---
OA_place: repository
OA_type: green
_id: '21737'
abstract:
- lang: eng
  text: In calculus, l'Hopital's rule provides a simple way to evaluate the limits
    of quotient functions when both the numerator and denominator vanish. But what
    happens when we move beyond real functions on a real interval? In this article,
    we study when the quotient of two complex-valued functions in higher dimension
    can be defined continuously at the points where both functions vanish. Surprisingly,
    the answer is far subtler than in the real-valued setting. We provide a complete
    characterization for the continuity of the quotient function. We also point out
    why extending this result to smoother quotients remains an intriguing challenge.
acknowledgement: "This project was funded in part by the European Research Council
  (ERC Consolidator Grant 101045083 CoDiNA) and the National Science Foundation CAREER
  Award 2239062.\r\n"
article_number: '2602.09958'
article_processing_charge: No
arxiv: 1
author:
- first_name: Albert
  full_name: Chern, Albert
  last_name: Chern
- first_name: Sadashige
  full_name: Ishida, Sadashige
  id: 6F7C4B96-A8E9-11E9-A7CA-09ECE5697425
  last_name: Ishida
  orcid: 0000-0002-3121-3100
citation:
  ama: Chern A, Ishida S. L’Hopital rules for complex-valued functions in higher dimensions.
    <i>arXiv</i>. doi:<a href="https://doi.org/10.48550/ARXIV.2602.09958">10.48550/ARXIV.2602.09958</a>
  apa: Chern, A., &#38; Ishida, S. (n.d.). L’Hopital rules for complex-valued functions
    in higher dimensions. <i>arXiv</i>. <a href="https://doi.org/10.48550/ARXIV.2602.09958">https://doi.org/10.48550/ARXIV.2602.09958</a>
  chicago: Chern, Albert, and Sadashige Ishida. “L’Hopital Rules for Complex-Valued
    Functions in Higher Dimensions.” <i>ArXiv</i>, n.d. <a href="https://doi.org/10.48550/ARXIV.2602.09958">https://doi.org/10.48550/ARXIV.2602.09958</a>.
  ieee: A. Chern and S. Ishida, “L’Hopital rules for complex-valued functions in higher
    dimensions,” <i>arXiv</i>. .
  ista: Chern A, Ishida S. L’Hopital rules for complex-valued functions in higher
    dimensions. arXiv, 2602.09958.
  mla: Chern, Albert, and Sadashige Ishida. “L’Hopital Rules for Complex-Valued Functions
    in Higher Dimensions.” <i>ArXiv</i>, 2602.09958, doi:<a href="https://doi.org/10.48550/ARXIV.2602.09958">10.48550/ARXIV.2602.09958</a>.
  short: A. Chern, S. Ishida, ArXiv (n.d.).
corr_author: '1'
date_created: 2026-04-15T16:28:24Z
date_published: 2026-02-10T00:00:00Z
date_updated: 2026-04-28T10:56:30Z
day: '10'
ddc:
- '510'
department:
- _id: GradSch
- _id: ChWo
doi: 10.48550/ARXIV.2602.09958
external_id:
  arxiv:
  - '2602.09958'
file:
- access_level: open_access
  checksum: 6a76591c723d3e949ad5afa9f7dbb2ee
  content_type: application/pdf
  creator: dernst
  date_created: 2026-04-28T10:53:27Z
  date_updated: 2026-04-28T10:53:27Z
  file_id: '21771'
  file_name: 2026_arXiv_2602.09958.pdf
  file_size: 867109
  relation: main_file
  success: 1
file_date_updated: 2026-04-28T10:53:27Z
fulldoi: https://doi.org/10.48550/ARXIV.2602.09958
has_accepted_license: '1'
keyword:
- l’Hopital theorem
- complex functions
language:
- iso: eng
month: '02'
oa: 1
oa_version: Preprint
project:
- _id: 34bc2376-11ca-11ed-8bc3-9a3b3961a088
  grant_number: '101045083'
  name: Computational Discovery of Numerical Algorithms for Animation and Simulation
    of Natural Phenomena
publication: arXiv
publication_status: submitted
status: public
title: L'Hopital rules for complex-valued functions in higher dimensions
tmp:
  image: /images/cc_by.png
  legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
  name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
  short: CC BY (4.0)
type: preprint
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2026'
...
---
OA_place: publisher
OA_type: hybrid
PlanS_conform: '1'
_id: '21743'
abstract:
- lang: eng
  text: We present symplectic structures on the shape space of unparameterized space
    curves that generalize the classical Marsden–Weinstein structure. Our method integrates
    the Liouville 1-form of the Marsden–Weinstein structure with Riemannian structures
    that have been introduced in mathematical shape analysis. We also derive Hamiltonian
    vector fields for several classical Hamiltonian functions with respect to these
    new symplectic structures.
acknowledgement: The authors are grateful to Boris Khesin for valuable comments on
  the MW symplectic structure and S. Ishida thanks Albert Chern for insightful discussions
  on space curves and Chris Wojtan for his continuous support. M. Bauer was partially
  supported by NSF grant DMS-1953244 and by the Binational Science Foundation (BSF).
  S. Ishida was partially supported by ERC Consolidator Grant 101045083 “CoDiNA” funded
  by the European Research Council. Some figures were generated by the software Houdini
  and its education license was provided by SideFX. Open access funding provided by
  University of Vienna.
article_number: '45'
article_processing_charge: Yes (via OA deal)
article_type: original
arxiv: 1
author:
- first_name: Martin
  full_name: Bauer, Martin
  last_name: Bauer
- first_name: Sadashige
  full_name: Ishida, Sadashige
  id: 6F7C4B96-A8E9-11E9-A7CA-09ECE5697425
  last_name: Ishida
  orcid: 0000-0002-3121-3100
- first_name: Peter W.
  full_name: Michor, Peter W.
  last_name: Michor
citation:
  ama: Bauer M, Ishida S, Michor PW. Symplectic structures on the space of space curves.
