Schedule for: 23w5015 - Applied and Computational Differential Geometry and Geometric PDEs

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

PDC lounge

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

We will discuss some of the recent work on the discrete conformal geometry of polyhedral surfaces, particularly focusing on the notion of discrete conformality, discrete uniformization, the prescribing curvature problem, and rigidity. We will also address some open problems that relate discrete conformal geometry to the Weyl problem on convex surfaces, the Koebe circle domain conjecture, and the Cauchy-Pogorelov rigidity theorem. This is a joint work with D. Gu, J. Sun, T. Wu, and Y. Luo.

We study the prescribed curvature problem for discrete conformal transformation of spherical cone-metrics and prove an existence and uniqueness theorem under elementary necessary assumptions. This can be viewed as a discrete version of a 1992 result by Luo and Tian. The talk is based on a joint work with Roman Prosanov and Tianqi Wu.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Meet in the PDC front desk for a guided tour of The Banff Centre campus.

With Liu, Yang, Siting Liu and Tingwei Meng, we present In Context Operator Networks (ICON), novel neural networks that can learn new operators from prompted states during the inference stage. ICON trains a single neural network as an operator learner. The approach seems promising with applications to physical systems governed by PDE's using artificial general intelligence. With Yat Tin Chow and Samy Wu Fung, we devise new, apparently successful optimization methods for minimizing non smooth, non convex functions. This is based on minimizing the Moreau envelope using stochastic variance gradient reduction, ptychography ideas, our previous work with Howard Heaton using Hamilton-Jacobi equations and a new convergence proof. Applications include phase retrieval.

Using Andrews' result on the contracting of anisotropy Gauss curvature flow, we re-establish existence of weak solutions to the Lutwak's $L^p$-Minkowski problem for the prescribed measures. The key is the study of the associated entropy and its monotonicity along the flow. This is a joint work with Karoly Boroczky.

We propose accurate, efficient, and convergent numerical methods for solving optimal mass transport equations arising from non-rigid image registration. To solve the model equation, we first transform the nonlinear PDEs into a Hamilton-Jacobi-Bellman (HJB) equation. We apply a mixed standard 7-point stencil and semi-Lagrangian wide stencil discretization, such that the numerical solution is guaranteed to converge to the viscosity solution of the Monge-Ampere equation. We design a numerical scheme that converges to the optimal transformation between the target and template images. Finally, we develop fast multigrid methods for solving the discrete nonlinear system. In particular, we use a four-directional alternating line relaxation scheme as smoother, a new coarsening strategy where wide stencil points are set as coarse grid points. Linear interpolation and injection are used in prolongation and restriction, respectively. Our numerical results show that the numerical solution yield good quality transformations for non-rigid image registration and the convergence rates of the proposed multigrid methods are mesh-independent.

Different Laplacian operators appear naturally in many fields. We will discuss some of the connections here, centered on the well-known connection between weighted graph Laplacians and discrete conformal variations that give rise to a class of finite volume Laplacians. Recent progress on expressing the determinant of finite volume Laplacians of a simplex, which can be used to determine nondegeneracy of these Laplacians on triangulated piecewise Euclidean manifolds, will be presented.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

The stable Bernstein problem for minimal hypersurfaces asks whether a complete, two-sided, stable minimal hypersurface in $R^{n+1}$ is flat. This problem is a geometric generalization of the classical Bernstein problem for the minimal surface equation, and has played a key role in the application of minimal surfaces in Riemannian geometry. When $n=2$, the question was answered in the affirmative in 1979. I will discuss recent solutions of this problem when $n=3$. The proof relies on an intriguing relation between the stability condition and the geometry of positive scalar curvature. If time permits, I will also talk about an extension to anisotropic minimal hypersurfaces. This is based on joint work with Otis Chodosh.

In this talk, we will address the problem of shape prior image segmentation, which aims to incorporate geometric information into the segmentation model to enhance the accuracy of the segmentation results. Prescribing priori shape information into the model can be beneficial, but the challenge is how to effectively integrate geometric information into the imaging model. We will discuss several shape prior image segmentation models that incorporate different geometric priors, such as topology, convexity, and general shape priors, and explore how computational quasiconformal geometry can be used to integrate these priors. We will examine various geometric quantities, including quasiconformality, discrete conformality structures and harmonic beltrami signature, to be used in the imaging models. Finally, we will explore the incorporation of deep neural networks into the segmentation model to achieve more accurate and efficient segmentation results.

