:00Z
date_updated: 2026-04-07T11:42:49Z
day: '21'
ddc:
- '514'
- '516'
degree_awarded: PhD
department:
- _id: GradSch
- _id: HeEd
- _id: UlWa
doi: 10.15479/AT-ISTA-21021
file:
- access_level: open_access
checksum: 4c0889130095c31d4e5088c5b8dfd607
content_type: application/pdf
creator: cfillmor
date_created: 2026-01-26T19:44:46Z
date_updated: 2026-01-30T11:40:09Z
file_id: '21046'
file_name: 2025_Fillmore_Christopher_Thesis.pdf
file_size: 55954297
relation: main_file
- access_level: closed
checksum: d69afb71d82ab98f856886126ee7303a
content_type: application/x-zip-compressed
creator: cfillmor
date_created: 2026-01-26T19:46:20Z
date_updated: 2026-01-26T19:46:20Z
file_id: '21047'
file_name: Thesis.zip
file_size: 166080788
relation: source_file
file_date_updated: 2026-01-30T11:40:09Z
has_accepted_license: '1'
language:
- iso: eng
month: '01'
oa: 1
oa_version: Published Version
page: '122'
publication_identifier:
issn:
- 2663-337X
publication_status: published
publisher: Institute of Science and Technology Austria
related_material:
record:
- id: '20260'
relation: part_of_dissertation
status: public
- id: '21050'
relation: part_of_dissertation
status: public
- id: '21051'
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
- first_name: Uli
full_name: Wagner, Uli
id: 36690CA2-F248-11E8-B48F-1D18A9856A87
last_name: Wagner
orcid: 0000-0002-1494-0568
title: Braiding geometry and topology to study shapes and data
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: '2026'
...
---
OA_place: repository
_id: '21051'
abstract:
- lang: eng
text: 'In this work, we introduce and study what we believe is an intriguing and,
to the best of our knowledge, previously unknown connection between two areas
in computational topology, topological data analysis (TDA) and knot theory. Given
a function from a topological space to $\mathbb{R}$, TDA provides tools to simplify
and study the importance of topological features: in particular, the $l^{th}$-dimensional
persistence diagram encodes the $l$-homology in the sublevel set as the function
value increases as a set of points in the plane. Given a continuous one-parameter
family of such functions, we can combine the persistence diagrams into an object
known as a vineyard, which track the evolution of points in the persistence diagram.
If we further restrict that family of functions to be periodic, we identify the
two ends of the vineyard, yielding a closed vineyard. This allows the study of
monodromy, which in this context means that following the family of functions
for a period permutes the set of points in a non-trivial way. In this work, given
a link and value $l$, we construct a topological space and periodic family of
functions such that the closed $l$-vineyard contains this link. This shows that
vineyards are topologically as rich as one could possibly hope. Importantly, it
has at least two immediate consequences: First, monodromy of any periodicity can
occur in a $l$-vineyard, answering a variant of a question by [Arya et al 2024].
To exhibit this, we also reformulate monodromy in a more geometric way, which
may be of interest in itself. Second, distinguishing vineyards is likely to be
difficult given the known difficulty of knot and link recognition, which have
strong connections to many NP-hard problems.'
article_processing_charge: No
arxiv: 1
author:
- first_name: Erin
full_name: ' Chambers, Erin'
last_name: ' Chambers'
- first_name: Christopher D
full_name: Fillmore, Christopher D
id: 35638A5C-AAC7-11E9-B0BF-5503E6697425
last_name: Fillmore
- first_name: Elizabeth R
full_name: Stephenson, Elizabeth R
id: 2D04F932-F248-11E8-B48F-1D18A9856A87
last_name: Stephenson
orcid: 0000-0002-6862-208X
- first_name: Mathijs
full_name: Wintraecken, Mathijs
id: 307CFBC8-F248-11E8-B48F-1D18A9856A87
last_name: Wintraecken
orcid: 0000-0002-7472-2220
citation:
ama: Chambers E, Fillmore CD, Stephenson ER, Wintraecken M. Braiding vineyards.
arXiv. doi:10.48550/ARXIV.2504.11203
apa: Chambers, E., Fillmore, C. D., Stephenson, E. R., & Wintraecken, M. (n.d.).
Braiding vineyards. arXiv. https://doi.org/10.48550/ARXIV.2504.11203
chicago: Chambers, Erin, Christopher D Fillmore, Elizabeth R Stephenson, and Mathijs
Wintraecken. “Braiding Vineyards.” ArXiv, n.d. https://doi.org/10.48550/ARXIV.2504.11203.
ieee: E. Chambers, C. D. Fillmore, E. R. Stephenson, and M. Wintraecken, “Braiding
vineyards,” arXiv. .
ista: Chambers E, Fillmore CD, Stephenson ER, Wintraecken M. Braiding vineyards.
arXiv, 10.48550/ARXIV.2504.11203.
mla: Chambers, Erin, et al. “Braiding Vineyards.” ArXiv, doi:10.48550/ARXIV.2504.11203.
short: E. Chambers, C.D. Fillmore, E.R. Stephenson, M. Wintraecken, ArXiv (n.d.).
corr_author: '1'
date_created: 2026-01-27T14:41:44Z
date_published: 2026-01-02T00:00:00Z
date_updated: 2026-04-07T11:42:48Z
day: '02'
department:
- _id: HeEd
doi: 10.48550/ARXIV.2504.11203
external_id:
arxiv:
- '2504.11203'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://doi.org/10.48550/arXiv.2504.11203
month: '01'
oa: 1
oa_version: Preprint
publication: arXiv
publication_status: draft
related_material:
record:
- id: '21056'
relation: later_version
status: public
- id: '21021'
relation: dissertation_contains
status: public
status: public
title: Braiding vineyards
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: '2026'
...
---
OA_place: repository
OA_type: green
_id: '21781'
abstract:
- lang: eng
text: "Given a set A of n points (vertices) in general position in the plane, the
complete geometric graph \r\nKn[A] consists of all (n2) segments (edges) between
the elements of A. It is known that the edge set of every complete geometric graph
on n vertices can be partitioned into O(n3∕2) crossing-free paths (or matchings).
We strengthen this result under various additional assumptions on the point set.