    <i>Journal of Nonlinear Science</i>. 2026;36(2). doi:<a href="https://doi.org/10.1007/s00332-026-10266-8">10.1007/s00332-026-10266-8</a>
  apa: Bauer, M., Ishida, S., &#38; Michor, P. W. (2026). Symplectic structures on
    the space of space curves. <i>Journal of Nonlinear Science</i>. Springer Nature.
    <a href="https://doi.org/10.1007/s00332-026-10266-8">https://doi.org/10.1007/s00332-026-10266-8</a>
  chicago: Bauer, Martin, Sadashige Ishida, and Peter W. Michor. “Symplectic Structures
    on the Space of Space Curves.” <i>Journal of Nonlinear Science</i>. Springer Nature,
    2026. <a href="https://doi.org/10.1007/s00332-026-10266-8">https://doi.org/10.1007/s00332-026-10266-8</a>.
  ieee: M. Bauer, S. Ishida, and P. W. Michor, “Symplectic structures on the space
    of space curves,” <i>Journal of Nonlinear Science</i>, vol. 36, no. 2. Springer
    Nature, 2026.
  ista: Bauer M, Ishida S, Michor PW. 2026. Symplectic structures on the space of
    space curves. Journal of Nonlinear Science. 36(2), 45.
  mla: Bauer, Martin, et al. “Symplectic Structures on the Space of Space Curves.”
    <i>Journal of Nonlinear Science</i>, vol. 36, no. 2, 45, Springer Nature, 2026,
    doi:<a href="https://doi.org/10.1007/s00332-026-10266-8">10.1007/s00332-026-10266-8</a>.
  short: M. Bauer, S. Ishida, P.W. Michor, Journal of Nonlinear Science 36 (2026).
date_created: 2026-04-16T07:29:17Z
date_published: 2026-04-15T00:00:00Z
date_updated: 2026-04-28T09:59:01Z
day: '15'
ddc:
- '510'
department:
- _id: GradSch
- _id: ChWo
doi: 10.1007/s00332-026-10266-8
external_id:
  arxiv:
  - '2407.19908'
file:
- access_level: open_access
  checksum: 760de2631b6fd7d57bcd5115ed36c0a2
  content_type: application/pdf
  creator: dernst
  date_created: 2026-04-28T09:55:32Z
  date_updated: 2026-04-28T09:55:32Z
  file_id: '21770'
  file_name: 2026_JourNonlinearScience_Bauer.pdf
  file_size: 1108518
  relation: main_file
  success: 1
file_date_updated: 2026-04-28T09:55:32Z
fulldoi: https://doi.org/10.1007/s00332-026-10266-8
has_accepted_license: '1'
intvolume: '        36'
issue: '2'
language:
- iso: eng
month: '04'
oa: 1
oa_version: Published Version
project:
- _id: 34bc2376-11ca-11ed-8bc3-9a3b3961a088
  grant_number: '101045083'
  name: Computational Discovery of Numerical Algorithms for Animation and Simulation
    of Natural Phenomena
publication: Journal of Nonlinear Science
publication_identifier:
  eissn:
  - 1432-1467
  issn:
  - 0938-8974
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
related_material:
  record:
  - id: '17361'
    relation: earlier_version
    status: public
scopus_import: '1'
status: public
title: Symplectic structures on the space of space curves
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: 36
year: '2026'
...
---
OA_place: publisher
OA_type: hybrid
PlanS_conform: '1'
_id: '14703'
abstract:
- lang: eng
  text: We present a discretization of the dynamic optimal transport problem for which
    we can obtain the convergence rate for the value of the transport cost to its
    continuous value when the temporal and spatial stepsize vanish. This convergence
    result does not require any regularity assumption on the measures, though experiments
    suggest that the rate is not sharp. Via an analysis of the duality gap we also
    obtain the convergence rates for the gradient of the optimal potentials and the
    velocity field under mild regularity assumptions. To obtain such rates we discretize
    the dual formulation of the dynamic optimal transport problem and use the mature
    literature related to the error due to discretizing the Hamilton-Jacobi equation.
acknowledgement: 'The authors would like to thank Chris Wojtan for his continuous
  support and several interesting discussions. Part of this research was performed
  during two visits: one of SI to the BIDSA research center at Bocconi University,
  and one of HL to the Institute of Science and Technology Austria. Both host institutions
  are warmly acknowledged for the hospitality. HL is partially supported by the MUR-Prin
  2022-202244A7YL “Gradient Flows and Non-Smooth Geometric Structures with Applications
  to Optimization and Machine Learning”, funded by the European Union - Next Generation
  EU. SI is supported in part by ERC Consolidator Grant 101045083 “CoDiNA” funded
  by the European Research Council. Open access funding provided by Institute of Science
  and Technology (IST Austria).'
