Study on recovering Lorentzian metrics from scattering data.
problem Recovering Lorentzian metrics from scattering data on a boundary.
method Analyzing the role of boundary distance functions and linearizing the light ray transform.
result Scattering rigidity can be reduced to boundary rigidity of magnetic systems.
Study recovers Lorentzian metrics from boundary data, proving local rigidity.
problem Recovering a Lorentzian metric from scattering data on a boundary.
method Analyzes jet and real analyticity of metrics near lightlike points.
result Metric can be recovered up to gauge transformations near lightlike strictly convex points.
Study proves rigidity results for analytic spacetimes without boundary or timelike boundary.
problem Determining compact subsets and spacetimes from scattering data.
method Analytic spacetimes, thin exterior layers, non-trapping lightlike geodesics.
result Time separation and scattering relations uniquely determine spacetimes.
Study shows essential self-adjointness of wave operators in Lorentzian settings.
problem Essential self-adjointness of wave operators in Lorentzian scattering spaces.
method Using a Fredholm framework to invert the spectral family and refine conclusions.
result Dense range in L^2 for the wave operator acting on an appropriate subdomain.
Study scattering rigidity on stationary manifolds using geodesics.
problem Scattering rigidity on standard stationary manifolds.
method Use Hamiltonian reduction to relate to MP-systems. result New rigidity results for stationary manifolds.
We show that conformally compact, globally hyperbolic, Lorentzian Einstein-Weyl 3-manifolds are in natural one-to-one correspondence with orientation-reversing diffeomorphisms of the 2-sphere. The proof hinges on a holomorphic-disk analog of Hitchin's mini-twistor correspondence.
I introduce a family of closeness functions between causal Lorentzian geometries of finite volume and arbitrary underlying topology. When points are randomly scattered in a Lorentzian manifold, with uniform density according to the volume element, some information on the topology and metric is encoded in the partial or…
In this paper we obtain the asymptotic behavior of solutions of the Klein-Gordon equation on Lorentzian manifolds (X∘,g) which are de Sitter-like at infinity. Such manifolds are Lorentzian analogues of the so-called Riemannian conformally compact (or asymptotically hyperbolic) spaces. Under global assumptions on…
Study reveals how boundary wave operator determines metric properties on anti-de Sitter spacetimes.
problem Determining metric properties on anti-de Sitter spacetimes.
method Analysis of Klein-Gordon equation and Dirichlet-to-Neumann map.
result Determines the Taylor series of the bulk metric at the boundary.
In this paper we describe the behavior of solutions of the Klein-Gordon equation, (Box_g+lambda)u=f, on Lorentzian manifolds (X^o,g) which are anti-de Sitter-like (AdS-like) at infinity. Such manifolds are Lorentzian analogues of the so-called Riemannian conformally compact (or asymptotically hyperbolic) spaces, in the…
Proves two non-trapping obstacles coincide if scattering rays have similar travelling times or scattering length spectra.
problem Identifying non-trapping obstacles based on scattering properties.
method Proves two obstacles coincide if their scattering rays have similar travelling times or scattering length spectra under weak non-degeneracy conditions.
result Two non-trapping obstacles coincide if their scattering rays have similar travelling times or scattering length spectra.
In recent years, Teichmüller theory, which is the study of moduli spaces of marked Riemann surfaces, has come to be considered more and more from the point of view of actions of surface groups inside certain semi-simple Lie groups. In particular, we consider the case where the Lie groups in question have symmetric spac…
Geometric scattering on manifolds improves neural network performance.
problem Improving neural network performance on manifold data.
method Generalized Euclidean scattering transform to compact manifolds.
result Geometric scattering provides localized isometry invariant descriptions of manifold signals.
Geometric wavelet scattering on manifolds improves neural network understanding.
problem Improving neural network understanding on manifold and graph domains.
method Defining a geometric scattering transform based on wavelet filters and nonlinearities.
result Generalizes deformation stability and local translation invariance to manifolds.