In particular, we prove that for a set A of n randomly selected points, uniformly
distributed in [0,1]2, with probability tending to 1 as n→∞, the edge set of Kn[A]
can be covered by O(nlogn) crossing-free paths and by O(n√logn) crossing-free
matchings. On the other hand, we construct n-element point sets such that covering
the edge set of Kn[A] requires a quadratic number of monotone paths."
acknowledgement: "Research partially supported by ERC Advanced Grant \"GeoScape\",
no. 882971 and\r\nHungarian NKFIH grant no. K-131529. Work by the third author is
supported by EPSRC grant\r\nEP/X013642/1. Work by the third author 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."
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Adrian
full_name: Dumitrescu, Adrian
last_name: Dumitrescu
- first_name: János
full_name: Pach, János
last_name: Pach
- first_name: Morteza
full_name: Saghafian, Morteza
id: f86f7148-b140-11ec-9577-95435b8df824
last_name: Saghafian
- first_name: Alex
full_name: Scott, Alex
last_name: Scott
citation:
ama: Dumitrescu A, Pach J, Saghafian M, Scott A. Covering complete geometric graphs
by monotone paths. Combinatorics and Number Theory. 2026;15(1):73-82. doi:10.2140/cnt.2026.15.73
apa: Dumitrescu, A., Pach, J., Saghafian, M., & Scott, A. (2026). Covering complete
geometric graphs by monotone paths. Combinatorics and Number Theory. Mathematical
Sciences Publishers. https://doi.org/10.2140/cnt.2026.15.73
chicago: Dumitrescu, Adrian, János Pach, Morteza Saghafian, and Alex Scott. “Covering
Complete Geometric Graphs by Monotone Paths.” Combinatorics and Number Theory.
Mathematical Sciences Publishers, 2026. https://doi.org/10.2140/cnt.2026.15.73.
ieee: A. Dumitrescu, J. Pach, M. Saghafian, and A. Scott, “Covering complete geometric
graphs by monotone paths,” Combinatorics and Number Theory, vol. 15, no.
1. Mathematical Sciences Publishers, pp. 73–82, 2026.
ista: Dumitrescu A, Pach J, Saghafian M, Scott A. 2026. Covering complete geometric
graphs by monotone paths. Combinatorics and Number Theory. 15(1), 73–82.
mla: Dumitrescu, Adrian, et al. “Covering Complete Geometric Graphs by Monotone
Paths.” Combinatorics and Number Theory, vol. 15, no. 1, Mathematical Sciences
Publishers, 2026, pp. 73–82, doi:10.2140/cnt.2026.15.73.
short: A. Dumitrescu, J. Pach, M. Saghafian, A. Scott, Combinatorics and Number
Theory 15 (2026) 73–82.
date_created: 2026-05-03T22:01:37Z
date_published: 2026-04-17T00:00:00Z
date_updated: 2026-05-07T07:45:24Z
day: '17'
department:
- _id: HeEd
doi: 10.2140/cnt.2026.15.73
ec_funded: 1
external_id:
arxiv:
- '2507.10840'
intvolume: ' 15'
issue: '1'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://doi.org/10.48550/arXiv.2507.10840
month: '04'
oa: 1
oa_version: Preprint
page: 73-82
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: Combinatorics and Number Theory
publication_identifier:
eissn:
- 2996-220X
issn:
- 2996-2196
publication_status: published
publisher: Mathematical Sciences Publishers
quality_controlled: '1'
scopus_import: '1'
status: public
title: Covering complete geometric graphs by monotone paths
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 15
year: '2026'
...
---
OA_place: repository
OA_type: green
_id: '18626'
abstract:
- lang: eng
text: "The local angle property of the (order-1) Delaunay triangulations of a generic
set in R2\r\n asserts that the sum of two angles opposite a common edge is less
than π. This paper extends this property to higher order and uses it to generalize
two classic properties from order-1 to order-2: (1) among the complete level-2
hypertriangulations of a generic point set in R2, the order-2 Delaunay triangulation
lexicographically maximizes the sorted angle vector; (2) among the maximal level-2
hypertriangulations of a generic point set in R2, the order-2 Delaunay triangulation
is the only one that has the local angle property. We also use our method of establishing
(2) to give a new short proof of the angle vector optimality for the (order-1)
Delaunay triangulation. For order-1, both properties have been instrumental in
numerous applications of Delaunay triangulations, and we expect that their generalization
will make order-2 Delaunay triangulations more attractive to applications as well."
acknowledgement: Work by the first and third authors is partially supported by the
European Research Council (ERC), grant no. 788183, by the Wittgenstein Prize, Austrian
Science Fund (FWF), grant no. Z 342-N31, and by the DFG Collaborative Research Center
TRR 109, Austrian Science Fund (FWF), grant no. I 02979-N35. Work by the second
author is partially supported by the Alexander von Humboldt Foundation.
article_number: '110055'
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: Morteza
full_name: Saghafian, Morteza
id: f86f7148-b140-11ec-9577-95435b8df824
last_name: Saghafian
citation:
ama: Edelsbrunner H, Garber A, Saghafian M. Order-2 Delaunay triangulations optimize
angles. Advances in Mathematics. 2025;461. doi:10.1016/j.aim.2024.110055
apa: Edelsbrunner, H., Garber, A., & Saghafian, M. (2025). Order-2 Delaunay
triangulations optimize angles. Advances in Mathematics. Elsevier. https://doi.org/10.1016/j.aim.2024.110055
chicago: Edelsbrunner, Herbert, Alexey Garber, and Morteza Saghafian. “Order-2 Delaunay
Triangulations Optimize Angles.” Advances in Mathematics. Elsevier, 2025.
https://doi.org/10.1016/j.aim.2024.110055.
ieee: H. Edelsbrunner, A. Garber, and M. Saghafian, “Order-2 Delaunay triangulations
optimize angles,” Advances in Mathematics, vol. 461. Elsevier, 2025.
ista: Edelsbrunner H, Garber A, Saghafian M. 2025. Order-2 Delaunay triangulations
optimize angles. Advances in Mathematics. 461, 110055.
mla: Edelsbrunner, Herbert, et al. “Order-2 Delaunay Triangulations Optimize Angles.”
