Finite singular times for symmetric network curvature flow.
problem Formation of singularities in network curvature flow.
method Curvature flow of networks with symmetric initial data and two triple junctions.
result The set of singular times is finite.
Describes spectral data for singular fibres of a specific Hitchin system.
problem Characterizing singular fibres of the SL(2,C)-Hitchin system. method Using Hecke transformations and analysis of parameter spaces, the paper stratifies and compactifies the singular spaces.
result Large classes of singular fibres are shown to be fibre bundles over Prym varieties.
HADES detects data singularities quickly and accurately.
problem Detecting singularities in data efficiently.
method Kernel goodness-of-fit test based on differential geometry and optimal transport theory.
result Correctly detects singularities with high probability.
Detects singularities in complex data to improve machine learning models.
problem Real-world data often contains non-manifold structures (singularities) that can mislead machine learning models.
method Develops a topological framework to quantify local intrinsic dimension and Euclidicity score for multiple scales.
result Identifies singularities and captures local geometric complexity in image data.
Maxfaces can have cuspidal edges near certain singularities.
problem Characterizing singularities on maxfaces.
method Analyzing singular Björling data and proving geometric properties.
result Near a maxface with a specific type of singularity, there exists another maxface with a cuspidal edge.
Paper tackles singularity detection in PDEs using data-driven self-supervised learning.
problem Detecting singularities in PDE solutions for efficient numerical methods.
method Data-driven self-supervised learning framework with filtering tasks.
result Proposes filtering methods for raw unlabeled data to improve singularity detection.
Unified view on big bang singularities from initial data.
problem Understanding quiescent big bang singularities from initial data.
method Unified geometric perspective on various results.
result Solutions of Oude Groeniger et al. induce data on the singularity.
The Positive Mass Theorem for special singular initial data.
problem Proving the positive mass theorem for data with a codimension one singularity.
method Using asymptotically flat spin initial data sets with matching Bartnik data condition involving spacetime rotations.
result Established a spacetime positive mass theorem and rigidity statement.
In this paper we investigate the singularities of Lagrangian mean curvature flows in Cm by means of smooth singularity models. Type I singularities can only occur at certain times determined by invariants in the cohomology of the initial data. In the type II case, these smooth singularity models are asympto…
This work removes logarithmic singularities from hyperboloidal initial data without creating new ones.
problem Logarithmic singularities in hyperboloidal initial data sets.
method Evolutionary framework of the constraint equations and generalization of Beyer and Ritchie's result.
result Generic solutions of the constraint equations are free of logarithmic singularities.
Proves Riemannian positive mass theorem with singularities.
problem Proves Riemannian positive mass theorem for specific types of singular manifolds.
method Uses initial data sets with a second fundamental form to transfer convexity between different singularity components.
result Proves the theorem for manifolds with some mean-concave components and others mean-convex.
We use a knot invariant, namely the Tristram--Levine signature to study deformations of singular points of plane curves. We find a bound on the sum of M numbers over all singularities of a generic fiber in terms of the M number of the singularity at the central fiber and some topological data.
New technique stabilizes singular values in concatenated matrices.
problem How singular values of concatenated matrices relate to individual components.
method Developed perturbation technique extending classical results to concatenated matrices.
result Dominant singular values remain stable under small perturbations in submatrices.
We study mean curvature flow of smooth, axially symmetric surfaces in R3 with Neumann boundary data. We show that all singularities at the first singular time must be of type I.
Study proves well-posedness and scattering for wave equations on hyperbolic spaces with singular data.
problem Proving well-posedness and scattering for wave equations on hyperbolic spaces with singular initial data.
method Using weak-Lp spaces and dispersive estimates on Lorentz spaces, the study establishes global well-posedness and exponential asymptotic stability. result Developed a scattering theory and constructed wave operators in a singular framework.
Proves initial data on big bang singularities for Einstein-nonlinear scalar field equations lead to unique solutions.
problem Initial data on big bang singularities for Einstein equations.
method Geometric formulation of initial data, proving existence and uniqueness of solutions.
result Initial data on the singularity for the Einstein-nonlinear scalar field equations in 4 spacetime dimensions lead to a unique development of the data.
Two new methods improve forecasting of functional time series data.
problem Forecasting of functional time-dependent data.
method Functional Singular Spectrum Analysis (FSFA) based forecasting methods.
result Our methods outperform existing algorithms for periodic stochastic processes.
