[{"license":"https://creativecommons.org/licenses/by/4.0/","OA_place":"repository","date_created":"2026-04-15T16:28:24Z","fulldoi":"https://doi.org/10.48550/ARXIV.2602.09958","file":[{"success":1,"checksum":"6a76591c723d3e949ad5afa9f7dbb2ee","file_size":867109,"date_updated":"2026-04-28T10:53:27Z","file_name":"2026_arXiv_2602.09958.pdf","relation":"main_file","date_created":"2026-04-28T10:53:27Z","file_id":"21771","access_level":"open_access","content_type":"application/pdf","creator":"dernst"}],"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","abstract":[{"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.","lang":"eng"}],"ddc":["510"],"OA_type":"green","author":[{"last_name":"Chern","full_name":"Chern, Albert","first_name":"Albert"},{"full_name":"Ishida, Sadashige","last_name":"Ishida","id":"6F7C4B96-A8E9-11E9-A7CA-09ECE5697425","orcid":"0000-0002-3121-3100","first_name":"Sadashige"}],"oa":1,"file_date_updated":"2026-04-28T10:53:27Z","status":"public","title":"L'Hopital rules for complex-valued functions in higher dimensions","arxiv":1,"project":[{"name":"Computational Discovery of Numerical Algorithms for Animation and Simulation of Natural Phenomena","grant_number":"101045083","_id":"34bc2376-11ca-11ed-8bc3-9a3b3961a088"}],"month":"02","_id":"21737","date_published":"2026-02-10T00:00:00Z","corr_author":"1","date_updated":"2026-04-28T10:56:30Z","day":"10","oa_version":"Preprint","external_id":{"arxiv":["2602.09958"]},"keyword":["l’Hopital theorem","complex functions"],"citation":{"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>.","ieee":"A. Chern and S. Ishida, “L’Hopital rules for complex-valued functions in higher dimensions,” <i>arXiv</i>. .","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>","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>.","short":"A. Chern, S. Ishida, ArXiv (n.d.).","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>"},"publication":"arXiv","article_processing_charge":"No","tmp":{"short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png"},"has_accepted_license":"1","department":[{"_id":"GradSch"},{"_id":"ChWo"}],"year":"2026","type":"preprint","doi":"10.48550/ARXIV.2602.09958","publication_status":"submitted","language":[{"iso":"eng"}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87"},{"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."}],"article_type":"original","article_number":"45","publication_identifier":{"eissn":["1432-1467"],"issn":["0938-8974"]},"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.","related_material":{"record":[{"relation":"earlier_version","status":"public","id":"17361"}]},"fulldoi":"https://doi.org/10.1007/s00332-026-10266-8","file":[{"success":1,"file_size":1108518,"checksum":"760de2631b6fd7d57bcd5115ed36c0a2","date_updated":"2026-04-28T09:55:32Z","file_name":"2026_JourNonlinearScience_Bauer.pdf","relation":"main_file","date_created":"2026-04-28T09:55:32Z","access_level":"open_access","file_id":"21770","creator":"dernst","content_type":"application/pdf"}],"OA_place":"publisher","date_created":"2026-04-16T07:29:17Z","author":[{"first_name":"Martin","full_name":"Bauer, Martin","last_name":"Bauer"},{"full_name":"Ishida, Sadashige","id":"6F7C4B96-A8E9-11E9-A7CA-09ECE5697425","last_name":"Ishida","orcid":"0000-0002-3121-3100","first_name":"Sadashige"},{"full_name":"Michor, Peter W.","last_name":"Michor","first_name":"Peter W."}],"OA_type":"hybrid","ddc":["510"],"month":"04","project":[{"name":"Computational Discovery of Numerical Algorithms for Animation and Simulation of Natural Phenomena","grant_number":"101045083","_id":"34bc2376-11ca-11ed-8bc3-9a3b3961a088"}],"arxiv":1,"title":"Symplectic structures on the space of space curves","volume":36,"status":"public","file_date_updated":"2026-04-28T09:55:32Z","oa":1,"date_updated":"2026-04-28T09:59:01Z","quality_controlled":"1","date_published":"2026-04-15T00:00:00Z","scopus_import":"1","_id":"21743","PlanS_conform":"1","oa_version":"Published Version","day":"15","intvolume":"        36","external_id":{"arxiv":["2407.19908"]},"has_accepted_license":"1","article_processing_charge":"Yes (via OA deal)","tmp":{"short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png"},"publication":"Journal of Nonlinear Science","citation":{"ista":"Bauer M, Ishida S, Michor PW. 2026. Symplectic structures on the space of space curves. Journal of Nonlinear Science. 