Advances in Mathematics, vol. 461, 110055, Elsevier, 2025, doi:10.1016/j.aim.2024.110055.
short: H. Edelsbrunner, A. Garber, M. Saghafian, Advances in Mathematics 461 (2025).
corr_author: '1'
date_created: 2024-12-08T23:01:54Z
date_published: 2025-02-01T00:00:00Z
date_updated: 2025-04-15T07:16:53Z
day: '01'
department:
- _id: HeEd
doi: 10.1016/j.aim.2024.110055
ec_funded: 1
external_id:
arxiv:
- '2310.18238'
isi:
- '001370682500001'
intvolume: ' 461'
isi: 1
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://doi.org/10.48550/arXiv.2310.18238
month: '02'
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: Advances in Mathematics
publication_identifier:
eissn:
- 1090-2082
issn:
- 0001-8708
publication_status: published
publisher: Elsevier
quality_controlled: '1'
scopus_import: '1'
status: public
title: Order-2 Delaunay triangulations optimize angles
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 461
year: '2025'
...
---
OA_type: closed access
_id: '19937'
abstract:
- lang: eng
text: Simplets are elementary units within simplicial complexes and are fundamental
for analyzing the structure of simplicial complexes. Previous efforts have mainly
focused on accurately counting or approximating the number of simplets rather
than studying their frequencies. However, analyzing simplet frequencies is more
practical for large-scale simplicial complexes. This paper introduces the Simplet
Frequency Distribution (SFD) vector, which enables the analysis of simplet frequencies
in simplicial complexes. Additionally, we provide a bound on the sample complexity
required to approximate the SFD vector using any uniform sampling-based algorithm
accurately. We extend the definition of simplet frequency distribution to encompass
simplices, allowing for the analysis of simplet frequencies within simplices of
simplicial complexes. This paper introduces the Simplet Degree Vector (SDV) and
the Simplet Degree Centrality (SDC), facilitating this analysis for each simplex.
Furthermore, we present a bound on the sample complexity required for accurately
approximating the SDV and SDC for a set of simplices using any uniform sampling-based
algorithm. We also introduce algorithms for approximating SFD, geometric SFD,
SDV, and SDC. We also validate the theoretical bounds with experiments on random
simplicial complexes and demonstrate the practical application through a case
study.
acknowledgement: "The authors would like to thank the anonymous reviewers for their
valuable comments and suggestions, which improved this paper.\r\nWork by the first
and fourth authors is partially 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."
article_number: '122425'
article_processing_charge: No
article_type: original
author:
- first_name: Mohammad
full_name: Mahini, Mohammad
last_name: Mahini
- first_name: Hamid
full_name: Beigy, Hamid
last_name: Beigy
- first_name: Salman
full_name: Qadami, Salman
last_name: Qadami
- first_name: Morteza
full_name: Saghafian, Morteza
id: f86f7148-b140-11ec-9577-95435b8df824
last_name: Saghafian
citation:
ama: 'Mahini M, Beigy H, Qadami S, Saghafian M. Simplet-based signatures and approximation
in simplicial complexes: Frequency, degree, and centrality. Information Sciences.
2025;719(11). doi:10.1016/j.ins.2025.122425'
apa: 'Mahini, M., Beigy, H., Qadami, S., & Saghafian, M. (2025). Simplet-based
signatures and approximation in simplicial complexes: Frequency, degree, and centrality.
Information Sciences. Elsevier. https://doi.org/10.1016/j.ins.2025.122425'
chicago: 'Mahini, Mohammad, Hamid Beigy, Salman Qadami, and Morteza Saghafian. “Simplet-Based
Signatures and Approximation in Simplicial Complexes: Frequency, Degree, and Centrality.”
Information Sciences. Elsevier, 2025. https://doi.org/10.1016/j.ins.2025.122425.'
ieee: 'M. Mahini, H. Beigy, S. Qadami, and M. Saghafian, “Simplet-based signatures
and approximation in simplicial complexes: Frequency, degree, and centrality,”
Information Sciences, vol. 719, no. 11. Elsevier, 2025.'
ista: 'Mahini M, Beigy H, Qadami S, Saghafian M. 2025. Simplet-based signatures
and approximation in simplicial complexes: Frequency, degree, and centrality.
Information Sciences. 719(11), 122425.'
mla: 'Mahini, Mohammad, et al. “Simplet-Based Signatures and Approximation in Simplicial
Complexes: Frequency, Degree, and Centrality.” Information Sciences, vol.
719, no. 11, 122425, Elsevier, 2025, doi:10.1016/j.ins.2025.122425.'
short: M. Mahini, H. Beigy, S. Qadami, M. Saghafian, Information Sciences 719 (2025).
corr_author: '1'
date_created: 2025-06-30T08:48:48Z
date_published: 2025-11-01T00:00:00Z
date_updated: 2025-12-30T09:05:32Z
day: '01'
department:
- _id: HeEd
doi: 10.1016/j.ins.2025.122425
ec_funded: 1
external_id:
isi:
- '001516170500002'
intvolume: ' 719'
isi: 1
issue: '11'
language:
- iso: eng
month: '11'
oa_version: None
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: Information Sciences
publication_identifier:
issn:
- 0020-0255
publication_status: published
publisher: Elsevier
quality_controlled: '1'
scopus_import: '1'
status: public
title: 'Simplet-based signatures and approximation in simplicial complexes: Frequency,
degree, and centrality'
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 719
year: '2025'
...
---
OA_place: publisher
OA_type: gold
_id: '20005'
abstract:
- lang: eng
text: "We generalize a classical result by Boris Delaunay that introduced Delaunay
triangulations. In particular, we prove that for a locally finite and coarsely
dense generic point set A in ℝ^d, every generic point of ℝ^d belongs to exactly
binom(d+k,d) simplices whose vertices belong to A and whose circumspheres enclose
exactly k points of A. We extend this result to the cases in which the points
are weighted, and when A contains only finitely many points in ℝ^d or in \U0001D54A^d.
