A spacetime can be embedded in an enveloping space with all its extensions.
problem Existence and uniqueness of C0-maximal extensions in globally hyperbolic conformally flat spacetimes.
method Proving conformal embedding into an enveloping space containing all extensions.
result Existence and uniqueness of C0-maximal extensions proven.
New conformally Einstein metrics on Heisenberg group found.
problem Finding new conformally Einstein metrics on specific Lie groups.
method Described Lorentzian semi-direct extensions of the Heisenberg group.
result Classified Bach-flat left-invariant Lorentzian metrics.
Study connects weighted isoperimetric problems to nonlocal elliptic operator extensions.
problem Sharp inequalities for weighted Poisson integrals and their extremizers.
method Formulates variational problem on conformal metric measure space.
result Sharp inequalities are linked to variational problem on CCE manifolds.
Projective geodesic extensions for nonholonomic systems are derived under conformal transformations.
problem Deriving conditions for projective geodesic extensions in nonholonomic mechanics.
method Analyzing necessary and sufficient conditions for existence under conformal modifications.
result Conditions for existence of projective geodesic extensions in nonholonomic systems under conformal transformations.
The paper extends fractional Laplacian theory using symmetric spaces and extension problems.
problem Developing a new framework for fractional Laplacians.
method Using representation theory on symmetric spaces and extension problems.
result Constructed new boundary operators with conformal properties.
The notion of maximal extension of a globally hyperbolic space-time arises from the notion of maximal solutions of the Cauchy problem associated to the Einstein's equations of general relativity. In 1969 Choquet-Bruhat and Geroch proved that if the Cauchy problem has a local solution, this solution has a unique maximal…
We consider the future causal boundary as a tool to find obstructions to conformal extensions, the latter being a slight generalization to conformal compactifications.
Study indefinite bi-invariant metrics and their conformal properties.
problem Characterize indefinite bi-invariant metrics and their conformal relationships.
method Analyzes Lie algebras under different cases and structures.
result Conformally Einstein properties for various Lie algebras are determined.
Study locally conformal symplectic manifolds on compact nilmanifolds.
problem Characterize locally conformal symplectic structures on compact nilmanifolds.
method Left-invariant structures, foliation analysis, Lie group theory.
result Structure theorem for locally conformal symplectic manifolds of the first kind.
Improved multivariate conformal prediction by standardizing residuals.
problem Weak conditional coverage in heteroskedastic multivariate settings.
method Natural extension of univariate normalization to multivariate setting, whitening residuals and standardizing local variance.
result Standardized residuals yield asymptotic conditional coverage under certain distributions.
We consider the problem of extending a conformal metric of negative curvature, given outside a neighbourhood of 0 in the unit disk $\DD$, to a conformal metric of negative curvature in $\DD$. We give conditions under which such an extension is possible, and also give obstructions to such an extension. The methods we us…
Jet isomorphism theorems for conformal geometry are discussed. A new proof of the jet isomorphism theorem for odd-dimensional conformal geometry is outlined, using an ambient realization of the conformal deformation complex. An infinite order ambient lift for conformal densities in the case in which harmonic extension …
This paper classifies Blaschke para-umbilical hypersurfaces in conformal space.
problem Classifying Blaschke para-umbilical hypersurfaces in conformal space.
method Classification through mathematical analysis and extension of previous classifications.
result Classification of Blaschke para-umbilical hypersurfaces in conformal space.
Proves conformal equivalence of visual metrics in smooth pseudoconvex domains.
problem Establishing conformal equivalence of visual metrics in pseudoconvex domains.
method Refined estimates and asymptotic hyperbolic character of metrics, inspired by Mostow's rigidity theorem.
result Boundary extensions of isometries are conformal with respect to the sub-Riemannian metric.
Generalizing Riemannian theorems of Anderson-Herzlich and Biquard, we show that two (n+1)-dimensional stationary vacuum space-times (possibly with cosmological constant Λ∈R) that coincide up to order one along a timelike hypersurface $\mycal T$ are isometric in a neighbourhood of $\mycal T$. We further prove th…
This study develops a NURBS-based method for conformal surface flattening without singularities.
problem Flatten surfaces conformally without singularities.
method NURBS-based approach with iterative refinement of input and flattening surfaces, leveraging nonlinear extension of VarPro.
result Developed a singularity-free NURBS-based method for conformal surface flattening.