Researchers prove Fredholm property for Dirac operator on specific spacetimes.
problem Proving Fredholm property for Dirac operator on asymptotically static spacetimes.
method Combining time-dependent scattering theory and Egorov's theorem for pseudo-differential hyperbolic systems.
result The Dirac operator is Fredholm under Atiyah-Patodi-Singer boundary conditions.
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.
problem Reconstruction problem in mirror symmetry.
method Introduction of scattering diagrams and their role in SYZ and HMS conjectures.
result Scattering diagrams help in understanding mirror symmetry.
GSAN learns adaptive node representations using geometric scattering and attention.
problem Oversmoothing in node representation learning.
method Attention-based architecture integrating geometric scattering and GCN channels.
result GSAN outperforms previous networks in semi-supervised node classification.
New method uses broken scattering to uniquely identify Finsler manifolds.
problem Identifying Finsler manifolds from scattering data.
method Uses broken scattering relation to compare geodesics.
result Two reversible Finsler manifolds with the same broken scattering relation are isometric.
New method learns soliton dynamics from scattering data without assuming known equations.
problem Deriving soliton dynamics from scattering data without prior knowledge.
method Combining IST with weak-form system identification for data-driven discovery.
result Effective soliton dynamics models derived from observed scattering data.
Unified graph scattering transforms improve theoretical properties of graph neural networks.
problem Improving theoretical guarantees for graph neural networks.
method Introducing windowed and non-windowed geometric scattering transforms for graphs.
result Unified family of graph scattering transforms with provable stability and invariance.
Scattering representations simplify SBI for images without extra compression.
problem Efficiently performing simulation-based inference on images with limited data.
method Use scattering representations for compression and learning, combined with spatial averaging and expressive density estimators.
result Scattering representations provide more information than traditional methods, without requiring additional simulations.
The paper establishes scattering theory for wave equations on Schwarzschild spacetime.
problem Defocusing semilinear wave equations on Schwarzschild spacetime.
method Combining energy and pointwise decay results with Sobolev embedding, constructing scattering operator.
result Construction of a scattering operator mapping past to future scattering data.
Paper develops formulas for shape derivatives in wave scattering.
problem Computing high order shape derivatives for wave scattering is challenging.
method Introduces elegant recurrence formulas using differential forms and Lie derivatives.
result Unified framework for computing high order shape perturbations in scattering problems.
Kymatio simplifies scattering transforms for Python.
problem Signal processing and machine learning applications.
method Wavelet scattering transform implemented in Python.
result Efficient, GPU-accelerated implementation.
MODWST improves classification tasks with wavelet scattering.
problem Signal classification challenges.
method Combines MODWT and WST for feature extraction.
result MODWST outperforms CNNs in limited data scenarios.
Study on scattering geodesics on modular surface and their sojourn times.
problem Distribution of scattering geodesics and their sojourn times on modular surface.
method Analysis of scattering geodesics in modular surface, establishing connection to prime divisors in arithmetic progression.
result Established a connection between scattering geodesics and prime divisors in arithmetic progression.
Graph scattering transforms are stable to metric perturbations of network topology.
problem Stability of graph data representations under metric perturbations.
method Extending scattering transforms to network data using multiresolution graph wavelets and graph convolutions.
result Graph scattering transforms are stable to metric perturbations of the underlying network topology.
Scattering theory on Riemann surfaces with explicit matrix and generalized period mappings.
problem Scattering theory for harmonic one-forms on Riemann surfaces.
method Construction of scattering theory from boundary value problems involving systems of curves and jump problems. Explicit expression for scattering matrix using Schiffer operators.
result Unitary scattering matrix and general association of polarizing Lagrangian spaces.
Scattering networks improve image representation learning without deep learning.
problem Improving image representation learning without deep learning.
method Scattering networks as generic representations in scattering space.
result Scattering networks achieve competitive results in supervised and unsupervised learning.