Furthermore, we use the result to give a new geometric proof for the fact that
volumes of hypersimplices are Eulerian numbers."
acknowledgement: "Herbert Edelsbrunner: partially supported by the Wittgenstein Prize,
Austrian Science\r\nFund (FWF), grant no. Z 342-N31, and by the DFG Collaborative
Research Center TRR 109,\r\nAustrian Science Fund (FWF), grant no. I 02979-N35.\r\nAlexey
Garber: partially supported by the Simons Foundation.\r\nMorteza Saghafian: partially
supported by the Wittgenstein Prize, Austrian Science Fund (FWF),\r\ngrant no. Z
342-N31, and by the DFG Collaborative Research Center TRR 109, Austrian Science\r\nFund
(FWF), grant no. I 02979-N35"
alternative_title:
- LIPIcs
article_number: '43'
article_processing_charge: Yes
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: Morteza
full_name: Saghafian, Morteza
id: f86f7148-b140-11ec-9577-95435b8df824
last_name: Saghafian
citation:
ama: 'Edelsbrunner H, Garber A, Saghafian M. On spheres with k points inside. In:
41st International Symposium on Computational Geometry. Vol 332. Schloss
Dagstuhl - Leibniz-Zentrum für Informatik; 2025. doi:10.4230/LIPIcs.SoCG.2025.43'
apa: 'Edelsbrunner, H., Garber, A., & Saghafian, M. (2025). On spheres with
k points inside. In 41st International Symposium on Computational Geometry
(Vol. 332). Kanazawa, Japan: Schloss Dagstuhl - Leibniz-Zentrum für Informatik.
https://doi.org/10.4230/LIPIcs.SoCG.2025.43'
chicago: Edelsbrunner, Herbert, Alexey Garber, and Morteza Saghafian. “On Spheres
with k Points Inside.” In 41st International Symposium on Computational Geometry,
Vol. 332. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2025. https://doi.org/10.4230/LIPIcs.SoCG.2025.43.
ieee: H. Edelsbrunner, A. Garber, and M. Saghafian, “On spheres with k points inside,”
in 41st International Symposium on Computational Geometry, Kanazawa, Japan,
2025, vol. 332.
ista: 'Edelsbrunner H, Garber A, Saghafian M. 2025. On spheres with k points inside.
41st International Symposium on Computational Geometry. SoCG: Symposium on Computational
Geometry, LIPIcs, vol. 332, 43.'
mla: Edelsbrunner, Herbert, et al. “On Spheres with k Points Inside.” 41st International
Symposium on Computational Geometry, vol. 332, 43, Schloss Dagstuhl - Leibniz-Zentrum
für Informatik, 2025, doi:10.4230/LIPIcs.SoCG.2025.43.
short: H. Edelsbrunner, A. Garber, M. Saghafian, in:, 41st International Symposium
on Computational Geometry, Schloss Dagstuhl - Leibniz-Zentrum für Informatik,
2025.
conference:
end_date: 2025-06-27
location: Kanazawa, Japan
name: 'SoCG: Symposium on Computational Geometry'
start_date: 2025-06-23
corr_author: '1'
date_created: 2025-07-13T22:01:22Z
date_published: 2025-06-20T00:00:00Z
date_updated: 2025-07-14T07:26:14Z
day: '20'
ddc:
- '510'
department:
- _id: HeEd
doi: 10.4230/LIPIcs.SoCG.2025.43
external_id:
arxiv:
- '2410.21204'
file:
- access_level: open_access
checksum: b5313ed8575ea87913c71a6e3c7513c8
content_type: application/pdf
creator: dernst
date_created: 2025-07-14T07:24:22Z
date_updated: 2025-07-14T07:24:22Z
file_id: '20016'
file_name: 2025_LIPIcs.SoCG_Edelsbrunner.pdf
file_size: 661893
relation: main_file
success: 1
file_date_updated: 2025-07-14T07:24:22Z
has_accepted_license: '1'
intvolume: ' 332'
language:
- iso: eng
month: '06'
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: 41st International Symposium on Computational Geometry
publication_identifier:
eissn:
- 1868-8969
isbn:
- '9783959773706'
publication_status: published
publisher: Schloss Dagstuhl - Leibniz-Zentrum für Informatik
quality_controlled: '1'
scopus_import: '1'
status: public
title: On spheres with k points inside
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: 332
year: '2025'
...
---
OA_place: publisher
OA_type: gold
_id: '20006'
abstract:
- lang: eng
text: In numerous fields, dynamic time series data require continuous updates, necessitating
efficient data processing techniques for accurate analysis. This paper examines
the banana tree data structure, specifically designed to efficiently maintain
the multi-scale topological descriptor commonly known as persistent homology for
dynamically changing time series data. We implement this data structure and conduct
an experimental study to assess its properties and runtime for update operations.
Our findings indicate that banana trees are highly effective with unbiased random
data, outperforming state-of-the-art static algorithms in these scenarios. Additionally,
our results show that real-world time series share structural properties with
unbiased random walks, suggesting potential practical utility for our implementation.
acknowledgement: "Lara Ost: Supported by the Vienna Graduate School on Computational
Optimization\r\n(VGSCO), FWF project no. W1260-N35.\r\nSebastiano Cultrera di Montesano:
Supported by the Eric and Wendy Schmidt Center at the Broad Institute of MIT and
Harvard.\r\nHerbert Edelsbrunner: Partially supported by the Wittgenstein Prize,
FWF grant no. Z 342-N31,\r\nand by the DFG Collaborative Research Center TRR 109,
FWF grant no. I 02979-N35."
alternative_title:
- LIPIcs
article_number: '71'
article_processing_charge: Yes
arxiv: 1
author:
- first_name: Lara
full_name: Ost, Lara
last_name: Ost
- 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
citation:
ama: 'Ost L, Cultrera di Montesano S, Edelsbrunner H. Banana trees for the persistence
in time series experimentally. In: 41st International Symposium on Computational
Geometry. Vol 332. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2025.
doi:10.4230/LIPIcs.SoCG.2025.71'
apa: 'Ost, L., Cultrera di Montesano, S., & Edelsbrunner, H. (2025). Banana
trees for the persistence in time series experimentally. In 41st International
Symposium on Computational Geometry (Vol. 332). Kanazawa, Japan: Schloss Dagstuhl
- Leibniz-Zentrum für Informatik. https://doi.org/10.4230/LIPIcs.SoCG.2025.71'
chicago: Ost, Lara, Sebastiano Cultrera di Montesano, and Herbert Edelsbrunner.