Develops methods to analyze manifold singularities using graph Laplacian.
problem Analyzing geometric properties of singularities in datasets.
method Theory and methods using the graph Laplacian to provide explicit bounds on manifold singularities.
result Explicit bounds on the graph Laplacian for functions near manifold singularities.
In previous work, Angenent, Isenberg, and Knopf created type-II Ricci flow neckpinch singularities. In this paper we construct solutions to Ricci flow whose initial data is the singular metric resulting from these singularities. We show in particular that the curvature decreases at the same rate at which it blew up. Th…
The study uses a ReLU network to discern geometric structure in data via the Data Information Matrix.
problem Understanding the geometric structure of real data in high-dimensional spaces.
method Employing a ReLU neural network trained as a classifier and the Data Information Matrix (DIM) to discern a singular foliation structure.
result The singular points of the foliation are measure zero, and a local regular foliation exists almost everywhere.
Generic low-entropy hypersurfaces in 4-6D flow with only generic singularities.
problem Analyzing mean curvature flow of low-entropy hypersurfaces.
method Proving flow encounters only generic singularities for specific entropy conditions.
result Proves flow encounters only generic singularities for low-entropy initial data.
The Kähler-Ricci flow near conical singularities is described with a C/t curvature bound.
problem Describing the Kähler-Ricci flow near conical singularities.
method Showed a C/t curvature bound and used the unique Kähler-Ricci expander. result The flow near each singular point is modelled on the unique Kähler-Ricci expander.
A low rank matrix X has been contaminated by uniformly distributed noise, missing values, outliers and corrupt entries. Reconstruction of X from the singular values and singular vectors of the contaminated matrix Y is a key problem in machine learning, computer vision and data science. In this paper we show that common…
Defines renormalised energies for singular harmonic maps into compact manifolds.
problem Analyzing harmonic maps with singularities in planar domains.
method Introduces renormalised energies and synharmony to study singularities and minimising configurations.
result Renormalised energies are coercive and Lipschitz-continuous, and associated with minimising singular harmonic maps.
STEEL tackles batch RL with singularity, improving policy optimization.
problem Existing RL methods assume absolutely continuous data, but STEEL handles non-overlapping regions.
method Proposes STEEL algorithm using maximum mean discrepancy and distributionally robust optimization.
result First finite-sample regret guarantee for batch RL with singularity.
The paper proves conditions for curvature blow-up in quiescent big bang singularities.
problem Understanding the nature of big bang singularities in cosmological models.
method Analyzing initial data sets with positive mean curvature and proving curvature blow-up conditions.
result Proves the formation of quiescent big bang singularities under certain conditions.
Improved likelihood estimation for singular distributions using deep models.
problem Estimating singular distributions using deep generative models.
method Data perturbation to avoid singularity issues in likelihood estimation.
result Consistent estimation of target distribution with desirable rates.
Study on hyperbolic elastic flow, proving convergence and quantifying singularities.
problem Understanding singularities and convergence of elastic flow in hyperbolic plane.
method Analyzes closed and open curves with clamped boundary conditions, proving convergence without small energy assumption.
result Each singularity carries an energy cost of at least 8, and blow-ups are explicitly classified.
We study singularities of spacelike, constant (non-zero) mean curvature (CMC) surfaces in the Lorentz-Minkowski 3-space L3. We show how to solve the singular Björling problem for such surfaces, which is stated as follows: given a real analytic null-curve f0(x), and a real analytic null vector field v(x) paralle…
The paper proves approximation and interpolation theorems for maxfaces with singularities.
problem Proving approximation and interpolation theorems for maxfaces with singularities.
method Surveying and applying Enneper--Weierstrass representation formula methods to maxfaces, incorporating singularity criteria.
result Existence of maxfaces with prescribed singularities and maxfaces with dense image singular set.
The paper tackles singularities in diffusion models on submanifolds.
problem Analyzing singularities in diffusion models on lower-dimensional submanifolds.
method Small-time approximations of the Green's function and derivation of a new target function.
result The new target function remains bounded for singular data distributions.
Study on prescribing positive curvature with conical singularities on a sphere.
problem Prescribing positive curvature with conical singularities on a sphere.
method Fine analysis of bubble trees and an area identity in the convergence process.
result Criterion for nonexistence in an open region of the prescribing data.