article_processing_charge: Yes (via OA deal)
article_type: original
arxiv: 1
author:
- first_name: Sadashige
  full_name: Ishida, Sadashige
  id: 6F7C4B96-A8E9-11E9-A7CA-09ECE5697425
  last_name: Ishida
  orcid: 0000-0002-3121-3100
- first_name: Hugo
  full_name: Lavenant, Hugo
  last_name: Lavenant
citation:
  ama: Ishida S, Lavenant H. Quantitative convergence of a discretization of dynamic
    optimal transport using the dual formulation. <i>Foundations of Computational
    Mathematics</i>. 2026;26:349-384. doi:<a href="https://doi.org/10.1007/s10208-024-09686-3">10.1007/s10208-024-09686-3</a>
  apa: Ishida, S., &#38; Lavenant, H. (2026). Quantitative convergence of a discretization
    of dynamic optimal transport using the dual formulation. <i>Foundations of Computational
    Mathematics</i>. Springer Nature. <a href="https://doi.org/10.1007/s10208-024-09686-3">https://doi.org/10.1007/s10208-024-09686-3</a>
  chicago: Ishida, Sadashige, and Hugo Lavenant. “Quantitative Convergence of a Discretization
    of Dynamic Optimal Transport Using the Dual Formulation.” <i>Foundations of Computational
    Mathematics</i>. Springer Nature, 2026. <a href="https://doi.org/10.1007/s10208-024-09686-3">https://doi.org/10.1007/s10208-024-09686-3</a>.
  ieee: S. Ishida and H. Lavenant, “Quantitative convergence of a discretization of
    dynamic optimal transport using the dual formulation,” <i>Foundations of Computational
    Mathematics</i>, vol. 26. Springer Nature, pp. 349–384, 2026.
  ista: Ishida S, Lavenant H. 2026. Quantitative convergence of a discretization of
    dynamic optimal transport using the dual formulation. Foundations of Computational
    Mathematics. 26, 349–384.
  mla: Ishida, Sadashige, and Hugo Lavenant. “Quantitative Convergence of a Discretization
    of Dynamic Optimal Transport Using the Dual Formulation.” <i>Foundations of Computational
    Mathematics</i>, vol. 26, Springer Nature, 2026, pp. 349–84, doi:<a href="https://doi.org/10.1007/s10208-024-09686-3">10.1007/s10208-024-09686-3</a>.
  short: S. Ishida, H. Lavenant, Foundations of Computational Mathematics 26 (2026)
    349–384.
corr_author: '1'
das_tickbox: '0'
date_created: 2023-12-21T10:14:37Z
date_published: 2026-02-01T00:00:00Z
date_updated: 2026-07-23T05:39:38Z
day: '01'
ddc:
- '000'
department:
- _id: GradSch
- _id: ChWo
doi: 10.1007/s10208-024-09686-3
external_id:
  arxiv:
  - '2312.12213'
  isi:
  - '001352503300001'
file:
- access_level: open_access
  checksum: 30671f88e792e8b75ae3e698ac4c131c
  content_type: application/pdf
  creator: dernst
  date_created: 2026-07-23T05:37:52Z
  date_updated: 2026-07-23T05:37:52Z
  file_id: '22384'
  file_name: 2026_FoundCompMath_Ishida.pdf
  file_size: 1240012
  relation: main_file
  success: 1
file_date_updated: 2026-07-23T05:37:52Z
fulldoi: https://doi.org/10.1007/s10208-024-09686-3
has_accepted_license: '1'
intvolume: '        26'
isi: 1
keyword:
- Optimal transport
- Hamilton-Jacobi equation
- convex optimization
language:
- iso: eng
month: '02'
oa: 1
oa_version: Published Version
page: 349-384
project:
- _id: 34bc2376-11ca-11ed-8bc3-9a3b3961a088
  grant_number: '101045083'
  name: Computational Discovery of Numerical Algorithms for Animation and Simulation
    of Natural Phenomena
publication: Foundations of Computational Mathematics
publication_identifier:
  eissn:
  - 1615-3383
  issn:
  - 1615-3375
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
researchdata_availability: no
scopus_import: '1'
status: public
supplementarymaterial: no
title: Quantitative convergence of a discretization of dynamic optimal transport using
  the dual formulation
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: 26
year: '2026'
...
---
OA_place: publisher
_id: '20551'
abstract:
- lang: eng
  text: "The space of codimension-2 shapes, such as curves in 3D and surfaces in 4D,
    is an infinite-dimensional manifold. This thesis explores geometric structures
    and dynamics on this space, with emphasis on their implications for physics, particularly
    hydrodynamics.\r\n\r\nOur investigation ranges from theoretical studies of infinite-dimensional
    symplectic and prequantum geometry to numerical computation of the time evolution
    of shapes. The thesis presents four main contributions.\r\n\r\nIn the first part,
    we introduce implicit representations of codimension-2 shapes using a class of
    complex-valued functions, and prove that the space of these implicit representations
    forms a prequantum bundle over the codimension-2 shape space. This reveals a new
    geometric interpretation of the canonical symplectic structure on the codimension-2
    shape space.\r\n\r\nIn the second part, we use implicit representations to develop
    a simulation method for the dynamics of space curves. To handle chaotic systems
    such as vortex filaments in hydrodynamics, we exploit the infinite degrees of
    freedom, hidden in both the configuration and dynamics of implicit representations.\r\n\r\nIn
    the third part, we introduce new symplectic structures on the space of space curves,
    which generalize the only previously known symplectic structure on this space,
    allowing for new Hamiltonian dynamics of space curves.\r\n\r\nIn the fourth part,
    we apply a symplectic viewpoint to a differential geometric problem with practical
    applications. We derive a new area formula for spherical polygons via prequantization. "
acknowledged_ssus:
- _id: CampIT
acknowledgement: "Projects contained in this thesis were financially supported in
  part by the\r\nEuropean Research Council with grants 1. ERC Consolidator Grant 101045083
  CoDiNA,\r\nand 2. the European Union’s Horizon 2020 research and innovation programme
  under grant\r\nagreement No. 638176."
alternative_title:
- ISTA Thesis
article_processing_charge: No
author:
- first_name: Sadashige
  full_name: Ishida, Sadashige
  id: 6F7C4B96-A8E9-11E9-A7CA-09ECE5697425
  last_name: Ishida
  orcid: 0000-0002-3121-3100
citation:
  ama: Ishida S. Symplectic-prequantum structures and dynamics on the codimension-2
    shape space. 2025. doi:<a href="https://doi.org/10.15479/AT-ISTA-20551">10.15479/AT-ISTA-20551</a>
  apa: Ishida, S. (2025). <i>Symplectic-prequantum structures and dynamics on the
    codimension-2 shape space</i>. Institute of Science and Technology Austria. <a
    href="https://doi.org/10.15479/AT-ISTA-20551">https://doi.org/10.15479/AT-ISTA-20551</a>
  chicago: Ishida, Sadashige. “Symplectic-Prequantum Structures and Dynamics on the
    Codimension-2 Shape Space.” Institute of Science and Technology Austria, 2025.