“Banana Trees for the Persistence in Time Series Experimentally.” In 41st International
Symposium on Computational Geometry, Vol. 332. Schloss Dagstuhl - Leibniz-Zentrum
für Informatik, 2025. https://doi.org/10.4230/LIPIcs.SoCG.2025.71.
ieee: L. Ost, S. Cultrera di Montesano, and H. Edelsbrunner, “Banana trees for the
persistence in time series experimentally,” in 41st International Symposium
on Computational Geometry, Kanazawa, Japan, 2025, vol. 332.
ista: 'Ost L, Cultrera di Montesano S, Edelsbrunner H. 2025. Banana trees for the
persistence in time series experimentally. 41st International Symposium on Computational
Geometry. SoCG: Symposium on Computational Geometry, LIPIcs, vol. 332, 71.'
mla: Ost, Lara, et al. “Banana Trees for the Persistence in Time Series Experimentally.”
41st International Symposium on Computational Geometry, vol. 332, 71, Schloss
Dagstuhl - Leibniz-Zentrum für Informatik, 2025, doi:10.4230/LIPIcs.SoCG.2025.71.
short: L. Ost, S. Cultrera di Montesano, H. Edelsbrunner, in:, 41st International
Symposium on Computational Geometry, Schloss Dagstuhl - Leibniz-Zentrum für Informatik,
2025.
conference:
end_date: 2025-06-27
location: Kanazawa, Japan
name: 'SoCG: Symposium on Computational Geometry'
start_date: 2025-06-23
corr_author: '1'
date_created: 2025-07-13T22:01:22Z
date_published: 2025-06-20T00:00:00Z
date_updated: 2025-12-30T11:04:33Z
day: '20'
ddc:
- '000'
department:
- _id: HeEd
doi: 10.4230/LIPIcs.SoCG.2025.71
external_id:
arxiv:
- '2405.17920'
file:
- access_level: open_access
checksum: 3a4a7a707a56e0cfdf51428782dee55a
content_type: application/pdf
creator: dernst
date_created: 2025-07-14T08:23:38Z
date_updated: 2025-07-14T08:23:38Z
file_id: '20017'
file_name: 2025_LIPIcs.SoCG_Ost.pdf
file_size: 834623
relation: main_file
success: 1
file_date_updated: 2025-07-14T08:23:38Z
has_accepted_license: '1'
intvolume: ' 332'
language:
- iso: eng
month: '06'
oa: 1
oa_version: Published Version
project:
- _id: 9B9290DE-BA93-11EA-9121-9846C619BF3A
grant_number: W1260-N35
name: Vienna Graduate School on Computational Optimization
- _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: 41st International Symposium on Computational Geometry
publication_identifier:
eissn:
- 1868-8969
isbn:
- '9783959773706'
publication_status: published
publisher: Schloss Dagstuhl - Leibniz-Zentrum für Informatik
quality_controlled: '1'
related_material:
link:
- relation: software
url: https://github.com/laraost/BananaPersist
scopus_import: '1'
status: public
title: Banana trees for the persistence in time series experimentally
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: 332
year: '2025'
...
---
DOAJ_listed: '1'
OA_place: publisher
OA_type: gold
PlanS_conform: '1'
_id: '20293'
abstract:
- lang: eng
text: Motivated by questions arising at the intersection of information theory and
geometry, we compare two dissimilarity measures between finite categorical distributions.
One is the well-known Jensen–Shannon divergence, which is easy to compute and
whose square root is a proper metric. The other is what we call the minmax divergence,
which is harder to compute. Just like the Jensen–Shannon divergence, it arises
naturally from the Kullback–Leibler divergence. The main contribution of this
paper is a proof showing that the minmax divergence can be tightly approximated
by the Jensen–Shannon divergence. The bounds suggest that the square root of the
minmax divergence is a metric, and we prove that this is indeed true in the one-dimensional
case. The general case remains open. Finally, we consider analogous questions
in the context of another Bregman divergence and the corresponding Burbea–Rao
(Jensen–Bregman) divergence.
acknowledgement: "This research received partial funding from the European Research
Council (ERC) under\r\nthe European Union’s Horizon 2020 research and innovation
programme, grant no. 788183, the\r\nWittgenstein Prize, Austrian Science Fund (FWF),
grant no. Z 342-N31, the DFG Collaborative\r\nResearch Center TRR 109, ‘Discretization
in Geometry and Dynamics’, Austrian Science Fund (FWF), grant no. I 02979-N35, and
the 2022 Google Research Scholar Award for project ‘Algorithms for Topological Analysis
of Neural Networks’. The APC was waived."
article_number: '854'
article_processing_charge: Yes
article_type: original
author:
- first_name: Arseniy
full_name: Akopyan, Arseniy
id: 430D2C90-F248-11E8-B48F-1D18A9856A87
last_name: Akopyan
orcid: 0000-0002-2548-617X
- first_name: Herbert
full_name: Edelsbrunner, Herbert
id: 3FB178DA-F248-11E8-B48F-1D18A9856A87
last_name: Edelsbrunner
orcid: 0000-0002-9823-6833
- first_name: Ziga
full_name: Virk, Ziga
id: 2E36B656-F248-11E8-B48F-1D18A9856A87
last_name: Virk
- first_name: Hubert
full_name: Wagner, Hubert
id: 379CA8B8-F248-11E8-B48F-1D18A9856A87
last_name: Wagner
citation:
ama: Akopyan A, Edelsbrunner H, Virk Z, Wagner H. Tight bounds between the Jensen–Shannon
divergence and the minmax divergence. Entropy. 2025;27(8). doi:10.3390/e27080854
apa: Akopyan, A., Edelsbrunner, H., Virk, Z., & Wagner, H. (2025). Tight bounds
between the Jensen–Shannon divergence and the minmax divergence. Entropy.
MDPI. https://doi.org/10.3390/e27080854
chicago: Akopyan, Arseniy, Herbert Edelsbrunner, Ziga Virk, and Hubert Wagner. “Tight
Bounds between the Jensen–Shannon Divergence and the Minmax Divergence.” Entropy.
MDPI, 2025. https://doi.org/10.3390/e27080854.
ieee: A. Akopyan, H. Edelsbrunner, Z. Virk, and H. Wagner, “Tight bounds between
the Jensen–Shannon divergence and the minmax divergence,” Entropy, vol.