36(2), 45.","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>.","short":"M. Bauer, S. Ishida, P.W. Michor, Journal of Nonlinear Science 36 (2026).","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>","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>","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.","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>."},"publication_status":"published","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","language":[{"iso":"eng"}],"doi":"10.1007/s00332-026-10266-8","publisher":"Springer Nature","type":"journal_article","department":[{"_id":"GradSch"},{"_id":"ChWo"}],"issue":"2","year":"2026"},{"researchdata_availability":"no","ddc":["000"],"OA_type":"hybrid","author":[{"first_name":"Sadashige","orcid":"0000-0002-3121-3100","id":"6F7C4B96-A8E9-11E9-A7CA-09ECE5697425","last_name":"Ishida","full_name":"Ishida, Sadashige"},{"first_name":"Hugo","full_name":"Lavenant, Hugo","last_name":"Lavenant"}],"supplementarymaterial":"no","OA_place":"publisher","date_created":"2023-12-21T10:14:37Z","file":[{"date_updated":"2026-07-23T05:37:52Z","success":1,"file_size":1240012,"checksum":"30671f88e792e8b75ae3e698ac4c131c","access_level":"open_access","file_id":"22384","content_type":"application/pdf","creator":"dernst","date_created":"2026-07-23T05:37:52Z","file_name":"2026_FoundCompMath_Ishida.pdf","relation":"main_file"}],"fulldoi":"https://doi.org/10.1007/s10208-024-09686-3","das_tickbox":"0","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_type":"original","publication_identifier":{"issn":["1615-3375"],"eissn":["1615-3383"]},"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."}],"_id":"14703","scopus_import":"1","date_published":"2026-02-01T00:00:00Z","quality_controlled":"1","corr_author":"1","date_updated":"2026-07-23T05:39:38Z","oa":1,"file_date_updated":"2026-07-23T05:37:52Z","page":"349-384","status":"public","volume":26,"title":"Quantitative convergence of a discretization of dynamic optimal transport using the dual formulation","arxiv":1,"project":[{"name":"Computational Discovery of Numerical Algorithms for Animation and Simulation of Natural Phenomena","_id":"34bc2376-11ca-11ed-8bc3-9a3b3961a088","grant_number":"101045083"}],"month":"02","external_id":{"arxiv":["2312.12213"],"isi":["001352503300001"]},"intvolume":"        26","oa_version":"Published Version","day":"01","PlanS_conform":"1","department":[{"_id":"GradSch"},{"_id":"ChWo"}],"year":"2026","isi":1,"type":"journal_article","publisher":"Springer Nature","doi":"10.1007/s10208-024-09686-3","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","language":[{"iso":"eng"}],"publication_status":"published","keyword":["Optimal transport","Hamilton-Jacobi equation","convex optimization"],"citation":{"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>.","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>","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.","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>.","short":"S. Ishida, H. Lavenant, Foundations of Computational Mathematics 26 (2026) 349–384.","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>"},"publication":"Foundations of Computational Mathematics","article_processing_charge":"Yes (via OA deal)","tmp":{"short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png"},"has_accepted_license":"1"},{"citation":{"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>.","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>","ieee":"S. Ishida, “Symplectic-prequantum structures and dynamics on the codimension-2 shape space,” Institute of Science and Technology Austria, 2025.","short":"S. Ishida, Symplectic-Prequantum Structures and Dynamics on the Codimension-2 Shape Space, Institute of Science and Technology Austria, 2025.","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>.","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>","ista":"Ishida S. 2025. Symplectic-prequantum structures and dynamics on the codimension-2 shape space. Institute of Science and Technology Austria."