    <a href="https://doi.org/10.15479/AT-ISTA-20551">https://doi.org/10.15479/AT-ISTA-20551</a>.
  ieee: S. Ishida, “Symplectic-prequantum structures and dynamics on the codimension-2
    shape space,” Institute of Science and Technology Austria, 2025.
  ista: Ishida S. 2025. Symplectic-prequantum structures and dynamics on the codimension-2
    shape space. Institute of Science and Technology Austria.
  mla: Ishida, Sadashige. <i>Symplectic-Prequantum Structures and Dynamics on the
    Codimension-2 Shape Space</i>. Institute of Science and Technology Austria, 2025,
    doi:<a href="https://doi.org/10.15479/AT-ISTA-20551">10.15479/AT-ISTA-20551</a>.
  short: S. Ishida, Symplectic-Prequantum Structures and Dynamics on the Codimension-2
    Shape Space, Institute of Science and Technology Austria, 2025.
corr_author: '1'
date_created: 2025-10-27T10:28:52Z
date_published: 2025-10-31T00:00:00Z
date_updated: 2026-04-07T12:02:23Z
day: '31'
ddc:
- '516'
degree_awarded: PhD
department:
- _id: GradSch
- _id: ChWo
doi: 10.15479/AT-ISTA-20551
ec_funded: 1
file:
- access_level: open_access
  checksum: 4eef80afcb67691cbb6549c4756fa534
  content_type: application/zip
  creator: sishida
  date_created: 2025-11-01T18:26:14Z
  date_updated: 2025-11-01T18:26:14Z
  file_id: '20583'
  file_name: Thesis_tex.zip
  file_size: 72487812
  relation: source_file
- access_level: open_access
  checksum: 1e5a557900bf2dce01966b211b15d0fe
  content_type: application/pdf
  creator: sishida
  date_created: 2025-11-10T08:45:05Z
  date_updated: 2025-11-10T08:45:05Z
  file_id: '20623'
  file_name: Thesis_Sadashige_Ishida_PDFA.pdf
  file_size: 8945141
  relation: main_file
  success: 1
file_date_updated: 2025-11-10T08:45:05Z
fulldoi: https://doi.org/10.15479/AT-ISTA-20551
has_accepted_license: '1'
language:
- iso: eng
month: '10'
oa: 1
oa_version: Published Version
page: '141'
project:
- _id: 2533E772-B435-11E9-9278-68D0E5697425
  call_identifier: H2020
  grant_number: '638176'
  name: 'Big Splash: Efficient Simulation of Natural Phenomena at Extremely Large
    Scales'
- _id: 34bc2376-11ca-11ed-8bc3-9a3b3961a088
  grant_number: '101045083'
  name: Computational Discovery of Numerical Algorithms for Animation and Simulation
    of Natural Phenomena
publication_identifier:
  isbn:
  - 978-3-99078-070-1
  issn:
  - 2663-337X
publication_status: published
publisher: Institute of Science and Technology Austria
related_material:
  record:
  - id: '12846'
    relation: part_of_dissertation
    status: public
  - id: '12431'
    relation: part_of_dissertation
    status: public
  - id: '17361'
    relation: part_of_dissertation
    status: public
  - id: '20580'
    relation: part_of_dissertation
    status: public
status: public
supervisor:
- first_name: Christopher J
  full_name: Wojtan, Christopher J
  id: 3C61F1D2-F248-11E8-B48F-1D18A9856A87
  last_name: Wojtan
  orcid: 0000-0001-6646-5546
- first_name: Albert
  full_name: Chern, Albert
  last_name: Chern
title: Symplectic-prequantum structures and dynamics on the codimension-2 shape space
tmp:
  image: /images/cc_by.png
  legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
  name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
  short: CC BY (4.0)
type: dissertation
user_id: ba8df636-2132-11f1-aed0-ed93e2281fdd
year: '2025'
...
---
OA_place: repository
_id: '20580'
abstract:
- lang: eng
  text: This paper explores the geometry of the space of codimension-2 submanifolds.
    We implicitly represent these submanifolds by a class of complex-valued functions.
    This reveals a prequantum bundle structure over the space of submanifolds, equipped
    with the well-known Marsden-Weinstein symplectic structure. This bundle allows
    a new physical interpretation of the Marsden-Weinstein structure as the curvature
    of a connection form, which measures the average of volumes swept by the deformation
    of the S^1-family of hypersurfaces, defined as the phases of a complex function
    implicitly representing a submanifold.