27, no. 8. MDPI, 2025.
ista: Akopyan A, Edelsbrunner H, Virk Z, Wagner H. 2025. Tight bounds between the
Jensen–Shannon divergence and the minmax divergence. Entropy. 27(8), 854.
mla: Akopyan, Arseniy, et al. “Tight Bounds between the Jensen–Shannon Divergence
and the Minmax Divergence.” Entropy, vol. 27, no. 8, 854, MDPI, 2025, doi:10.3390/e27080854.
short: A. Akopyan, H. Edelsbrunner, Z. Virk, H. Wagner, Entropy 27 (2025).
corr_author: '1'
date_created: 2025-09-07T22:01:33Z
date_published: 2025-08-01T00:00:00Z
date_updated: 2025-09-30T14:32:31Z
day: '01'
ddc:
- '500'
department:
- _id: HeEd
doi: 10.3390/e27080854
ec_funded: 1
external_id:
isi:
- '001557476000001'
pmid:
- '40870326'
file:
- access_level: open_access
checksum: 65c5399c4015d9c8abb8c7a96f3d7836
content_type: application/pdf
creator: dernst
date_created: 2025-09-08T07:55:48Z
date_updated: 2025-09-08T07:55:48Z
file_id: '20309'
file_name: 2025_Entropy_Akopyan.pdf
file_size: 379340
relation: main_file
success: 1
file_date_updated: 2025-09-08T07:55:48Z
has_accepted_license: '1'
intvolume: ' 27'
isi: 1
issue: '8'
language:
- iso: eng
month: '08'
oa: 1
oa_version: Published Version
pmid: 1
project:
- _id: 266A2E9E-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '788183'
name: Alpha Shape Theory Extended
- _id: 268116B8-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: Z00342
name: Mathematics, Computer Science
- _id: 2561EBF4-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: I02979-N35
name: Persistence and stability of geometric complexes
publication: Entropy
publication_identifier:
eissn:
- 1099-4300
publication_status: published
publisher: MDPI
quality_controlled: '1'
scopus_import: '1'
status: public
title: Tight bounds between the Jensen–Shannon divergence and the minmax divergence
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
short: CC BY (4.0)
type: journal_article
user_id: 317138e5-6ab7-11ef-aa6d-ffef3953e345
volume: 27
year: '2025'
...
---
OA_place: publisher
OA_type: hybrid
PlanS_conform: '1'
_id: '20323'
abstract:
- lang: eng
text: We establish several results combining discrete Morse theory and microlocal
sheaf theory in the setting of finite posets and simplicial complexes. Our primary
tool is a computationally tractable description of the bounded derived category
of sheaves on a poset with the Alexandrov topology. We prove that each bounded
complex of sheaves on a finite poset admits a unique (up to isomorphism of complexes)
minimal injective resolution, and we provide algorithms for computing minimal
injective resolution of an injective complex, as well as several useful functors
between derived categories of sheaves. For the constant sheaf on a simplicial
complex, we give asymptotically tight bounds on the complexity of computing the
minimal injective resolution using those algorithms. Our main result is a novel
definition of the discrete microsupport of a bounded complex of sheaves on a finite
poset. We detail several foundational properties of the discrete microsupport,
as well as a microlocal generalization of the discrete homological Morse theorem
and Morse inequalities.
acknowledgement: This project has received funding from the European Research Council
(ERC) under the European Union’s Horizon 2020 research and innovation programme,
grant no. 788183, from the Wittgenstein Prize, Austrian Science Fund (FWF), grant
no. Z342-N31, and from the DFG Collaborative Research Center TRR 109, ‘Discretization
in Geometry and Dynamics’, Austrian Science Fund (FWF), grant no. I 02979-N35
article_number: '108068'
article_processing_charge: Yes (via OA deal)
article_type: original
arxiv: 1
author:
- first_name: Adam
full_name: Brown, Adam
id: 70B7FDF6-608D-11E9-9333-8535E6697425
last_name: Brown
- first_name: Ondrej
full_name: Draganov, Ondrej
id: 2B23F01E-F248-11E8-B48F-1D18A9856A87
last_name: Draganov
orcid: 0000-0003-0464-3823
citation:
ama: Brown A, Draganov O. Discrete microlocal Morse theory. Journal of Pure and
Applied Algebra. 2025;229(10). doi:10.1016/j.jpaa.2025.108068
apa: Brown, A., & Draganov, O. (2025). Discrete microlocal Morse theory. Journal
of Pure and Applied Algebra. Elsevier. https://doi.org/10.1016/j.jpaa.2025.108068
chicago: Brown, Adam, and Ondrej Draganov. “Discrete Microlocal Morse Theory.” Journal
of Pure and Applied Algebra. Elsevier, 2025. https://doi.org/10.1016/j.jpaa.2025.108068.
ieee: A. Brown and O. Draganov, “Discrete microlocal Morse theory,” Journal of
Pure and Applied Algebra, vol. 229, no. 10. Elsevier, 2025.
ista: Brown A, Draganov O. 2025. Discrete microlocal Morse theory. Journal of Pure
and Applied Algebra. 229(10), 108068.
mla: Brown, Adam, and Ondrej Draganov. “Discrete Microlocal Morse Theory.” Journal
of Pure and Applied Algebra, vol. 229, no. 10, 108068, Elsevier, 2025, doi:10.1016/j.jpaa.2025.108068.
short: A. Brown, O. Draganov, Journal of Pure and Applied Algebra 229 (2025).
corr_author: '1'
date_created: 2025-09-10T05:40:09Z
date_published: 2025-10-01T00:00:00Z
date_updated: 2025-12-30T07:55:21Z
day: '01'
ddc:
- '510'
department:
- _id: HeEd
doi: 10.1016/j.jpaa.2025.108068
ec_funded: 1
external_id:
arxiv:
- '2209.14993'
file:
- access_level: open_access
checksum: 39bcad462278c9322ef810af7db67f56
content_type: application/pdf
creator: dernst
date_created: 2025-12-30T07:55:08Z
date_updated: 2025-12-30T07:55:08Z
file_id: '20886'
file_name: 2025_JourPureAppliedAlgebra_Brown.pdf
file_size: 3090836
relation: main_file
success: 1
file_date_updated: 2025-12-30T07:55:08Z
has_accepted_license: '1'
intvolume: ' 229'
issue: '10'
language:
- iso: eng
month: '10'
oa: 1
oa_version: Published Version
project:
- _id: 266A2E9E-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '788183'
name: Alpha Shape Theory Extended
- _id: 268116B8-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: Z00342
name: Mathematics, Computer Science
- _id: 2561EBF4-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: I02979-N35
name: Persistence and stability of geometric complexes
publication: Journal of Pure and Applied Algebra
publication_identifier:
issn:
- 0022-4049
publication_status: published
publisher: Elsevier
quality_controlled: '1'
related_material:
record:
- id: '18981'
relation: earlier_version
status: public
scopus_import: '1'
status: public
title: Discrete microlocal Morse theory
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
short: CC BY (4.0)
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 229
year: '2025'
...