Proves conditions for black hole formation in Einstein-Maxwell theory.
problem Existence of black holes in Einstein-Maxwell theory.
method Singularity theorems ensuring finite lifetime regions, requiring conformal extension.
result Conditions for the existence of finite lifetime black hole regions.
Unified study of nine multi-output conformal methods with generalized scores.
problem Challenges in extending conformal prediction to multi-output problems.
method Nine conformal methods with generalized multi-output conformity scores.
result Generalized scores ensure asymptotic conditional coverage and exact finite-sample marginal coverage.
New types of Einstein manifolds discovered from homogeneous surfaces.
problem Understanding quasi-Einstein equations on homogeneous surfaces.
method Modified Riemannian extension and warped product construction.
result Discovery of new Einstein manifolds.
Classifies 1-connected Lorentzian manifolds with essential conformal groups.
problem Classifying Lorentzian manifolds with specific conformal groups.
method Proves existence of a metric making the manifold homogeneous and plane wave.
result Completes the classification of 1-connected Lorentzian manifolds with transitive conformal groups.
The local classification of conformally flat Lorentzian manifolds with special holonomy groups is obtained. The corresponding local metrics are certain extensions of Riemannian spaces of constant sectional curvature to Walker metrics.
Smooth Riemannian manifolds can be embedded without boundary.
problem Embedding smooth Riemannian manifolds with boundary into complete manifolds without boundary.
method General gluing and conformal-deformation construction.
result Any smooth, metrically complete Riemannian manifold with smooth boundary can be realized as a closed domain into a smooth, geodesically complete Riemannian manifold without boundary.
We study first and second order conformal symmetries of the Yamabe Laplacian on a general pseudo-Riemannian manifold and of the Paneitz operator on Einstein spaces. We show that first order conformal symmetries of the Yamabe operator induce second order conformal symmetries. We show that on an Einstein space every conf…
Graph conformal prediction predicts future power outages with high confidence.
problem Accurately predicting future power outages to enable rapid recovery.
method Developed a graph conformal prediction method for quarter-hourly outage data.
result Graph conformal prediction method delivers accurate prediction regions for future outage numbers.
Extends Gromov non-squeezing to locally conformally symplectic structures.
problem Generalizing Gromov non-squeezing to new geometric structures.
method Deformation theory applied to locally conformally symplectic structures.
result Proves a new extension of the Gromov non-squeezing phenomenon.
Improves classifier accuracy in ambiguous data settings.
problem Training classifiers with partially labeled data.
method Incremental pruning of candidate labels using conformal prediction.
result Significantly improves test set accuracies of PLL classifiers.
This work extends locally conformal analysis to multi-Hamiltonian settings, providing new geometric structures and Hamiltonian dynamics.
problem Globalization problem in multi-Hamiltonian formalisms due to incompatibilities on chart overlaps.
method Investigation of locally conformally Nambu--Poisson and locally conformally generalized Poisson manifolds, constructing Hamiltonian-type evolution equations.
result Unified framework for classical, Nambu--Poisson, and generalized Poisson manifolds within a locally conformal context.
We review previous work of Alain Connes, and its extension by the author, on some conformal invariants obtained from the noncommutative residue on even dimensional compact manifolds without boundary. Inspired by recent work of Yong Wang, we also address possible generalizations of these conformal invariants to the sett…
New geometric variant of factorization homology for conformally flat manifolds.
problem Defining invariants of conformally flat manifolds.
method Introducing a metric-dependent geometric variant of factorization homology.
result Left Kan extensions of conformally flat d-disk algebras define invariants of conformally flat manifolds. The paper studies conformally Einstein-Maxwell Kähler metrics and automorphism group structure.
problem Analyzing conformally Einstein-Maxwell Kähler metrics and their automorphism groups.
method Using a Hessian formula for the Calabi functional and extending the Lichnerowicz-Matsushima Theorem.
result Proves a reductiveness result of the reduced Lie algebra of holomorphic vector fields for conformally Einstein-Maxwell Kähler manifolds.
Modified construction for conformal structures with twistor spinors.
problem Geometric construction and characterization of conformal structures.
method Geometric construction and characterization of 2n-dimensional split-signature conformal structures. result Explicit geometrically constructed Fefferman-Graham ambient metric with vanishing Q-curvature. Solves modified Schouten tensor problems in conformal metric classes.
problem Prescribed problems for modified Schouten tensors in conformal classes of metrics.
method Uniform ellipticity confirmation under topological and functional constraints.
result Extends results from previous work on smooth complete metrics.