},"article_processing_charge":"No","tmp":{"short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png"},"has_accepted_license":"1","department":[{"_id":"GradSch"},{"_id":"ChWo"}],"year":"2025","type":"dissertation","publisher":"Institute of Science and Technology Austria","ec_funded":1,"doi":"10.15479/AT-ISTA-20551","alternative_title":["ISTA Thesis"],"acknowledged_ssus":[{"_id":"CampIT"}],"publication_status":"published","user_id":"ba8df636-2132-11f1-aed0-ed93e2281fdd","language":[{"iso":"eng"}],"supervisor":[{"orcid":"0000-0001-6646-5546","first_name":"Christopher J","full_name":"Wojtan, Christopher J","id":"3C61F1D2-F248-11E8-B48F-1D18A9856A87","last_name":"Wojtan"},{"first_name":"Albert","last_name":"Chern","full_name":"Chern, Albert"}],"oa_version":"Published Version","day":"31","oa":1,"file_date_updated":"2025-11-10T08:45:05Z","page":"141","status":"public","title":"Symplectic-prequantum structures and dynamics on the codimension-2 shape space","project":[{"name":"Big Splash: Efficient Simulation of Natural Phenomena at Extremely Large Scales","grant_number":"638176","call_identifier":"H2020","_id":"2533E772-B435-11E9-9278-68D0E5697425"},{"name":"Computational Discovery of Numerical Algorithms for Animation and Simulation of Natural Phenomena","_id":"34bc2376-11ca-11ed-8bc3-9a3b3961a088","grant_number":"101045083"}],"month":"10","_id":"20551","date_published":"2025-10-31T00:00:00Z","corr_author":"1","date_updated":"2026-04-07T12:02:23Z","date_created":"2025-10-27T10:28:52Z","OA_place":"publisher","related_material":{"record":[{"relation":"part_of_dissertation","id":"12846","status":"public"},{"relation":"part_of_dissertation","id":"12431","status":"public"},{"relation":"part_of_dissertation","id":"17361","status":"public"},{"id":"20580","status":"public","relation":"part_of_dissertation"}]},"fulldoi":"https://doi.org/10.15479/AT-ISTA-20551","file":[{"relation":"source_file","file_name":"Thesis_tex.zip","date_created":"2025-11-01T18:26:14Z","content_type":"application/zip","creator":"sishida","file_id":"20583","access_level":"open_access","checksum":"4eef80afcb67691cbb6549c4756fa534","file_size":72487812,"date_updated":"2025-11-01T18:26:14Z"},{"creator":"sishida","content_type":"application/pdf","access_level":"open_access","file_id":"20623","date_created":"2025-11-10T08:45:05Z","relation":"main_file","file_name":"Thesis_Sadashige_Ishida_PDFA.pdf","date_updated":"2025-11-10T08:45:05Z","checksum":"1e5a557900bf2dce01966b211b15d0fe","file_size":8945141,"success":1}],"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.","publication_identifier":{"isbn":["978-3-99078-070-1"],"issn":["2663-337X"]},"abstract":[{"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. ","lang":"eng"}],"ddc":["516"],"author":[{"full_name":"Ishida, Sadashige","last_name":"Ishida","id":"6F7C4B96-A8E9-11E9-A7CA-09ECE5697425","orcid":"0000-0002-3121-3100","first_name":"Sadashige"}],"degree_awarded":"PhD"},{"main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2507.11727","open_access":"1"}],"author":[{"full_name":"Chern, Albert","last_name":"Chern","first_name":"Albert"},{"orcid":"0000-0002-3121-3100","first_name":"Sadashige","full_name":"Ishida, Sadashige","id":"6F7C4B96-A8E9-11E9-A7CA-09ECE5697425","last_name":"Ishida"}],"external_id":{"arxiv":["2507.11727"]},"OA_place":"repository","date_created":"2025-10-30T18:36:56Z","fulldoi":"https://doi.org/10.48550/ARXIV.2507.11727","related_material":{"record":[{"relation":"dissertation_contains","id":"20551","status":"public"}]},"oa_version":"Preprint","day":"15","abstract":[{"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.","lang":"eng"}],"_id":"20580","department":[{"_id":"GradSch"},{"_id":"ChWo"}],"year":"2025","date_published":"2025-07-15T00:00:00Z","type":"preprint","corr_author":"1","date_updated":"2026-04-07T12:02:23Z","doi":"10.48550/ARXIV.2507.11727","language":[{"iso":"eng"}],"publication_status":"draft","user_id":"8b945eb4-e2f2-11eb-945a-df72226e66a9","oa":1,"citation":{"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.).","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>.","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>","ieee":"A. Chern and S. Ishida, “Implicit representations of codimension-2 submanifolds and their prequantum structure,” <i>arXiv</i>. ."},"status":"public","title":"Implicit representations of codimension-2 submanifolds and their prequantum structure","publication":"arXiv","arxiv":1,"article_processing_charge":"No","tmp":{"short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png"},"month":"07"},{"date_updated":"2026-04-07T12:02:22Z","quality_controlled":"1","corr_author":"1","_id":"12846","date_published":"2024-09-23T00:00:00Z","scopus_import":"1","arxiv":1,"month":"09","project":[{"grant_number":"101045083","_id":"34bc2376-11ca-11ed-8bc3-9a3b3961a088","name":"Computational Discovery of Numerical Algorithms for Animation and Simulation of Natural Phenomena"}],"page":"782-796","oa":1,"title":"Area formula for spherical polygons via prequantization","volume":8,"status":"public","OA_type":"green","author":[{"first_name":"Albert","last_name":"Chern","full_name":"Chern, Albert"},{"orcid":"0000-0002-3121-3100","first_name":"Sadashige","full_name":"Ishida, Sadashige","id":"6F7C4B96-A8E9-11E9-A7CA-09ECE5697425","last_name":"Ishida"}],"ddc":["516"],"main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2303.14555","open_access":"1"}],"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.","related_material":{"record":[{"id":"20551","status":"public","relation":"dissertation_contains"}]},"fulldoi":"https://doi.org/10.1137/23M1565255","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."