Thurston's conjecture on the convergence of circle packings to the Riemann mapping is a constructive and geometric approach to the Riemann mapping theorem. The conjecture was solved elegantly by Rodin and Sullivan in 1987. In 2004, Bowers and Stephenson introduced the inversive distance circle packings as a natural generalization of Thurston's circle packings. They further conjectured that the discrete conformal maps induced by inversive distance circle packings converge to the Riemann mapping. In this talk, we will discuss some progress on Bowers-Stephenson's conjecture for Jordan domains. This is a joint work with Yuxiang Chen, Yanwen Luo and Siqi Zhang.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

This talk is about two apparently non-overlapping stories. One story is about shapes in space, their perimeter, their curvature, and about a notion of generalized Wasserstein barycenters. The other story is about machine learning, specifically about how to train learning models to be robust to adversarial perturbations of data. The bigger story in the talk will be about how these two stories interact with each other, how adversarial robustness motivates new notions of perimeter and curvature, and how geometry and optimal transport can cast new lights on and in this way reveal new faces of an important task in machine learning.

William Thurston proposed regarding the map induced from two circle packings with the same tangency pattern as a discrete holomorphic function. A discrete analogue of the Riemann mapping is deduced from Koebe-Andreev-Thurston theorem. One question is how to extend this theory to Riemann surfaces and relate classical conformal structures to discrete conformal structures. Since circles are preserved under complex projective transformations, it is natural to consider circle packings on surfaces with complex projective structures. Kojima, Mizushima and Tan conjectured that for a given combinatorics the deformation space of circle packings is homeomorphic to the Teichmueller space. In this talk, we consider the space of infinite circle patterns parametrised by discrete harmonic functions with finite energy, as well as its projection to the universal Teichmueller space.

Many computational algorithms in 3D require a mesh, or triangulation describing a surface, with high quality. Very small angles are problematic. I will describe an algorithm that efficiently creates a high quality mesh for surfaces in space. The algorithm gives guaranteed triangle angles between 35.2 and 101.5 degrees. Previous methods gave angles between 30 and 120 degrees. This is joint work with Maria Trnkova. The algorithm has been implemented and is available at gitlab.com/joelhass/midnormal.w

In this talk, we introduce a series of sampling-friendly, optimization-free methods for computing high-dimensional gradient flows and Hamiltonian flows on the Wasserstein probability manifold by leveraging generative models from deep learning. Such methods project the corresponding probability flows to parameter space and obtain finite-dimensional ordinary differential equations (ODEs) which can be directly solved by using classical numerical methods. Furthermore, the computed generative models can efficiently generate samples from the probability flows via pushforward maps.

In this talk I discuss recent joint work with Robert McCann and Kelvin Shuangjian Zhang in which we prove the $C^{1,1}$ regularity for solutions of principal agent problems. Principal agent problems are a class of variational problems arising in economics with strong links to optimal transport. Our proof relies on some classical ideas in elliptic regularity theory, as well as more recent concepts from the regularity theory for optimal transport.

We will introduce the spaces of geodesic triangulations of surfaces and explain their connection with diffeomorphisms groups of surfaces and applications in graph morphing problems.

We present a definition of stochastic Hamiltonian process on finite graph via its corresponding density dynamics in Wasserstein manifold. We demonstrate the existence of stochastic Hamiltonian process in many classical discrete problems, such as the optimal transport problem, Schr\"odinger equation and Schr\"odinger bridge problem (SBP). This is a joint work with Jianbo Cui (HK Poly) and Shu Liu (UCLA).

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

Beckner inequality is a higher dimensional counterpart of Moser-Trudinger-Onofri inequality on two dimensional sphere. We have obtained improved inequalities for axially symmetric functions with center of mass at the origin when the dimension $n=4, 6, 8$. Some computational questions remain for higher dimensions in order to extend our results.

In many machine learning tasks, the goal is to learn an unknown function which has some known group symmetries. Equivariant machine learning algorithms exploit this by devising architectures (=function spaces) which have these symmetries by construction. Examples include convolutional neural networks which respect translation symmetries, neural networks for graphs or sets which respect their permutation symmetries, or neural networks for 3D point sets which additionally respect Euclidean symmetries. A common theoretical requirement of symmetry based architecture is that they will be able to approximate any continuous function with the same symmetries. Using Stone-Weirstrass, this boils down to the ability of a function class to separate any two objects which are not related by a group symmetry . We will review results showing that under very general assumptions such a symmetry preserving separating mapping f exists, and the embedding dimension m can be taken to be roughly twice the dimension of the data. We will then propose a general methodology for efficient computation of such f using random invariants. This methodology is a generalization of the algebraic geometry argument used for the well known proof of phase retrieval injectivity. Finally, we will show how this result can be used to achieve efficient separation for point sets with respect to permutation and/or rotation actions. Based on work with Steven J. Gortler, Snir Hordan and Tal Amir and on the papers: [1] Low dimensional Invariant Embeddings for Universal Geometric Learning Nadav Dym and Steven J. Gortler [2] Complete Neural Networks for Euclidean Graphs Snir Hordan, Tal Amir, Steven J. Gortler and Nadav Dym

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

Standard numerical algorithms for differential equations typically apply also for geometric differential equations. There are however at least two aspects that require special attention. One is the desire for exact representation of some geometric property and the other is the handling of non-uniqueness and singularities based on geometric goals. We will exemplify the first aspect by the preservation of symplectic structures in dynamical systems. The second aspect will be exemplified by geometric optics as described by Hamilton-Jacobi equations, nonlinear conservation laws and the Monge–Ampère equation.