---
OA_place: repository
OA_type: green
_id: '20490'
abstract:
- lang: eng
text: "We study flips in hypertriangulations of planar points sets. Here a level-k
hypertriangulation of n\r\n points in the plane is a subdivision induced by the
projection of a k-hypersimplex, which is the convex hull of the barycenters of
the (k-1)-dimensional faces of the standard (n-1)-simplex. In particular, we introduce
four types of flips and prove that the level-2 hypertriangulations are connected
by these flips.\r\n"
acknowledgement: Work by all authors but the second is supported by the European Research
Council (ERC), grant no. 788183, by the Wittgenstein Prize, Austrian Science Fund
(FWF), grant no. Z 342-N31, and by the DFG Collaborative Research Center TRR 109,
Austrian Science Fund (FWF), grant no. I 02979-N35. Work by the second author is
partially supported by the Alexander von Humboldt Foundation and by the Simons Foundation
. The second author thanks Jesús A. De Loera for useful discussions on flips and
non-flips and Pavel Galashin and Alexey Balitskiy for useful discussions on plabic
graphs.
article_number: '104248'
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Herbert
full_name: Edelsbrunner, Herbert
id: 3FB178DA-F248-11E8-B48F-1D18A9856A87
last_name: Edelsbrunner
orcid: 0000-0002-9823-6833
- first_name: Alexey
full_name: Garber, Alexey
last_name: Garber
- first_name: Mohadese
full_name: Ghafari, Mohadese
last_name: Ghafari
- first_name: Teresa
full_name: Heiss, Teresa
id: 4879BB4E-F248-11E8-B48F-1D18A9856A87
last_name: Heiss
orcid: 0000-0002-1780-2689
- first_name: Morteza
full_name: Saghafian, Morteza
id: f86f7148-b140-11ec-9577-95435b8df824
last_name: Saghafian
citation:
ama: Edelsbrunner H, Garber A, Ghafari M, Heiss T, Saghafian M. Flips in two-dimensional
hypertriangulations. European Journal of Combinatorics. 2025;132. doi:10.1016/j.ejc.2025.104248
apa: Edelsbrunner, H., Garber, A., Ghafari, M., Heiss, T., & Saghafian, M. (2025).
Flips in two-dimensional hypertriangulations. European Journal of Combinatorics.
Elsevier. https://doi.org/10.1016/j.ejc.2025.104248
chicago: Edelsbrunner, Herbert, Alexey Garber, Mohadese Ghafari, Teresa Heiss, and
Morteza Saghafian. “Flips in Two-Dimensional Hypertriangulations.” European
Journal of Combinatorics. Elsevier, 2025. https://doi.org/10.1016/j.ejc.2025.104248.
ieee: H. Edelsbrunner, A. Garber, M. Ghafari, T. Heiss, and M. Saghafian, “Flips
in two-dimensional hypertriangulations,” European Journal of Combinatorics,
vol. 132. Elsevier, 2025.
ista: Edelsbrunner H, Garber A, Ghafari M, Heiss T, Saghafian M. 2025. Flips in
two-dimensional hypertriangulations. European Journal of Combinatorics. 132, 104248.
mla: Edelsbrunner, Herbert, et al. “Flips in Two-Dimensional Hypertriangulations.”
European Journal of Combinatorics, vol. 132, 104248, Elsevier, 2025, doi:10.1016/j.ejc.2025.104248.
short: H. Edelsbrunner, A. Garber, M. Ghafari, T. Heiss, M. Saghafian, European
Journal of Combinatorics 132 (2025).
corr_author: '1'
date_created: 2025-10-19T22:01:31Z
date_published: 2025-10-10T00:00:00Z
date_updated: 2025-12-01T12:57:29Z
day: '10'
department:
- _id: HeEd
doi: 10.1016/j.ejc.2025.104248
ec_funded: 1
external_id:
arxiv:
- '2212.11380'
isi:
- '001599061500002'
intvolume: ' 132'
isi: 1
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://doi.org/10.48550/arXiv.2212.11380
month: '10'
oa: 1
oa_version: Preprint
project:
- _id: 266A2E9E-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '788183'
name: Alpha Shape Theory Extended
- _id: 268116B8-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: Z00342
name: Mathematics, Computer Science
- _id: 2561EBF4-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: I02979-N35
name: Persistence and stability of geometric complexes
publication: European Journal of Combinatorics
publication_identifier:
issn:
- 0195-6698
publication_status: epub_ahead
publisher: Elsevier
quality_controlled: '1'
scopus_import: '1'
status: public
title: Flips in two-dimensional hypertriangulations
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 132
year: '2025'
...
---
OA_place: repository
OA_type: green
_id: '20585'
abstract:
- lang: eng
text: Motivated by applications in medical sciences, we study finite chromatic sets
in Euclidean space from a topological perspective. Based on the persistent homology
for images, kernels and cokernels, we design provably stable homological quantifiers
that describe the geometric micro- and macro-structure of how the color classes
mingle. These can be efficiently computed using chromatic variants of Delaunay
and alpha complexes, and code that does these computations is provided.
acknowledgement: "This project has received funding from the European Research\r\nCouncil
(ERC) under the European Union’s Horizon 2020 research and innovation\r\nprogramme,
grant no. 788183, from the Wittgenstein Prize, Austrian Science Fund\r\n(FWF), grant
no. Z 342-N31, and from the DFG Collaborative Research Center TRR\r\n109, ‘Discretization
in Geometry and Dynamics’, Austrian Science Fund (FWF),\r\ngrant no. I 02979-N35."