}],"publication_identifier":{"eissn":["2470-6566"]},"article_type":"original","date_created":"2023-04-18T19:16:06Z","OA_place":"repository","doi":"10.1137/23M1565255","publisher":"Society for Industrial and Applied Mathematics","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","language":[{"iso":"eng"}],"publication_status":"published","issue":"3","year":"2024","department":[{"_id":"GradSch"},{"_id":"ChWo"}],"type":"journal_article","isi":1,"tmp":{"short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png"},"article_processing_charge":"No","publication":"SIAM Journal on Applied Algebra and Geometry","has_accepted_license":"1","citation":{"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.","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>.","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>","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."},"external_id":{"arxiv":["2303.14555"],"isi":["001342265800009"]},"oa_version":"Preprint","day":"23","intvolume":"         8"},{"title":"Symplectic structures on the space of space curves","status":"public","oa":1,"month":"07","project":[{"_id":"34bc2376-11ca-11ed-8bc3-9a3b3961a088","grant_number":"101045083","name":"Computational Discovery of Numerical Algorithms for Animation and Simulation of Natural Phenomena"}],"arxiv":1,"date_published":"2024-07-29T00:00:00Z","_id":"17361","date_updated":"2026-04-28T09:59:01Z","OA_place":"repository","date_created":"2024-08-01T06:34:08Z","abstract":[{"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.","lang":"eng"}],"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.","fulldoi":"https://doi.org/10.48550/arXiv.2407.19908","related_material":{"record":[{"status":"public","relation":"dissertation_contains","id":"20551"},{"id":"21743","status":"public","relation":"later_version"}]},"main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.2407.19908"}],"author":[{"last_name":"Bauer","full_name":"Bauer, Martin","first_name":"Martin"},{"full_name":"Ishida, Sadashige","id":"6F7C4B96-A8E9-11E9-A7CA-09ECE5697425","last_name":"Ishida","orcid":"0000-0002-3121-3100","first_name":"Sadashige"},{"last_name":"Michor","full_name":"Michor, Peter W.","first_name":"Peter W."}],"citation":{"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>.","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>","ieee":"M. Bauer, S. Ishida, and P. W. Michor, “Symplectic structures on the space of space curves,” <i>arXiv</i>. .","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>.","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>","short":"M. Bauer, S. Ishida, P.W. Michor, ArXiv (n.d.).","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>."},"keyword":["space of space curves","symplectic stuctures"],"article_processing_charge":"No","publication":"arXiv","type":"preprint","year":"2024","department":[{"_id":"GradSch"},{"_id":"ChWo"}],"publication_status":"draft","language":[{"iso":"eng"}],"user_id":"8b945eb4-e2f2-11eb-945a-df72226e66a9","doi":"10.48550/arXiv.2407.19908","oa_version":"Preprint","day":"29","external_id":{"arxiv":["2407.19908"]}},{"date_created":"2023-01-29T23:00:59Z","fulldoi":"https://doi.org/10.1145/3550454.3555459","related_material":{"record":[{"id":"20551","relation":"dissertation_contains","status":"public"}]},"file":[{"content_type":"application/pdf","creator":"dernst","access_level":"open_access","file_id":"12433","relation":"main_file","file_name":"2022_ACM_Ishida.pdf","date_created":"2023-01-30T07:15:48Z","date_updated":"2023-01-30T07:15:48Z","checksum":"a2fba257fdefe0e747182be6c0f7c70c","file_size":15551202,"success":1}],"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_type":"original","publication_identifier":{"eissn":["1557-7368"],"issn":["0730-0301"]},"article_number":"241","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."