Publications (25)
Penrose's eight-conic theorem
Russell Arnold, Albert Chern, Morten Eide +3
This article proves the following theorem, first enunciated by Roger Penrose about 70 years ago but never published: In , if conics are assigned to seven of the ve…
Elastic Curves via Geometric Mechanics
Oliver Gross, Rohit Jammula, Albert Chern
Elastic curves are the mathematical shapes of thin elastic rods in equilibrium, with deep connections to mechanics, geometry, and computer graphics. Traditionally described as stat…
A Reflectionless Discrete Perfectly Matched Layer
Albert Chern
Perfectly Matched Layer (PML) is a widely adopted non-reflecting boundary treatment for wave simulations. Reducing numerical reflections from a discretized PML has been a long last…
L'Hopital rules for complex-valued functions in higher dimensions
Albert Chern, Sadashige Ishida
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 beyon…
Fluid Implicit Particles on Coadjoint Orbits
Mohammad Sina Nabizadeh, Ritoban Roy-Chowdhury, Hang Yin +2
We propose Coadjoint Orbit FLIP (CO-FLIP), a high order accurate, structure preserving fluid simulation method in the hybrid Eulerian-Lagrangian framework. We start with a Hamilton…
Adaptive Surface Meshes from Harmonic Maps
Nicolas Nebel, Albert Chern
We present a novel shape-approximating anisotropic re-meshing algorithm as a geometric generalization of the adaptive moving mesh method. Conventional moving mesh methods reduce th…
Myopically Verifiable Probabilistic Certificate for Long-term Safety
Zhuoyuan Wang, Haoming Jing, Christian Kurniawan +2
In this paper, we consider the use of barrier function-based approaches for the safe control problem in stochastic systems. With the presence of stochastic uncertainties, a myopic…
Commuting Hamiltonian flows of curves in real space forms
Albert Chern, Felix Knöppel, Franz Pedit +1
Starting from the vortex filament flow introduced in 1906 by Da Rios, there is a hierarchy of commuting geometric flows on space curves. The traditional approach relates those flow…
Active nematic fluids on Riemannian 2-manifolds
Cuncheng Zhu, David Saintillan, Albert Chern
Recent advances in cell biology and experimental techniques using reconstituted cell extracts have generated significant interest in understanding how geometry and topology influen…
Wave Simulations in Infinite Spacetime
Chad McKell, Mohammad Sina Nabizadeh, Stephanie Wang +1
Solving the wave equation on an infinite domain has been an ongoing challenge in scientific computing. Conventional approaches to this problem only generate numerical solutions on…
Implicit representations of codimension-2 submanifolds and their prequantum structure
Albert Chern, Sadashige Ishida
This paper explores the geometry of the space of codimension-2 submanifolds. We implicitly represent these submanifolds by a class of complex-valued functions. We show that the spa…
DeepCurrents: Learning Implicit Representations of Shapes with Boundaries
David Palmer, Dmitriy Smirnov, Stephanie Wang +2
Recent techniques have been successful in reconstructing surfaces as level sets of learned functions (such as signed distance fields) parameterized by deep neural networks. Many of…
Lifting Directional Fields to Minimal Sections
David Palmer, Albert Chern, Justin Solomon
Directional fields, including unit vector, line, and cross fields, are essential tools in the geometry processing toolkit. The topology of directional fields is characterized by th…
Safe Control in the Presence of Stochastic Uncertainties
Albert Chern, Xiang Wang, Abhiram Iyer +1
Accurate quantification of safety is essential for the design of autonomous systems. In this paper, we present a methodology to characterize the exact probabilities associated with…
Center of circle after perspective transformation
Xi Wang, Albert Chern, Marc Alexa
Video-based glint-free eye tracking commonly estimates gaze direction based on the pupil center. The boundary of the pupil is fitted with an ellipse and the euclidean center of the…
Quadratic-Order Geodesics on Meshes
Yue Ruan, Albert Chern, Tzu-Mao Li +2
We introduce a novel representation and optimization framework for discrete geodesics on triangle meshes that reduces artifacts of linear methods on uneven and coarse discretizatio…
Stokes flow of an evolving fluid film with arbitrary shape and topology
Cuncheng Zhu, David Saintillan, Albert Chern
The dynamics of evolving fluid films in the viscous Stokes limit is relevant to various applications, such as the modeling of lipid bilayers in cells. While the governing equations…
Myopically Verifiable Probabilistic Certificates for Safe Control and Learning
Zhuoyuan Wang, Haoming Jing, Christian Kurniawan +2
This paper addresses the design of safety certificates for stochastic systems, with a focus on ensuring long-term safety through fast real-time control. In stochastic environments,…
Finding Conformal and Isometric Immersions of Surfaces
Albert Chern, Felix Knöppel, Franz Pedit +2
We introduce a family of variational functionals for spinor fields on a compact Riemann surface that can be used to find close-to-conformal immersions of into $\mathbb{R}^3…
Neural Representation of Minimal Surfaces
Jiayin Sun, Albert Chern
We propose a neural representation for minimal surfaces. Unlike prior approaches based on discretization or Physics-Informed Neural Networks (PINNs), where meshes or neural fields…
Area formula for spherical polygons via prequantization
Albert Chern, Sadashige Ishida
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 measurin…
Force-Free Fields are Conformally Geodesic
Albert Chern, Oliver Gross
In this paper, we establish an equivalence between force-free fields and conformally geodesic fields, and between harmonic fields and conformally eikonal fields in the context of c…
Hidden Degrees of Freedom in Implicit Vortex Filaments
Sadashige Ishida, Chris Wojtan, Albert Chern
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 geo…
Vakonomic Fluids
Ritoban Roy-Chowdhury, Mohammad Sina Nabizadeh, Oliver Gross +2
We introduce a novel discretization of the incompressible Euler equations based on their interpretation as geodesic equations on the Lie group of volume-preserving diffeomorphisms.…
Generalizable Physics-Informed Learning for Stochastic Safety-Critical Systems
Zhuoyuan Wang, Albert Chern, Yorie Nakahira
Accurate estimation of long-term risk is essential for the design and analysis of stochastic dynamical systems. Existing risk quantification methods typically rely on extensive dat…