---
OA_place: publisher
OA_type: gold
_id: '22002'
abstract:
- lang: eng
text: Topological simplification is the process of reducing complexity of a function
while maintaining its essential features. Its goal is to find a new filter function,
which reorders cells of the input complex in a way which eliminates some persistent
homological features, without affecting the rest. We present a new approach to
simplification based on the concept of forbidden regions and combinatorial dynamics.
It allows us to reorder and cancel critical values, whose cancellation is not
possible using existing methods because they are not consecutive in the total
order. Each such cancellation takes O(c⋅n) time in the worst case, where c is
the number of birth-death pairs and n is the size of the input complex.
acknowledgement: "Jakub Leśkiewicz wants to thank his supervisor, Prof. Marian Mrozek,
forscientific guidance, patience, and opportunity to delay the rest of his duties
while writing this work.\r\nThe author also extends thanks to his entire family,
to Zuzanna Świątek, and to Mikołaj Kardyś,\r\nBEng, MSc, for providing meals during
the most intensive periods of work. Jakub Leśkiewicz: The research was partially
funded by the Polish National Science Center under Opus Grant No. 2019/35/B/ST1/00874
and Opus Grant 2025/57/B/ST1/00550. Bartosz Furmanek: The research was partially
funded by the Polish National Science Center under Opus Grant No. 2019/35/B/ST1/00874
and Opus Grant 2025/57/B/ST1/00550. Michał Lipiński: This project has received funding
from the European Union’s Horizon 2020 research and innovation programme under the
Marie Skłodowska-Curie Grant Agreement No. 101034413. \r\nDmitriy Morozov: This
work was supported in part by the U.S. Department of Energy, Office\r\nof Science,
Office of Advanced Scientific Computing Research, under Contract No. DE-AC02-\r\n05CH11231."
alternative_title:
- LIPIcs
article_number: 72:1-72:17
article_processing_charge: No
arxiv: 1
author:
- first_name: Jakub
full_name: Leśkiewicz, Jakub
last_name: Leśkiewicz
- first_name: Bartosz
full_name: Furmanek, Bartosz
last_name: Furmanek
- first_name: Michał
full_name: Lipiński, Michał
id: dfffb474-4317-11ee-8f5c-fe3fc95a425e
last_name: Lipiński
orcid: 0000-0001-9789-9750
- first_name: Dmitriy
full_name: Morozov, Dmitriy
last_name: Morozov
citation:
ama: 'Leśkiewicz J, Furmanek B, Lipiński M, Morozov D. Topological simplification
guided by forbidden regions. In: 42nd International Symposium on Computational
Geometry. Vol 367. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2026.
doi:10.4230/LIPIcs.SoCG.2026.72'
apa: 'Leśkiewicz, J., Furmanek, B., Lipiński, M., & Morozov, D. (2026). Topological
simplification guided by forbidden regions. In 42nd International Symposium
on Computational Geometry (Vol. 367). New Brunswick, NJ, United States: Schloss
Dagstuhl - Leibniz-Zentrum für Informatik. https://doi.org/10.4230/LIPIcs.SoCG.2026.72'
chicago: Leśkiewicz, Jakub, Bartosz Furmanek, Michał Lipiński, and Dmitriy Morozov.
“Topological Simplification Guided by Forbidden Regions.” In 42nd International
Symposium on Computational Geometry, Vol. 367. Schloss Dagstuhl - Leibniz-Zentrum
für Informatik, 2026. https://doi.org/10.4230/LIPIcs.SoCG.2026.72.
ieee: J. Leśkiewicz, B. Furmanek, M. Lipiński, and D. Morozov, “Topological simplification
guided by forbidden regions,” in 42nd International Symposium on Computational
Geometry, New Brunswick, NJ, United States, 2026, vol. 367.
ista: 'Leśkiewicz J, Furmanek B, Lipiński M, Morozov D. 2026. Topological simplification
guided by forbidden regions. 42nd International Symposium on Computational Geometry.
