arXiv research

Study scattering rigidity on stationary manifolds using geodesics.

Study on recovering Lorentzian metrics from scattering data.

Researchers solve boundary and scattering rigidity problems for magnetic systems.

Study recovers Lorentzian metrics from boundary data, proving local rigidity.

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…

Study scattering rigidity for Hamiltonian systems, proving lens rigidity for non-trapping Finsler manifolds.

Analytic metrics are uniquely determined by their scattering map.

We consider the scattering and lens rigidity of compact surfaces with boundary that have a trapped geodesic. In particular we show that the flat cylinder and the flat Möbius strip are determined by their lens data. We also see by example that the flat Möbius strip is not determined by it's scattering data. We then cons…

We prove that if two non-trapping obstacles in $\mathbb{R}^n$ satisfy some rather weak non-degeneracy conditions and the scattering rays in their exteriors have (almost) the same travelling times or (almost) the same scattering length spectrum, then they coincide.

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…

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…

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…

Take two isomorphic convex co-compact co-infinite volume Kleinian groups, whose regular sets are diffeomorphic. The quotient of hyperbolic 3-space by these groups gives two hyperbolic 3-manifolds whose scattering operators may be compared. We prove that the operator norm of the difference between the scattering operato…

It has been shown in \cite{DPSU} that, under some additional assumptions, two simple domains with the same scattering data are equivalent. We show that the simplicity of a region can be read from the metric in the boundary and the scattering data. This lets us extend the results in \cite{DPSU} to regions with the same …

It was proved in \cite{NS1} that obstacles $K$ in $\R^d$ that are finite disjoint unions of strictly convex domains with $C^3$ boundaries are uniquely determined by the travelling times of billiard trajectories in their exteriors and also by their so called scattering length spectra. However the case $d = 2$ is not pro…

We consider the motion of a classical colored spinless particle under the influence of an external Yang-Mills potential $A$ on a compact manifold with boundary of dimension $\geq 3$. We show that under suitable convexity assumptions, we can recover the potential $A$, up to gauge transformations, from the lens data of t…

For a Riemannian manifold $(M,g)$ with strictly convex boundary $\partial M$, the lens data consists in the set of lengths of geodesics $γ$ with endpoints on $\partial M$, together with their endpoints $(x_-,x_+)\in \partial M\times \partial M$ and tangent exit vectors $(v_-,v_+)\in T_{x_-} M\times T_{x_+} M$. We show …

For a compact Riemannian manifold with boundary, endowed with a magnetic potential $α$, we consider the problem of restoring the metric $g$ and the magnetic potential $α$ from the values of the Mañé action potential between boundary points and the associated linearized problem. We study simple magnetic systems. In this…

Study travel time tomography for transversely isotropic media using modified pseudodifferential calculus.

Consider a broken geodesics $α([0,l])$ on a compact Riemannian manifold $(M,g)$ with boundary of dimension $n\geq 3$. The broken geodesics are unions of two geodesics with the property that they have a common end point. Assume that for every broken geodesic $α([0,l])$ starting at and ending to the boundary $\partial M$…

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.

Scattering representations simplify SBI for images without extra compression.

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…

MODWST improves classification tasks with wavelet scattering.

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

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

The Euclidean scattering transform was introduced nearly a decade ago to improve the mathematical understanding of convolutional neural networks. Inspired by recent interest in geometric deep learning, which aims to generalize convolutional neural networks to manifold and graph-structured domains, we define a geometric…

Bayesian Scattering offers a simple baseline for image data uncertainty.

Scattering theory for linearised gravity on Schwarzschild black hole exterior.

The scattering transform is a multilayered wavelet-based deep learning architecture that acts as a model of convolutional neural networks. Recently, several works have introduced generalizations of the scattering transform for non-Euclidean settings such as graphs. Our work builds upon these constructions by introducin…

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…

Scattering theory developed for linearised gravity near Schwarzschild black hole.

Establishes scattering theory for de Sitter vacuum solutions in even dimensions.

We develop a geometric scattering theory for a geometrically finite group acting on (a vector bundle over) a symmetric space of negative curvature. In particular, we obtain the meromorphic continuation of Eisenstein series and scattering matrices and their functional equations.

This paper summarizes closed-form relations for SE(3) maps and their derivatives.

We develop the scattering theory of general conformally compact metrics. For low frequencies, the domain of the scattering matrix is shown to be frequency dependent. In particular, generalized eigenfunctions exhibit L^2 decay in directions where the asymptotic curvature is sufficiently negative. The scattering matrix i…

Wavelet scattering spectra model non-Gaussian time-series, proving scale invariance for self-similar processes.