article_processing_charge: No
arxiv: 1
author:
- first_name: Albert
  full_name: Chern, Albert
  last_name: Chern
- first_name: Sadashige
  full_name: Ishida, Sadashige
  id: 6F7C4B96-A8E9-11E9-A7CA-09ECE5697425
  last_name: Ishida
  orcid: 0000-0002-3121-3100
citation:
  ama: Chern A, Ishida S. Implicit representations of codimension-2 submanifolds and
    their prequantum structure. <i>arXiv</i>. doi:<a href="https://doi.org/10.48550/ARXIV.2507.11727">10.48550/ARXIV.2507.11727</a>
  apa: Chern, A., &#38; Ishida, S. (n.d.). Implicit representations of codimension-2
    submanifolds and their prequantum structure. <i>arXiv</i>. <a href="https://doi.org/10.48550/ARXIV.2507.11727">https://doi.org/10.48550/ARXIV.2507.11727</a>
  chicago: Chern, Albert, and Sadashige Ishida. “Implicit Representations of Codimension-2
    Submanifolds and Their Prequantum Structure.” <i>ArXiv</i>, n.d. <a href="https://doi.org/10.48550/ARXIV.2507.11727">https://doi.org/10.48550/ARXIV.2507.11727</a>.
  ieee: A. Chern and S. Ishida, “Implicit representations of codimension-2 submanifolds
    and their prequantum structure,” <i>arXiv</i>. .
  ista: Chern A, Ishida S. Implicit representations of codimension-2 submanifolds
    and their prequantum structure. arXiv, <a href="https://doi.org/10.48550/ARXIV.2507.11727">10.48550/ARXIV.2507.11727</a>.
  mla: Chern, Albert, and Sadashige Ishida. “Implicit Representations of Codimension-2
    Submanifolds and Their Prequantum Structure.” <i>ArXiv</i>, doi:<a href="https://doi.org/10.48550/ARXIV.2507.11727">10.48550/ARXIV.2507.11727</a>.
  short: A. Chern, S. Ishida, ArXiv (n.d.).
corr_author: '1'
date_created: 2025-10-30T18:36:56Z
date_published: 2025-07-15T00:00:00Z
date_updated: 2026-04-07T12:02:23Z
day: '15'
department:
- _id: GradSch
- _id: ChWo
doi: 10.48550/ARXIV.2507.11727
external_id:
  arxiv:
  - '2507.11727'
fulldoi: https://doi.org/10.48550/ARXIV.2507.11727
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.2507.11727
month: '07'
oa: 1
oa_version: Preprint
publication: arXiv
publication_status: draft
related_material:
  record:
  - id: '20551'
    relation: dissertation_contains
    status: public
status: public
title: Implicit representations of codimension-2 submanifolds and their prequantum
  structure
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: '2025'
...
---
OA_place: repository
OA_type: green
_id: '12846'
abstract:
- lang: eng
  text: We present a formula for the signed area of a spherical polygon via prequantization.
    In contrast to the traditional formula based on the Gauss-Bonnet theorem that
    requires measuring angles, the new formula mimics Green's theorem and is applicable
    to a wider range of degenerate spherical curves and polygons.
acknowledgement: This work was funded by European Research Council Consolidator grant
  101045083 CoDiNA and National Science Foundation CAREER award 2239062. Some figures
  in the article were generated by the software Houdini and its education license
  was provided by SideFX. The authors acknowledge anonymous referees for their reviews
  and insightful suggestions, and Chris Wojtan for his continuous support through
  discussions. The second author thanks Anna Sisak for a fruitful discussion on prequantum
  bundles.
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Albert
  full_name: Chern, Albert
  last_name: Chern
- first_name: Sadashige
  full_name: Ishida, Sadashige
  id: 6F7C4B96-A8E9-11E9-A7CA-09ECE5697425
  last_name: Ishida
  orcid: 0000-0002-3121-3100
citation:
  ama: Chern A, Ishida S. Area formula for spherical polygons via prequantization.
    <i>SIAM Journal on Applied Algebra and Geometry</i>. 2024;8(3):782-796. doi:<a
    href="https://doi.org/10.1137/23M1565255">10.1137/23M1565255</a>
  apa: Chern, A., &#38; Ishida, S. (2024). Area formula for spherical polygons via
    prequantization. <i>SIAM Journal on Applied Algebra and Geometry</i>. Society
    for Industrial and Applied Mathematics. <a href="https://doi.org/10.1137/23M1565255">https://doi.org/10.1137/23M1565255</a>
  chicago: Chern, Albert, and Sadashige Ishida. “Area Formula for Spherical Polygons
    via Prequantization.” <i>SIAM Journal on Applied Algebra and Geometry</i>. Society
    for Industrial and Applied Mathematics, 2024. <a href="https://doi.org/10.1137/23M1565255">https://doi.org/10.1137/23M1565255</a>.
  ieee: A. Chern and S. Ishida, “Area formula for spherical polygons via prequantization,”
    <i>SIAM Journal on Applied Algebra and Geometry</i>, vol. 8, no. 3. Society for
    Industrial and Applied Mathematics, pp. 782–796, 2024.
  ista: Chern A, Ishida S. 2024. Area formula for spherical polygons via prequantization.
    SIAM Journal on Applied Algebra and Geometry. 8(3), 782–796.
  mla: Chern, Albert, and Sadashige Ishida. “Area Formula for Spherical Polygons via
    Prequantization.” <i>SIAM Journal on Applied Algebra and Geometry</i>, vol. 8,
    no. 3, Society for Industrial and Applied Mathematics, 2024, pp. 782–96, doi:<a
    href="https://doi.org/10.1137/23M1565255">10.1137/23M1565255</a>.
  short: A. Chern, S. Ishida, SIAM Journal on Applied Algebra and Geometry 8 (2024)
    782–796.