Bayesian Scattering offers a simple baseline for image data uncertainty.
problem Lack of interpretable, mathematically grounded uncertainty quantification methods for image data.
method Coupling wavelet scattering transform with a simple probabilistic head.
result Bayesian Scattering provides sensible uncertainty estimates under distribution shifts.
Scattering theory for linearised gravity on Schwarzschild black hole exterior.
problem Constructing a scattering theory for linearised gravity equations on Schwarzschild background.
method Building on previous work, constructing Hilbert space-isomorphisms for finite energy initial data and scattering states.
result Past and future linear memories are related by an antipodal map for Bondi-normalised solutions.
We construct continuous families of scattering manifolds with the same scattering phase. The manifolds are compactly supported metric perturbations of Euclidean Rn for n≥8. The metric perturbation may have arbitrarily small support.
Reconstructing a manifold from internal scattering data.
problem Reconstructing a compact Riemannian manifold from internal scattering data.
method Generic assumption of the metric allows reconstruction from boundary measurements of geodesic exit directions.
result Isometric reconstruction of the manifold from boundary scattering data.
Scattering theory for harmonic one-forms on Riemann surfaces.
problem Understanding harmonic one-forms on Riemann surfaces.
method Constructing scattering theory through boundary value problems and integral operators.
result Explicit expression for the scattering matrix and proof of unitarity.
Paper shows how scattering maps of Schrödinger equations relate to metrics.
problem Relating scattering maps of time-dependent Schrödinger equations to metrics.
method Analyzes scattering maps for specific classes of metrics and diffeomorphisms.
result Scattering maps differ by a compact operator if and only if metrics are related by diffeomorphism.
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…
Establishes scattering theory for de Sitter vacuum solutions in even dimensions.
problem Quantitative nonlinear scattering theory for asymptotically de Sitter vacuum solutions in even spatial dimensions.
method Geometric Littlewood-Paley decomposition of the solution, constructing the scattering map.
result Existence and uniqueness of scattering states, asymptotic completeness, and invertible scattering map with quantitative control.
Scattering transforms adapted for non-Euclidean domains using diffusion wavelets.
problem Stability of data representations in non-Euclidean domains.
method Generalization of scattering transforms to non-Euclidean domains using diffusion wavelets and diffusion maps.
result Stability of the representation to metric perturbations of the domain.
Scattering theory developed for linearised gravity near Schwarzschild black hole.
problem Linear stability of Schwarzschild spacetime and scattering of gravitational waves.
method Physical-space Chandrasekhar transformation and Teukolsky-Starobinsky correspondence.
result Construction of scattering theory for spin 2 Teukolsky equations.
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.
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.
problem Modeling non-Gaussian time-series with stationary increments.
method Complex wavelet transform for scale variations, joint correlation matrix for scale dependencies, second wavelet transform for diagonalization, maximum entropy models conditioned by scattering spectra coefficients.
result Scattering spectra of self-similar processes are scale invariant, allowing statistical testing and generation of new time-series.
Analytic metrics are uniquely determined by their scattering map.
problem Determining Riemannian manifolds from scattering data.
method Analytic negatively curved Riemannian manifolds with strictly convex boundary.
result The scattering map determines the manifold up to isometry.
Deep learning solves wave-based inverse problems, including super-resolution imaging.
problem Solving inverse wave scattering problems across all length scales.
method Wide-band butterfly network coupled with dynamic noise injection.
result Framework successfully solves super-resolution imaging problems.
Unified geometric scattering model for measure spaces.
problem Improving CNNs for non-Euclidean data.
method Unified geometric scattering model for measure spaces.
result Unified model includes previous work and applies to more general settings.
WideBNet learns inverse scattering from wide-band data efficiently and stably.
problem Learning the inverse scattering map from wide-band scattering data.
method Combines butterfly factorization, FFT, and deep learning.
result WideBNet requires fewer training points and has stable training dynamics.