}],"ddc":["000"],"author":[{"first_name":"Sadashige","orcid":"0000-0002-3121-3100","id":"6F7C4B96-A8E9-11E9-A7CA-09ECE5697425","last_name":"Ishida","full_name":"Ishida, Sadashige"},{"first_name":"Christopher J","orcid":"0000-0001-6646-5546","last_name":"Wojtan","id":"3C61F1D2-F248-11E8-B48F-1D18A9856A87","full_name":"Wojtan, Christopher J"},{"first_name":"Albert","full_name":"Chern, Albert","last_name":"Chern"}],"oa":1,"file_date_updated":"2023-01-30T07:15:48Z","status":"public","volume":41,"title":"Hidden degrees of freedom in implicit vortex filaments","project":[{"name":"Computational Discovery of Numerical Algorithms for Animation and Simulation of Natural Phenomena","_id":"34bc2376-11ca-11ed-8bc3-9a3b3961a088","grant_number":"101045083"}],"month":"12","_id":"12431","scopus_import":"1","date_published":"2022-12-01T00:00:00Z","quality_controlled":"1","date_updated":"2026-04-07T12:02:23Z","intvolume":"        41","day":"01","oa_version":"Published Version","external_id":{"isi":["000891651900061"]},"citation":{"ista":"Ishida S, Wojtan C, Chern A. 2022. Hidden degrees of freedom in implicit vortex filaments. ACM Transactions on Graphics. 41(6), 241.","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.","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>","short":"S. Ishida, C. Wojtan, A. Chern, ACM Transactions on Graphics 41 (2022).","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>.","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>."},"publication":"ACM Transactions on Graphics","tmp":{"short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png"},"article_processing_charge":"No","has_accepted_license":"1","department":[{"_id":"ChWo"}],"year":"2022","issue":"6","isi":1,"type":"journal_article","publisher":"Association for Computing Machinery","doi":"10.1145/3550454.3555459","publication_status":"published","user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","language":[{"iso":"eng"}]},{"intvolume":"        39","oa_version":"Submitted Version","day":"08","external_id":{"isi":["000583700300004"]},"citation":{"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).","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>.","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>","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>","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."},"has_accepted_license":"1","article_processing_charge":"No","publication":"ACM Transactions on Graphics","type":"journal_article","isi":1,"department":[{"_id":"ChWo"}],"issue":"4","year":"2020","publication_status":"published","user_id":"ba8df636-2132-11f1-aed0-ed93e2281fdd","language":[{"iso":"eng"}],"acknowledged_ssus":[{"_id":"ScienComp"}],"doi":"10.1145/3386569.3392405","ec_funded":1,"publisher":"Association for Computing Machinery","date_created":"2020-09-13T22:01:18Z","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."}],"article_number":"31","article_type":"original","publication_identifier":{"issn":["0730-0301"],"eissn":["1557-7368"]},"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.","related_material":{"record":[{"id":"19630","relation":"dissertation_contains","status":"public"}]},"fulldoi":"https://doi.org/10.1145/3386569.3392405","file":[{"date_created":"2020-11-23T09:03:19Z","file_name":"2020_soapfilm_submitted.pdf","relation":"main_file","file_id":"8795","access_level":"open_access","creator":"dernst","content_type":"application/pdf","success":1,"file_size":14935529,"checksum":"813831ca91319d794d9748c276b24578","date_updated":"2020-11-23T09:03:19Z"}],"ddc":["000"],"main_file_link":[{"url":"https://doi.org/10.1145/3386569.3392405","open_access":"1"}],"author":[{"orcid":"0000-0002-3121-3100","first_name":"Sadashige","full_name":"Ishida, Sadashige","id":"6F7C4B96-A8E9-11E9-A7CA-09ECE5697425","last_name":"Ishida"},{"first_name":"Peter","id":"331776E2-F248-11E8-B48F-1D18A9856A87","last_name":"Synak","full_name":"Synak, Peter"},{"first_name":"Fumiya","full_name":"Narita, Fumiya","last_name":"Narita"},{"first_name":"Toshiya","full_name":"Hachisuka, Toshiya","last_name":"Hachisuka"},{"orcid":"0000-0001-6646-5546","first_name":"Christopher J","full_name":"Wojtan, Christopher J","id":"3C61F1D2-F248-11E8-B48F-1D18A9856A87","last_name":"Wojtan"}],"title":"A model for soap film dynamics with evolving thickness","status":"public","volume":39,"file_date_updated":"2020-11-23T09:03:19Z","oa":1,"month":"07","project":[{"name":"Big Splash: Efficient Simulation of Natural Phenomena at Extremely Large Scales","grant_number":"638176","call_identifier":"H2020","_id":"2533E772-B435-11E9-9278-68D0E5697425"}],"date_published":"2020-07-08T00:00:00Z","scopus_import":"1","_id":"8384","date_updated":"2026-04-16T08:29:36Z","quality_controlled":"1"}]