corr_author: '1'
date_created: 2023-04-18T19:16:06Z
date_published: 2024-09-23T00:00:00Z
date_updated: 2026-04-07T12:02:22Z
day: '23'
ddc:
- '516'
department:
- _id: GradSch
- _id: ChWo
doi: 10.1137/23M1565255
external_id:
  arxiv:
  - '2303.14555'
  isi:
  - '001342265800009'
fulldoi: https://doi.org/10.1137/23M1565255
has_accepted_license: '1'
intvolume: '         8'
isi: 1
issue: '3'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.2303.14555
month: '09'
oa: 1
oa_version: Preprint
page: 782-796
project:
- _id: 34bc2376-11ca-11ed-8bc3-9a3b3961a088
  grant_number: '101045083'
  name: Computational Discovery of Numerical Algorithms for Animation and Simulation
    of Natural Phenomena
publication: SIAM Journal on Applied Algebra and Geometry
publication_identifier:
  eissn:
  - 2470-6566
publication_status: published
publisher: Society for Industrial and Applied Mathematics
quality_controlled: '1'
related_material:
  record:
  - id: '20551'
    relation: dissertation_contains
    status: public
scopus_import: '1'
status: public
title: Area formula for spherical polygons via prequantization
tmp:
  image: /images/cc_by.png
  legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
  name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
  short: CC BY (4.0)
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 8
year: '2024'
...
---
OA_place: repository
_id: '17361'
abstract:
- lang: eng
  text: We present symplectic structures on the shape space of unparameterized space
    curves that generalize the classical Marsden-Weinstein structure. Our method integrates
    the Liouville 1-form of the Marsden-Weinstein structure with Riemannian structures
    that have been introduced in mathematical shape analysis. We also derive Hamiltonian
    vector fields for several classical Hamiltonian functions with respect to these
    new symplectic structures.
acknowledgement: The authors are grateful to Boris Khesin for valuable comments on
  the MW symplectic structure and S. Ishida thanks Albert Chern for insightful discussions
  on space curves and Chris Wojtan for his continuous support. M. Bauer was partially
  supported by NSF grant DMS-1953244 and by the Binational Science Foundation (BSF).
  S. Ishida was partially supported by ERC Consolidator Grant 101045083 “CoDiNA” funded
  by the European Research Council. Some figures were generated by the software Houdini
  and its education license was provided by SideFX.
article_processing_charge: No
arxiv: 1
author:
- first_name: Martin
  full_name: Bauer, Martin
  last_name: Bauer
- first_name: Sadashige
  full_name: Ishida, Sadashige
  id: 6F7C4B96-A8E9-11E9-A7CA-09ECE5697425
  last_name: Ishida
  orcid: 0000-0002-3121-3100
- first_name: Peter W.
  full_name: Michor, Peter W.
  last_name: Michor
citation:
  ama: Bauer M, Ishida S, Michor PW. Symplectic structures on the space of space curves.
    <i>arXiv</i>. doi:<a href="https://doi.org/10.48550/arXiv.2407.19908">10.48550/arXiv.2407.19908</a>
  apa: Bauer, M., Ishida, S., &#38; Michor, P. W. (n.d.). Symplectic structures on
    the space of space curves. <i>arXiv</i>. <a href="https://doi.org/10.48550/arXiv.2407.19908">https://doi.org/10.48550/arXiv.2407.19908</a>
  chicago: Bauer, Martin, Sadashige Ishida, and Peter W. Michor. “Symplectic Structures
    on the Space of Space Curves.” <i>ArXiv</i>, n.d. <a href="https://doi.org/10.48550/arXiv.2407.19908">https://doi.org/10.48550/arXiv.2407.19908</a>.
  ieee: M. Bauer, S. Ishida, and P. W. Michor, “Symplectic structures on the space
    of space curves,” <i>arXiv</i>. .
  ista: Bauer M, Ishida S, Michor PW. Symplectic structures on the space of space
    curves. arXiv, <a href="https://doi.org/10.48550/arXiv.2407.19908">10.48550/arXiv.2407.19908</a>.
  mla: Bauer, Martin, et al. “Symplectic Structures on the Space of Space Curves.”
    <i>ArXiv</i>, doi:<a href="https://doi.org/10.48550/arXiv.2407.19908">10.48550/arXiv.2407.19908</a>.
  short: M. Bauer, S. Ishida, P.W. Michor, ArXiv (n.d.).
date_created: 2024-08-01T06:34:08Z
date_published: 2024-07-29T00:00:00Z
date_updated: 2026-04-28T09:59:01Z
day: '29'
department:
- _id: GradSch
- _id: ChWo
doi: 10.48550/arXiv.2407.19908
external_id:
  arxiv:
  - '2407.19908'
fulldoi: https://doi.org/10.48550/arXiv.2407.19908
keyword:
- space of space curves
- symplectic stuctures
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.2407.19908
month: '07'
oa: 1
oa_version: Preprint
project:
- _id: 34bc2376-11ca-11ed-8bc3-9a3b3961a088
  grant_number: '101045083'
  name: Computational Discovery of Numerical Algorithms for Animation and Simulation
    of Natural Phenomena
publication: arXiv
publication_status: draft
related_material:
  record:
  - id: '20551'
    relation: dissertation_contains
    status: public
  - id: '21743'
    relation: later_version
    status: public
status: public
title: Symplectic structures on the space of space curves
type: preprint
user_id: 8b945eb4-e2f2-11eb-945a-df72226e66a9
year: '2024'
...
---
_id: '12431'
abstract:
- lang: eng
  text: This paper presents a new representation of curve dynamics, with applications
    to vortex filaments in fluid dynamics. Instead of representing these filaments
    with explicit curve geometry and Lagrangian equations of motion, we represent
    curves implicitly with a new co-dimensional 2 level set description. Our implicit
    representation admits several redundant mathematical degrees of freedom in both
    the configuration and the dynamics of the curves, which can be tailored specifically
    to improve numerical robustness, in contrast to naive approaches for implicit
    curve dynamics that suffer from overwhelming numerical stability problems. Furthermore,
    we note how these hidden degrees of freedom perfectly map to a Clebsch representation
    in fluid dynamics. Motivated by these observations, we introduce untwisted level
    set functions and non-swirling dynamics which successfully regularize sources
    of numerical instability, particularly in the twisting modes around curve filaments.