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Sebastiano
full_name: Cultrera di Montesano, Sebastiano
id: 34D2A09C-F248-11E8-B48F-1D18A9856A87
last_name: Cultrera di Montesano
orcid: 0000-0001-6249-0832
- first_name: Ondrej
full_name: Draganov, Ondrej
id: 2B23F01E-F248-11E8-B48F-1D18A9856A87
last_name: Draganov
orcid: 0000-0003-0464-3823
- first_name: Herbert
full_name: Edelsbrunner, Herbert
id: 3FB178DA-F248-11E8-B48F-1D18A9856A87
last_name: Edelsbrunner
orcid: 0000-0002-9823-6833
- first_name: Morteza
full_name: Saghafian, Morteza
id: f86f7148-b140-11ec-9577-95435b8df824
last_name: Saghafian
citation:
ama: Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M. Chromatic
alpha complexes. Foundations of Data Science. 2025;8:30-62. doi:10.3934/fods.2025003
apa: Cultrera di Montesano, S., Draganov, O., Edelsbrunner, H., & Saghafian,
M. (2025). Chromatic alpha complexes. Foundations of Data Science. American
Institute of Mathematical Sciences. https://doi.org/10.3934/fods.2025003
chicago: Cultrera di Montesano, Sebastiano, Ondrej Draganov, Herbert Edelsbrunner,
and Morteza Saghafian. “Chromatic Alpha Complexes.” Foundations of Data Science.
American Institute of Mathematical Sciences, 2025. https://doi.org/10.3934/fods.2025003.
ieee: S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, and M. Saghafian,
“Chromatic alpha complexes,” Foundations of Data Science, vol. 8. American
Institute of Mathematical Sciences, pp. 30–62, 2025.
ista: Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M. 2025. Chromatic
alpha complexes. Foundations of Data Science. 8, 30–62.
mla: Cultrera di Montesano, Sebastiano, et al. “Chromatic Alpha Complexes.” Foundations
of Data Science, vol. 8, American Institute of Mathematical Sciences, 2025,
pp. 30–62, doi:10.3934/fods.2025003.
short: S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, M. Saghafian, Foundations
of Data Science 8 (2025) 30–62.
corr_author: '1'
date_created: 2025-11-02T23:01:33Z
date_published: 2025-03-01T00:00:00Z
date_updated: 2025-11-04T12:25:47Z
day: '01'
department:
- _id: HeEd
doi: 10.3934/fods.2025003
ec_funded: 1
external_id:
arxiv:
- '2212.03128'
intvolume: ' 8'
language:
- iso: eng
month: '03'
oa_version: Preprint
page: 30-62
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call_identifier: FWF
grant_number: Z00342
name: Mathematics, Computer Science
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call_identifier: FWF
grant_number: I02979-N35
name: Persistence and stability of geometric complexes
publication: Foundations of Data Science
publication_identifier:
eissn:
- 2639-8001
publication_status: epub_ahead
publisher: American Institute of Mathematical Sciences
quality_controlled: '1'
related_material:
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title: Chromatic alpha complexes
type: journal_article
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...
---
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abstract:
- lang: eng
text: 'The Upper Bound Theorem for convex polytopes implies that the p-th Betti
number of the Čech complex of any set of N points in ℝ^d and any radius satisfies
β_p = O(N^m), with m = min{p+1, ⌈d/2⌉}. We construct sets in even and odd dimensions,
which prove that this upper bound is asymptotically tight. For example, we describe
a set of N = 2(n+1) points in ℝ³ and two radii such that the first Betti number
of the Čech complex at one radius is (n+1)² - 1, and the second Betti number of
the Čech complex at the other radius is n². '
acknowledgement: The first author is supported by the European Research Council (ERC),
grant no. 788183, and by the DFG Collaborative Research Center TRR 109, Austrian
Science Fund (FWF), grant no. I 02979-N35. The second author is supported by the
European Research Council (ERC), grant “GeoScape” and by the Hungarian Science Foundation
(NKFIH), grant K-131529. Both authors are supported by the Wittgenstein Prize, Austrian
Science Fund (FWF), grant no. Z 342-N31.
article_processing_charge: Yes (via OA deal)
article_type: original
arxiv: 1
author:
- first_name: Herbert
full_name: Edelsbrunner, Herbert
id: 3FB178DA-F248-11E8-B48F-1D18A9856A87
last_name: Edelsbrunner
orcid: 0000-0002-9823-6833
- first_name: János
full_name: Pach, János
id: E62E3130-B088-11EA-B919-BF823C25FEA4
last_name: Pach
citation:
ama: Edelsbrunner H, Pach J. Maximum Betti numbers of Čech complexes. Discrete
& Computational Geometry. 2025. doi:10.1007/s00454-025-00796-5
apa: Edelsbrunner, H., & Pach, J. (2025). Maximum Betti numbers of Čech complexes.
Discrete & Computational Geometry. Springer Nature. https://doi.org/10.1007/s00454-025-00796-5
chicago: Edelsbrunner, Herbert, and János Pach. “Maximum Betti Numbers of Čech Complexes.”
Discrete & Computational Geometry. Springer Nature, 2025. https://doi.org/10.1007/s00454-025-00796-5.
ieee: H. Edelsbrunner and J. Pach, “Maximum Betti numbers of Čech complexes,” Discrete
& Computational Geometry. Springer Nature, 2025.
ista: Edelsbrunner H, Pach J. 2025. Maximum Betti numbers of Čech complexes. Discrete
& Computational Geometry.
mla: Edelsbrunner, Herbert, and János Pach. “Maximum Betti Numbers of Čech Complexes.”
Discrete & Computational Geometry, Springer Nature, 2025, doi:10.1007/s00454-025-00796-5.
short: H. Edelsbrunner, J. Pach, Discrete & Computational Geometry (2025).
corr_author: '1'
date_created: 2025-11-19T09:44:58Z
date_published: 2025-11-10T00:00:00Z
date_updated: 2025-12-01T15:19:21Z
day: '10'
ddc:
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department:
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doi: 10.1007/s00454-025-00796-5
ec_funded: 1
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language:
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url: https://doi.org/10.1007/s00454-025-00796-5
month: '11'
oa: 1
oa_version: Published Version
project:
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call_identifier: H2020
grant_number: '788183'
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- _id: 2561EBF4-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: I02979-N35
name: Persistence and stability of geometric complexes
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call_identifier: FWF
grant_number: Z00342
name: Mathematics, Computer Science
publication: Discrete & Computational Geometry
publication_identifier:
eissn:
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issn:
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publication_status: epub_ahead
publisher: Springer Nature
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title: Maximum Betti numbers of Čech complexes
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name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
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type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
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...