SoCG: Symposium on Computational Geometry, LIPIcs, vol. 367, 72:1-72:17.'
mla: Leśkiewicz, Jakub, et al. “Topological Simplification Guided by Forbidden Regions.”
42nd International Symposium on Computational Geometry, vol. 367, 72:1-72:17,
Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2026, doi:10.4230/LIPIcs.SoCG.2026.72.
short: J. Leśkiewicz, B. Furmanek, M. Lipiński, D. Morozov, in:, 42nd International
Symposium on Computational Geometry, Schloss Dagstuhl - Leibniz-Zentrum für Informatik,
2026.
conference:
end_date: 2026-06-05
location: New Brunswick, NJ, United States
name: 'SoCG: Symposium on Computational Geometry'
start_date: 2026-06-02
corr_author: '1'
das_tickbox: '0'
date_created: 2026-06-14T22:01:43Z
date_published: 2026-05-27T00:00:00Z
date_updated: 2026-06-22T07:45:36Z
day: '27'
ddc:
- '500'
department:
- _id: HeEd
doi: 10.4230/LIPIcs.SoCG.2026.72
ec_funded: 1
external_id:
arxiv:
- '2603.16416'
file:
- access_level: open_access
checksum: 3be91c06fdf716c8735b6af64a09a921
content_type: application/pdf
creator: dernst
date_created: 2026-06-22T07:39:21Z
date_updated: 2026-06-22T07:39:21Z
file_id: '22110'
file_name: 2026_LIPIcSSoCG_Leskiewicz.pdf
file_size: 2052749
relation: main_file
success: 1
file_date_updated: 2026-06-22T07:39:21Z
has_accepted_license: '1'
intvolume: ' 367'
keyword:
- persistent homology
- topological simplification
- depth posets
language:
- iso: eng
month: '05'
oa: 1
oa_version: Published Version
project:
- _id: fc2ed2f7-9c52-11eb-aca3-c01059dda49c
call_identifier: H2020
grant_number: '101034413'
name: 'IST-BRIDGE: International postdoctoral program'
publication: 42nd International Symposium on Computational Geometry
publication_identifier:
eissn:
- 1868-8969
isbn:
- '9783959774185'
publication_status: published
publisher: Schloss Dagstuhl - Leibniz-Zentrum für Informatik
quality_controlled: '1'
scopus_import: '1'
status: public
title: Topological simplification guided by forbidden regions
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: 367
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: publisher
OA_type: hybrid
_id: '18580'
abstract:
- lang: eng
text: Motivated by the study of recurrent orbits and dynamics within a Morse set
of a Morse decomposition we introduce the concept of Morse predecomposition of
an isolated invariant set within the setting of both combinatorial and classical
dynamical systems. While Morse decomposition summarizes solely the gradient part
of a dynamical system, the developed generalization extends to the recurrent component
as well. In particular, a chain recurrent set, which is indecomposable in terms
of Morse decomposition, can be represented more finely in the Morse predecomposition
framework. This generalization is achieved by forgoing the poset structure inherent
to Morse decomposition and relaxing the notion of connection between Morse sets
(elements of Morse decomposition) in favor of what we term ’links’. We prove that
a Morse decomposition is a special case of Morse predecomposition indexed by a
poset. Additionally, we show how a Morse predecomposition may be condensed back
to retrieve a Morse decomposition.
acknowledgement: 'M.L. acknowledge support by the Dioscuri program initiated by the
Max Planck Society, jointly managed with the National Science Centre (Poland), and
mutually funded by the Polish Ministry of Science and Higher Education and the German
Federal Ministry of Education and Research. M.L. also acknowledges that this project
has received funding from the European Union’s Horizon 2020 research and innovation
programme under the Marie Skłodowska-Curie Grant Agreement No. 101034413. Research
of M.M. is partially supported by the Polish National Science Center under Opus
Grant No. 2019/35/B/ST1/00874. The work of K.M. was partially supported by the National
Science Foundation under awards DMS-1839294 and HDR TRIPODS award CCF-1934924, DARPA
contract HR0011-16-2-0033, National Institutes of Health award R01 GM126555, Air
Force Office of Scientific Research under award numbers FA9550-23-1-0011, AWD00010853-MOD002
and MURI FA9550-23-1-0400. K.M. was also supported by a grant from the Simons Foundation.