    A consequence is a novel simulation method which produces stable dynamics for
    large numbers of interacting vortex filaments and effortlessly handles topological
    changes and re-connection events.
acknowledgement: We thank the visual computing group at IST Austria for their valuable
  discussions and feedback. Houdini Education licenses were provided by SideFX software.
  This project was funded in part by the European Research Council (ERC Consolidator
  Grant 101045083 CoDiNA).
article_number: '241'
article_processing_charge: No
article_type: original
author:
- first_name: Sadashige
  full_name: Ishida, Sadashige
  id: 6F7C4B96-A8E9-11E9-A7CA-09ECE5697425
  last_name: Ishida
  orcid: 0000-0002-3121-3100
- first_name: Christopher J
  full_name: Wojtan, Christopher J
  id: 3C61F1D2-F248-11E8-B48F-1D18A9856A87
  last_name: Wojtan
  orcid: 0000-0001-6646-5546
- first_name: Albert
  full_name: Chern, Albert
  last_name: Chern
citation:
  ama: Ishida S, Wojtan C, Chern A. Hidden degrees of freedom in implicit vortex filaments.
    <i>ACM Transactions on Graphics</i>. 2022;41(6). doi:<a href="https://doi.org/10.1145/3550454.3555459">10.1145/3550454.3555459</a>
  apa: Ishida, S., Wojtan, C., &#38; Chern, A. (2022). Hidden degrees of freedom in
    implicit vortex filaments. <i>ACM Transactions on Graphics</i>. Association for
    Computing Machinery. <a href="https://doi.org/10.1145/3550454.3555459">https://doi.org/10.1145/3550454.3555459</a>
  chicago: Ishida, Sadashige, Chris Wojtan, and Albert Chern. “Hidden Degrees of Freedom
    in Implicit Vortex Filaments.” <i>ACM Transactions on Graphics</i>. Association
    for Computing Machinery, 2022. <a href="https://doi.org/10.1145/3550454.3555459">https://doi.org/10.1145/3550454.3555459</a>.
  ieee: S. Ishida, C. Wojtan, and A. Chern, “Hidden degrees of freedom in implicit
    vortex filaments,” <i>ACM Transactions on Graphics</i>, vol. 41, no. 6. Association
    for Computing Machinery, 2022.
  ista: Ishida S, Wojtan C, Chern A. 2022. Hidden degrees of freedom in implicit vortex
    filaments. ACM Transactions on Graphics. 41(6), 241.
  mla: Ishida, Sadashige, et al. “Hidden Degrees of Freedom in Implicit Vortex Filaments.”
    <i>ACM Transactions on Graphics</i>, vol. 41, no. 6, 241, Association for Computing
    Machinery, 2022, doi:<a href="https://doi.org/10.1145/3550454.3555459">10.1145/3550454.3555459</a>.
  short: S. Ishida, C. Wojtan, A. Chern, ACM Transactions on Graphics 41 (2022).
date_created: 2023-01-29T23:00:59Z
date_published: 2022-12-01T00:00:00Z
date_updated: 2026-04-07T12:02:23Z
day: '01'
ddc:
- '000'
department:
- _id: ChWo
doi: 10.1145/3550454.3555459
external_id:
  isi:
  - '000891651900061'
file:
- access_level: open_access
  checksum: a2fba257fdefe0e747182be6c0f7c70c
  content_type: application/pdf
  creator: dernst
  date_created: 2023-01-30T07:15:48Z
  date_updated: 2023-01-30T07:15:48Z
  file_id: '12433'
  file_name: 2022_ACM_Ishida.pdf
  file_size: 15551202
  relation: main_file
  success: 1
file_date_updated: 2023-01-30T07:15:48Z
fulldoi: https://doi.org/10.1145/3550454.3555459
has_accepted_license: '1'
intvolume: '        41'
isi: 1
issue: '6'
language:
- iso: eng
month: '12'
oa: 1
oa_version: Published Version
project:
- _id: 34bc2376-11ca-11ed-8bc3-9a3b3961a088
  grant_number: '101045083'
  name: Computational Discovery of Numerical Algorithms for Animation and Simulation
    of Natural Phenomena
publication: ACM Transactions on Graphics
publication_identifier:
  eissn:
  - 1557-7368
  issn:
  - 0730-0301
publication_status: published
publisher: Association for Computing Machinery
quality_controlled: '1'
related_material:
  record:
  - id: '20551'
    relation: dissertation_contains
    status: public
scopus_import: '1'
status: public
title: Hidden degrees of freedom in implicit vortex filaments
tmp:
  image: /images/cc_by.png
  legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
  name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
  short: CC BY (4.0)
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 41
year: '2022'
...
---
_id: '8384'
abstract:
- lang: eng
  text: Previous research on animations of soap bubbles, films, and foams largely
    focuses on the motion and geometric shape of the bubble surface. These works neglect
    the evolution of the bubble’s thickness, which is normally responsible for visual
    phenomena like surface vortices, Newton’s interference patterns, capillary waves,
    and deformation-dependent rupturing of films in a foam. In this paper, we model
    these natural phenomena by introducing the film thickness as a reduced degree
    of freedom in the Navier-Stokes equations and deriving their equations of motion.
