@unpublished{21737, abstract = {In calculus, l'Hopital's rule provides a simple way to evaluate the limits of quotient functions when both the numerator and denominator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.}, author = {Chern, Albert and Ishida, Sadashige}, booktitle = {arXiv}, keywords = {l’Hopital theorem, complex functions}, title = {{L'Hopital rules for complex-valued functions in higher dimensions}}, doi = {10.48550/ARXIV.2602.09958}, year = {2026}, } @article{21743, abstract = {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.}, author = {Bauer, Martin and Ishida, Sadashige and Michor, Peter W.}, issn = {1432-1467}, journal = {Journal of Nonlinear Science}, number = {2}, publisher = {Springer Nature}, title = {{Symplectic structures on the space of space curves}}, doi = {10.1007/s00332-026-10266-8}, volume = {36}, year = {2026}, } @article{14703, abstract = {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.}, author = {Ishida, Sadashige and Lavenant, Hugo}, issn = {1615-3383}, journal = {Foundations of Computational Mathematics}, keywords = {Optimal transport, Hamilton-Jacobi equation, convex optimization}, pages = {349--384}, publisher = {Springer Nature}, title = {{Quantitative convergence of a discretization of dynamic optimal transport using the dual formulation}}, doi = {10.1007/s10208-024-09686-3}, volume = {26}, year = {2026}, } @phdthesis{20551, abstract = {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. Our 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. In 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. In 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. In 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. In 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. }, author = {Ishida, Sadashige}, isbn = {978-3-99078-070-1}, issn = {2663-337X}, pages = {141}, publisher = {Institute of Science and Technology Austria}, title = {{Symplectic-prequantum structures and dynamics on the codimension-2 shape space}}, doi = {10.15479/AT-ISTA-20551}, year = {2025}, } @unpublished{20580, abstract = {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.}, author = {Chern, Albert and Ishida, Sadashige}, booktitle = {arXiv}, title = {{Implicit representations of codimension-2 submanifolds and their prequantum structure}}, doi = {10.48550/ARXIV.2507.11727}, year = {2025}, } @article{12846, abstract = {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.}, author = {Chern, Albert and Ishida, Sadashige}, issn = {2470-6566}, journal = {SIAM Journal on Applied Algebra and Geometry}, number = {3}, pages = {782--796}, publisher = {Society for Industrial and Applied Mathematics}, title = {{Area formula for spherical polygons via prequantization}}, doi = {10.1137/23M1565255}, volume = {8}, year = {2024}, } @unpublished{17361, abstract = {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.}, author = {Bauer, Martin and Ishida, Sadashige and Michor, Peter W.}, booktitle = {arXiv}, keywords = {space of space curves, symplectic stuctures}, title = {{Symplectic structures on the space of space curves}}, doi = {10.48550/arXiv.2407.19908}, year = {2024}, } @article{12431, abstract = {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.}, author = {Ishida, Sadashige and Wojtan, Christopher J and Chern, Albert}, issn = {1557-7368}, journal = {ACM Transactions on Graphics}, number = {6}, publisher = {Association for Computing Machinery}, title = {{Hidden degrees of freedom in implicit vortex filaments}}, doi = {10.1145/3550454.3555459}, volume = {41}, year = {2022}, } @article{8384, abstract = {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.}, author = {Ishida, Sadashige and Synak, Peter and Narita, Fumiya and Hachisuka, Toshiya and Wojtan, Christopher J}, issn = {1557-7368}, journal = {ACM Transactions on Graphics}, number = {4}, publisher = {Association for Computing Machinery}, title = {{A model for soap film dynamics with evolving thickness}}, doi = {10.1145/3386569.3392405}, volume = {39}, year = {2020}, }