Lectures on Moebius-Lie geometry and its extension, including new geometric ensembles.
problem Classical Moebius-Lie geometry and its extension to ensembles of cycles.
method Reduces conformally invariant geometric relations to linear equations with a fixed quadratic relation.
result Efficient method implemented as a C++ library for numeric and symbolic data in arbitrary dimensions.
Extends conformal prediction for controlling expected risk of monotone loss functions.
problem Controlling expected risk of monotone loss functions.
method Generalizes split conformal prediction with coverage guarantee, extending to distribution shift, quantile risk, multiple, adversarial, and expectations of U-statistics.
result Tight up to an O(1/n) factor, with worked examples in computer vision and natural language processing. New heat equation method solves intertwining problems in CR geometry.
problem Intertwining problems in conformal CR geometry.
method Heat equation and extension problems approach.
result New intertwining formulas derived.
New method uses quantile regression for adaptive prediction intervals.
problem Constructing valid prediction intervals without distributional assumptions.
method Combines conformal prediction with classical quantile regression.
result Shows shorter, more efficient prediction intervals.
The paper examines Yamabe flow on modified Riemann extensions and curvature tensors.
problem Analyzing modified Riemann extensions under Yamabe flow.
method Study of rate relations of curvature tensors under Yamabe flow.
result Discussion on standard metrics in modified Riemann extensions.
We show that the conformally invariant fractional powers of the sub-Laplacian on the Heisenberg group are given in terms of the scattering operator for an extension problem to the Siegel upper halfspace. Remarkably, this extension problem is different from the one studied, among others, by Caffarelli and Silvestre.
Improved conformalized quantile regression for adaptive prediction intervals.
problem Lack of adaptiveness in the conformal step of conformalized quantile regression.
method Cluster explanatory variables by permutation importance and apply k conformal steps.
result Improved prediction intervals are more adaptive to heteroscedasticity.
Unique foliation of AdS3 domains by constant mean curvature surfaces.
problem Foliation of domains of dependence in AdS3. method Proving existence and uniqueness of constant mean curvature foliation.
result Existence and uniqueness of foliation by constant mean curvature surfaces.
The paper studies equations in conformal geometry with gradient and existence results.
problem Equations in conformal geometry on closed smooth Riemannian manifolds.
method Local gradient and second derivative estimates, existence result.
result Proved local gradient and second derivative estimates for solutions.
In this note we revisit the notion of conformal barycenter of a measure on $\SS^n$ as defined by Douady and Earle in Acta Math. Vol 157, 1986. The aim is to extend rational maps from the Riemann sphere $\Cbar\isom\SS^2$ to the (hyperbolic) three ball $\BB^3$ and thus to $\SS^3$ by reflection. The construction which was…
Researchers transform equations and define integral operators on a ball.
problem Transforming equations from half space to ball.
method Identify Poisson kernel, define extension operator, prove inequalities.
result Uniqueness of extremal functions in limit case.
Enhances machine learning performance predictions with transparency.
problem Providing accurate and practical performance guarantees for machine learning.
method Natural extension of conformal prediction framework.
result Valid and well-calibrated predictive statements about future performance.
New method for distribution-free regression prediction intervals.
problem Creating reliable prediction intervals for regression without distributional assumptions.
method Conformal inference, split conformal inference, jackknife method.
result Valid prediction intervals with finite-sample coverage guarantees.
Unique continuation results are proved for metrics with prescribed Ricci curvature in the setting of bounded metrics on compact manifolds with boundary, and in the setting of complete, conformally compact metrics. Related to this issue, an isometry extension property is proved: continuous groups of isometries at confor…
The classical Pohozaev identity constrains potential solutions of certain semilinear PDE boundary value problems. The Kazdan-Warner identity is a similar necessary condition important for the Nirenberg problem of conformally prescribing scalar curvature on the sphere. For dimensions n≥3 both identities are captur…
A new conformal prediction framework for graph-valued outputs using Z-Gromov-Wasserstein distances.
problem Lack of principled uncertainty quantification for graph-valued supervised prediction.
method Proposes a conformal prediction framework using Z-Gromov-Wasserstein distances for graph-valued outputs.
result Provides distribution-free coverage guarantees for graph-valued outputs.