Open access funding provided by Institute of Science and Technology (IST Austria). '
article_number: '5'
article_processing_charge: Yes (via OA deal)
article_type: original
arxiv: 1
author:
- first_name: Michał
full_name: Lipiński, Michał
id: dfffb474-4317-11ee-8f5c-fe3fc95a425e
last_name: Lipiński
orcid: 0000-0001-9789-9750
- first_name: Konstantin
full_name: Mischaikow, Konstantin
last_name: Mischaikow
- first_name: Marian
full_name: Mrozek, Marian
last_name: Mrozek
citation:
ama: Lipiński M, Mischaikow K, Mrozek M. Morse predecomposition of an invariant
set. Qualitative Theory of Dynamical Systems. 2025;24(1). doi:10.1007/s12346-024-01144-3
apa: Lipiński, M., Mischaikow, K., & Mrozek, M. (2025). Morse predecomposition
of an invariant set. Qualitative Theory of Dynamical Systems. Springer
Nature. https://doi.org/10.1007/s12346-024-01144-3
chicago: Lipiński, Michał, Konstantin Mischaikow, and Marian Mrozek. “Morse Predecomposition
of an Invariant Set.” Qualitative Theory of Dynamical Systems. Springer
Nature, 2025. https://doi.org/10.1007/s12346-024-01144-3.
ieee: M. Lipiński, K. Mischaikow, and M. Mrozek, “Morse predecomposition of an invariant
set,” Qualitative Theory of Dynamical Systems, vol. 24, no. 1. Springer
Nature, 2025.
ista: Lipiński M, Mischaikow K, Mrozek M. 2025. Morse predecomposition of an invariant
set. Qualitative Theory of Dynamical Systems. 24(1), 5.
mla: Lipiński, Michał, et al. “Morse Predecomposition of an Invariant Set.” Qualitative
Theory of Dynamical Systems, vol. 24, no. 1, 5, Springer Nature, 2025, doi:10.1007/s12346-024-01144-3.
short: M. Lipiński, K. Mischaikow, M. Mrozek, Qualitative Theory of Dynamical Systems
24 (2025).
corr_author: '1'
date_created: 2024-11-24T23:01:47Z
date_published: 2025-02-01T00:00:00Z
date_updated: 2025-04-14T07:54:56Z
day: '01'
ddc:
- '514'
- '510'
department:
- _id: UlWa
doi: 10.1007/s12346-024-01144-3
ec_funded: 1
external_id:
arxiv:
- '2312.08013'
isi:
- '001356000500005'
file:
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checksum: 73309a57cc798d696caa57b6aa1467d8
content_type: application/pdf
creator: mlipinsk
date_created: 2024-11-28T06:52:38Z
date_updated: 2024-11-28T06:52:38Z
file_id: '18595'
file_name: 2025_predecomposition.pdf
file_size: 1483668
relation: main_file
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file_date_updated: 2024-11-28T06:52:38Z
has_accepted_license: '1'
intvolume: ' 24'
isi: 1
issue: '1'
language:
- iso: eng
month: '02'
oa: 1
oa_version: Published Version
project:
- _id: fc2ed2f7-9c52-11eb-aca3-c01059dda49c
call_identifier: H2020
grant_number: '101034413'
name: 'IST-BRIDGE: International postdoctoral program'
publication: Qualitative Theory of Dynamical Systems
publication_identifier:
eissn:
- 1662-3592
issn:
- 1575-5460
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: Morse predecomposition of an invariant set
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: 24
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...