Unique continuation property is a fundamental property for harmonic functions, as well as a large class of elliptic and parabolic PDEs. It says that if a harmonic function vanishes at a point to infinite order, it must vanish everywhere. In the same spirit, we are interested in quantitative unique continuation problems, where we use the local growth rate of a harmonic function to deduce some global estimates, such as estimating the size of its singular set. In this talk, I will talk about some recent results together with C. Kenig on boundary unique continuation.

I will outline what we know about the homotopy-type of diffeomorphism groups of manifolds. In low dimensions we have fairly complete descriptions. In dimensions 4 and up there is significantly less known. Dimensions 4 and 5 are the most mysterious at present, but we have some uniform results that apply to all dimensions n>=4.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

We develop a novel computational framework to approximate solution operators of evolution partial differential equations (PDEs). By employing a general nonlinear reduced-order model, such as a deep neural network, to approximate the solution of a given PDE, we realize that the evolution of the model parameter is a control problem in the parameter space. Based on this observation, we propose to approximate the solution operator of the PDE by learning the control vector field in the parameter space. From any initial value, this control field can steer the parameter to generate a trajectory such that the corresponding reduced-order model solves the PDE. This allows for substantially reduced computational cost to solve the evolution PDE with arbitrary initial conditions. We also develop comprehensive error analysis for the proposed method when solving a large class of semilinear parabolic PDEs. Numerical experiments on different high-dimensional evolution PDEs with various initial conditions demonstrate the promising results of the proposed method.

In this talk, we will discuss curvature estimates for Hessian equations in hyperbolic space. A major difficulty of our problem is that the right hand side of the equation depends on the normal/gradient in a nontrivial way. We will discuss our recent progress in this direction. A crucial ingredient in our estimates is a concavity inequality for Hessian operator.

We focus on the transition path problem on manifold suggested by data points. Consider a Fokker-Planck equation on the manifold in terms of the reaction coordinates. We propose an implementable, unconditionally stable, data-driven finite volume scheme for this Fokker-Planck equation. This finite volume scheme defines a Markov chain on finite states. Using the Girsanov transformation for Markov chain, we choose the certain running cost and terminal cost for a stochastic optimal control problem in an infinite time horizon. We proved the discrete committor function gives an optimal control which drives the transition between local minimums efficiently. This optimal control also allows one to find a transition path on the finite states with minimum energy barrier efficiently.

In this talk we introduce a notion of discrete conformal equivalence for decorated piecewise euclidean surfaces (PE-surface), that is, PE-surfaces with a choice of circle about each vertex. It is closely related to inversive distance and hyperideal circle patterns. We discuss the corresponding discrete uniformization theorem which extends the known results for "undecorated" discrete conformal equivalence. Our proof relies on an intimate relationship between decorated PE-surfaces, canonical tessellations of hyperbolic surfaces and convex hyperbolic polyhedra. This is joint work with A. Bobenko.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

In this talk we will introduce some recent regularity results of free boundary in optimal transportation. Particularly for higher order regularity, when densities are $\alpha$-H\"older and domains are $C^2$, uniformly convex, we obtain the free boundary is $C^{2,\alpha}$. We also consider a model case that the target consists of two disjoint convex sets, in which singularities of optimal transport mapping arise. Under similar assumptions, we show that the singular set of the optimal mapping is an $(n-1)$-dimensional $C^{2,\alpha}$ regular sub-manifold of $\mathbb{R}^n$. These are based on a series of joint work with Shibing Chen and Xu-Jia Wang.

In order to compare and interpolate signals, we investigate a Riemannian geometry on the space of signals. The metric allows discontinuous signals and measures both horizontal (thus providing many benefits of the Wasserstein metric) and vertical deformations. Moreover, it allows for signed signals, which overcomes the main deficiency of optimal transportation-based metrics in signal processing. We characterize the metric properties of the space of signals and establish the regularity and stability of geodesics. Furthermore, we introduce an efficient numerical scheme to compute the geodesics and present several experiments which highlight the nature of the metric. This is joint work with Ruiyu Han (CMU) and Dejan Slepčev (CMU).

5-day workshop participants are welcome to use BIRS facilities (TCPL ) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 11AM.