In this paper we introduce the concept of singular Finsler foliation, which generalizes the concepts of Finsler actions, Finsler submersions and (regular) Finsler foliations. We show that if F is a singular Finsler foliation on a Randers manifold (M,Z) with Zermelo data (h,W), then $\mathcal{F}…
An analytic extension of the Reissner-Nordstrom solution at and beyond the singularity is presented. The extension is obtained by using new coordinates in which the metric becomes degenerate at r=0. The metric is still singular in the new coordinates, but its components become finite and smooth. Using this extension …
This is a continuation of our paper math.AG/0111298. We prove an explicit formula for the geometric genus p_g of a quasihomogeneous isolated surface singularity in terms of the Seiberg-Witten invariant of the link and other topological data which can be read from a resolution graph of the singularity. Moreover, we also…
We study non-variational degenerate elliptic equations with high order singular structures. No boundary data are imposed and singularities occur along an {\it a priori} unknown interior region. We prove that positive solutions have a universal modulus of continuity that does not depend on their infimum value. We furthe…
Study optimizes shared singular subspace estimation from noisy matrices.
problem Estimating shared singular subspaces across multiple noisy matrices.
method Low-rank matrix denoising framework with Stack-SVD and novel estimators.
result Stack-SVD achieves minimax rate-optimality for identical shared subspaces, and novel estimators for partial sharing.
Localized big bang singularities found without background solutions.
problem Proving localized big bang formation without proximity to background solutions.
method Introducing a new foliation by spacelike hypersurfaces and a time function to synchronize and stabilize the singularity.
result Maximally globally hyperbolic developments have local quiescent big bang singularities with curvature blow-up.
New method estimates high-dimensional GoM models efficiently.
problem Estimating GoM models for high-dimensional polytomous data.
method Flattening three-way quasi-tensor into a matrix, performing singular value decomposition.
result Established finite-sample error bounds for estimated parameters.
This study reduces Willmore flows of tori to simpler problems and finds new conformally constrained Willmore tori.
problem Analyzing singularities and existence of Willmore tori under specific constraints.
method Dimension reduction approach, strong relation with elastic flow, necessary condition for singularities, criterion for initial data.
result Existence of new conformally constrained Willmore tori and identification of inverted catenoid as a limit shape.
We implement methods from computational homology to obtain a topological signal of singularity formation in a selection of geometries evolved numerically by Ricci flow. Our approach, based on persistent homology, produces precise, quantitative measures describing the behavior of an entire collection of data across a di…
We introduce certain spherically symmetric singular Ricci solitons and study their stability under the Ricci flow from a dynamical PDE point of view. The solitons in question exist for all dimensions n+1≥3, and all have a point singularity where the curvature blows up; their evolution under the Ricci flow is in sh…
We construct smooth solutions to Ricci flow starting from a class of singular metrics and give asymptotics for the forward evolution. The singular metrics heal with a set of points (of codimension at least three) coming out of the singular point. We conjecture that these metrics arise as final-time limits of Ricci flow…
Study neckpinch singularities in Ricci flow with cylindrical symmetry.
problem Understanding the asymptotic behavior of neckpinch singularities in Ricci flow.
method Rigorous analysis under Type-I assumption for general symmetric initial data.
result Previously constructed asymptotic profiles are the only possibilities.
The paper connects orbifold singularities to higher symmetries in SQFTs.
problem Understanding higher symmetries in supersymmetric quantum field theories.
method Cutting and gluing of orbifold singularities to determine symmetries.
result Local orbifold singularities encode 0-form, 1-form, and 2-group symmetries.
We study surfaces of constant positive Gauss curvature in Euclidean 3-space via the harmonicity of the Gauss map. Using the loop group representation, we solve the regular and the singular geometric Cauchy problems for these surfaces, and use these solutions to compute several new examples. We give the criteria on the …
Low-rank matrix regression refers to the instances of recovering a low-rank matrix based on specially designed measurements and the corresponding noisy outcomes. In the last decade, numerous statistical methodologies have been developed for efficiently recovering the unknown low-rank matrices. However, in some applicat…
Under mean curvature flow, a closed, embedded hypersurface M(t) becomes singular in finite time. For certain classes of mean-convex mean curvature flows, we show the continuity of the first singular time T and the limit set "M(T)", with respect to initial data. We employ an Angenent-like neck-pinching argument to…