arXiv research

Study fixed angle inverse scattering with Riemannian metrics, proving uniqueness and stability.

Study uses Lagrangian approach to prove limiting absorption principle on Riemannian spaces.

The scattering data of a Riemannian manifold with boundary record the incoming and outgoing directions of each geodesic passing through. We show that the scattering data of a generic Riemannian surface with no trapped geodesics and no conjugate points determine the lengths of geodesics. Counterexamples exists when trap…

Analytic metrics are uniquely determined by their scattering map.

Sharp inequalities and symmetries on Riemannian surfaces quantified.

Scattering rigidity of a Riemannian manifold allows one to tell the metric of a manifold with boundary by looking at the directions of geodesics at the boundary. Lens rigidity allows one to tell the metric of a manifold with boundary from the same information plus the length of geodesics. There are a variety of results…

Geometric wavelet scattering on manifolds improves neural network understanding.

Study on recovering Lorentzian metrics from scattering data.

Researchers solve boundary and scattering rigidity problems for magnetic systems.

We analyze the inverse problem, originally formulated by Dix in geophysics, of reconstructing the wave speed inside a domain from boundary measurements associated with the single scattering of seismic waves. We consider a domain $\tilde M$ with a varying and possibly anisotropic wave speed which we model as a Riemannia…

Given a smooth non-trapping compact manifold with strictly con- vex boundary, we consider an inverse problem of reconstructing the manifold from the scattering data initiated from internal sources. This data consist of the exit directions of geodesics that are emaneted from interior points of the manifold. We show that…

We prove that the flat product metric on $D^n\times S^1$ is scattering rigid where $D^n$ is the unit ball in $\R^n$ and $n\geq 2$. The scattering data (loosely speaking) of a Riemannian manifold with boundary is map $S:U^+\partial M\to U^-\partial M$ from unit vectors $V$ at the boundary that point inward to unit vecto…

Consider a compact Riemannian manifold with boundary endowed with a magnetic field. A path taken by a particle of unit charge, mass, and energy is called a magnetic geodesic. It is shown that if everything is real-analytic, the topology, metric, and magnetic field are uniquely determined by the scattering relation of t…

New methods implement manifold scattering transform for high-dimensional point cloud data.

Unified geometric scattering model for measure spaces.

In this paper we introduce a notion of scattering theory for the Laplace-Beltrami operator on non-compact, connected and complete Riemannian manifolds. A principal condition is given by a certain positive lower bound of the second fundamental form of angular submanifolds at infinity. Another condition is certain bounds…

The nonlinear equations describing all the nonsingular pencils of metrics of constant Riemannian curvature are derived and the integrability of these nonlinear equations by the method of inverse scattering problem is proved. It is proved that all the nonsingular pairs of compatible metrics of constant Riemannian curvat…

We consider an inverse problem associated with some 2-dimensional non-compact surfaces with conical singularities, cusps and regular ends. Our motivating example is a Riemann surface $\mathcal M = Γ\backslash{\bf H}^2$ associated with a Fuchsian group of the 1st kind $Γ$ containing parabolic elements. $\mathcal M$ is t…

Study proves rigidity results for analytic spacetimes without boundary or timelike boundary.

In this paper, we consider a compact Riemannian manifold with boundary, endowed with a magnetic potential $α$ and a potential $U$. For brevity, this type of systems are called $\MP$-systems. On simple $\MP$-systems, we consider both the boundary rigidity problem and scattering rigidity problem, see the introduction for…

New proofs confirm travel time data determine simple metrics on a disc.

Two operators are equivalent in geometric scattering theory under certain conditions.

We present a riemannian structure on the disk that has a remarkably rich structure. Geodesics are hypocycloids and the (negative of the) laplacian has integer spectrum with multiplicity the Dirichlet divisor function. Eigenfunctions of the laplacian are orthogonal polynomials naturally suited to the analysis of acousti…

For a given smooth compact manifold $M$, we introduce an open class $\mathcal G(M)$ of Riemannian metrics, which we call \emph{metrics of the gradient type}. For such metrics $g$, the geodesic flow $v^g$ on the spherical tangent bundle $SM \to M$ admits a Lyapunov function (so the $v^g$-flow is traversing). It turns ou…

We introduce scattering-symplectic manifolds, manifolds with a type of minimally degenerate Poisson structure that is not too restrictive so as to have a large class of examples, yet restrictive enough for standard Poisson invariants to be computable. This paper will demonstrate the potential of the scattering symplect…

We introduce general scattering transforms as mathematical models of deep neural networks with l2 pooling. Scattering networks iteratively apply complex valued unitary operators, and the pooling is performed by a complex modulus. An expected scattering defines a contractive representation of a high-dimensional probabil…

Paper explains scattering diagrams' role in mirror symmetry.

New method uses broken scattering to uniquely identify Finsler manifolds.

GSAN learns adaptive node representations using geometric scattering and attention.

New method learns soliton dynamics from scattering data without assuming known equations.

Unified graph scattering transforms improve theoretical properties of graph neural networks.

Scattering representations simplify SBI for images without extra compression.

New criterion for wave operators on Kato-Ricci manifolds.

The paper establishes scattering theory for wave equations on Schwarzschild spacetime.

Paper develops formulas for shape derivatives in wave scattering.

The Euclidean scattering transform was introduced nearly a decade ago to improve the mathematical understanding of the success of convolutional neural networks (ConvNets) in image data analysis and other tasks. Inspired by recent interest in geometric deep learning, which aims to generalize ConvNets to manifold and gra…

Let $g$ and $\tilde{g}$ be Riemannian metrics on a noncompact manifold $M$, which are conformally equivalent. We show that under a very mild \emph{first order} control on the conformal factor, the wave operators corresponding to the Hodge-Laplacians $Δ_g$ and $Δ_{\tilde{g}}$ acting on differential forms exist and are c…

Study scattering rigidity on stationary manifolds using geodesics.

MODWST improves classification tasks with wavelet scattering.

Study on scattering geodesics on modular surface and their sojourn times.

Graph scattering transforms are stable to metric perturbations of network topology.

Scattering theory on Riemann surfaces with explicit matrix and generalized period mappings.

Bayesian Scattering offers a simple baseline for image data uncertainty.

Scattering theory for linearised gravity on Schwarzschild black hole exterior.

We construct continuous families of scattering manifolds with the same scattering phase. The manifolds are compactly supported metric perturbations of Euclidean $\mathbf{R}^{n}$ for $n\geq8$. The metric perturbation may have arbitrarily small support.

Scattering theory for harmonic one-forms on Riemann surfaces.

Paper shows how scattering maps of Schrödinger equations relate to metrics.

The problem of recovering the asymptotics of a short range perturbation of the Euclidean metric on R^n from fixed energy scattering data is studied. It is shown that if two such metrics, g1, g2, have scattering data at some fixed energy which are equal up to smoothing, then there exists a diffeomorphism ψ`fixing infini…