arXiv research

New method uses broken scattering to uniquely identify Finsler manifolds.

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

Paper explains scattering diagrams' role in mirror symmetry.

Describes geodesic scattering on hyperboloids using quadrics results.

Study scattering rigidity on stationary manifolds using geodesics.

Paper proves DN map determination for simple surfaces with low regularity metrics.

Relation between generalized Weierstrass representation for conformal immersion of generic surfaces into three-dimensional space and Lax-Phillips scattering theory for automorphic functions is considered.

Scattering theory for harmonic one-forms on Riemann surfaces.

Study on recovering Lorentzian metrics from scattering data.

Inverse scattering result on AH manifolds determines metric up to diffeo and conformal factor.

We explore the generalization of scattering transforms from traditional (e.g., image or audio) signals to graph data, analogous to the generalization of ConvNets in geometric deep learning, and the utility of extracted graph features in graph data analysis. In particular, we focus on the capacity of these features to r…

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

Researchers solve boundary and scattering rigidity problems for magnetic systems.

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…

Scattering theory for linearised gravity on Schwarzschild black hole exterior.

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…

A new GNN module learns geometric scattering features for better graph classification and feature exploration.

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

Scattering networks are a class of designed Convolutional Neural Networks (CNNs) with fixed weights. We argue they can serve as generic representations for modelling images. In particular, by working in scattering space, we achieve competitive results both for supervised and unsupervised learning tasks, while making pr…

This paper addresses classification tasks on a particular target domain in which labeled training data are only available from source domains different from (but related to) the target. Two closely related frameworks, domain adaptation and domain generalization, are concerned with such tasks, where the only difference …

Study applies inverse scattering to BKM systems, linking spectra and integrable systems.

In this paper we continue our program of extending the methods of geometric scattering theory to encompass the analysis of the Laplacian on symmetric spaces of rank greater than one and their geometric perturbations. Our goal here is to explain how analysis of the Laplacian on the globally symmetric space $\SL(3,\RR)/\…

Transformers predict scattering amplitudes in theoretical physics.

System helps scientists visualize deep learning model of x-ray images.

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

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…

Geometric interpretation of 2d-4d wall-crossing formulas.

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…

We construct a large class of dynamical vacuum black hole spacetimes whose exterior geometry asymptotically settles down to a fixed Schwarzschild or Kerr metric. The construction proceeds by solving a backwards scattering problem for the Einstein vacuum equations with characteristic data prescribed on the event horizon…

Compactifications of N-body problems help in spectral theory and symmetry analysis.

Novel method uses Gaussian process to estimate particle sizes from scattering data.

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…

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…

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.

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

Paper develops formulas for shape derivatives in wave scattering.

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…

We discuss positivity properties of `distinguished propagators', i.e. distinguished inverses of operators that frequently occur in scattering theory and wave propagation. We relate this to the work of Duistermaat and Hörmander on distinguished parametrices (approximate inverses), which has played a major role in quantu…

Bayesian Scattering offers a simple baseline for image data uncertainty.

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.

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…