    We discretize the equations on a nonmanifold triangle mesh surface and couple
    it to an existing bubble solver. In doing so, we also introduce an incompressible
    fluid solver for 2.5D films and a novel advection algorithm for convecting fields
    across non-manifold surface junctions. Our simulations enhance state-of-the-art
    bubble solvers with additional effects caused by convection, rippling, draining,
    and evaporation of the thin film.
acknowledged_ssus:
- _id: ScienComp
acknowledgement: "We wish to thank the anonymous reviewers and the members of the
  Visual Computing Group at IST Austria for their valuable feedback, especially Camille
  Schreck for her help in rendering. This research was supported by the Scientific
  Service Units (SSU) of IST Austria through resources provided by Scientific Computing.
  We would like to thank the authors of [Belcour and Barla 2017] for providing their
  implementation, the authors of [Atkins and Elliott 2010] and [Seychelles et al.
  2008] for allowing us to use their results, and Rok Grah for helpful discussions.
  Finally, we thank Ryoichi Ando for many discussions from the beginning of the project
  that resulted in important contents of the paper including our formulation, numerical
  scheme, and initial implementation. This project has received funding from the\r\nEuropean
  Research Council (ERC) under the European Union’s Horizon 2020 research and innovation
  programme under grant agreement No. 638176."
article_number: '31'
article_processing_charge: No
article_type: original
author:
- first_name: Sadashige
  full_name: Ishida, Sadashige
  id: 6F7C4B96-A8E9-11E9-A7CA-09ECE5697425
  last_name: Ishida
  orcid: 0000-0002-3121-3100
- first_name: Peter
  full_name: Synak, Peter
  id: 331776E2-F248-11E8-B48F-1D18A9856A87
  last_name: Synak
- first_name: Fumiya
  full_name: Narita, Fumiya
  last_name: Narita
- first_name: Toshiya
  full_name: Hachisuka, Toshiya
  last_name: Hachisuka
- first_name: Christopher J
  full_name: Wojtan, Christopher J
  id: 3C61F1D2-F248-11E8-B48F-1D18A9856A87
  last_name: Wojtan
  orcid: 0000-0001-6646-5546
citation:
  ama: Ishida S, Synak P, Narita F, Hachisuka T, Wojtan C. A model for soap film dynamics
    with evolving thickness. <i>ACM Transactions on Graphics</i>. 2020;39(4). doi:<a
    href="https://doi.org/10.1145/3386569.3392405">10.1145/3386569.3392405</a>
  apa: Ishida, S., Synak, P., Narita, F., Hachisuka, T., &#38; Wojtan, C. (2020).
    A model for soap film dynamics with evolving thickness. <i>ACM Transactions on
    Graphics</i>. Association for Computing Machinery. <a href="https://doi.org/10.1145/3386569.3392405">https://doi.org/10.1145/3386569.3392405</a>
  chicago: Ishida, Sadashige, Peter Synak, Fumiya Narita, Toshiya Hachisuka, and Chris
    Wojtan. “A Model for Soap Film Dynamics with Evolving Thickness.” <i>ACM Transactions
    on Graphics</i>. Association for Computing Machinery, 2020. <a href="https://doi.org/10.1145/3386569.3392405">https://doi.org/10.1145/3386569.3392405</a>.
  ieee: S. Ishida, P. Synak, F. Narita, T. Hachisuka, and C. Wojtan, “A model for
    soap film dynamics with evolving thickness,” <i>ACM Transactions on Graphics</i>,
    vol. 39, no. 4. Association for Computing Machinery, 2020.
  ista: Ishida S, Synak P, Narita F, Hachisuka T, Wojtan C. 2020. A model for soap
    film dynamics with evolving thickness. ACM Transactions on Graphics. 39(4), 31.
  mla: Ishida, Sadashige, et al. “A Model for Soap Film Dynamics with Evolving Thickness.”
    <i>ACM Transactions on Graphics</i>, vol. 39, no. 4, 31, Association for Computing
    Machinery, 2020, doi:<a href="https://doi.org/10.1145/3386569.3392405">10.1145/3386569.3392405</a>.
  short: S. Ishida, P. Synak, F. Narita, T. Hachisuka, C. Wojtan, ACM Transactions
    on Graphics 39 (2020).
date_created: 2020-09-13T22:01:18Z
date_published: 2020-07-08T00:00:00Z
date_updated: 2026-04-16T08:29:36Z
day: '08'
ddc:
- '000'
department:
- _id: ChWo
doi: 10.1145/3386569.3392405
ec_funded: 1
external_id:
  isi:
  - '000583700300004'
file:
- access_level: open_access
  checksum: 813831ca91319d794d9748c276b24578
  content_type: application/pdf
  creator: dernst
  date_created: 2020-11-23T09:03:19Z
  date_updated: 2020-11-23T09:03:19Z
  file_id: '8795'
  file_name: 2020_soapfilm_submitted.pdf
  file_size: 14935529
  relation: main_file
  success: 1
file_date_updated: 2020-11-23T09:03:19Z
fulldoi: https://doi.org/10.1145/3386569.3392405
has_accepted_license: '1'
intvolume: '        39'
isi: 1
issue: '4'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.1145/3386569.3392405
month: '07'
oa: 1
oa_version: Submitted Version
project:
- _id: 2533E772-B435-11E9-9278-68D0E5697425
  call_identifier: H2020
  grant_number: '638176'
  name: 'Big Splash: Efficient Simulation of Natural Phenomena at Extremely Large
    Scales'
publication: ACM Transactions on Graphics
publication_identifier:
  eissn:
  - 1557-7368
  issn:
  - 0730-0301
publication_status: published
publisher: Association for Computing Machinery
quality_controlled: '1'
related_material:
  record:
  - id: '19630'
    relation: dissertation_contains
    status: public
scopus_import: '1'
status: public
title: A model for soap film dynamics with evolving thickness
type: journal_article
user_id: ba8df636-2132-11f1-aed0-ed93e2281fdd
volume: 39
year: '2020'
...