Unified treatment of stability problems in geometry and analysis.
problem Spherical closeness of hypersurfaces under geometric constraints.
method Estimate relating distance to geodesic spheres with norms of traceless Hessian operator.
result Unified treatment of stability problems in geometry and analysis.
Survey on soap bubble partitions and their stability.
problem Characterizing and stabilizing soap bubble partitions.
method Survey and analysis of recent research.
result Recent advancements in multi-bubble isoperimetric minimizers and stability.
Stability results for geometric equations in warped product spaces.
problem Geometric partial differential equations in warped product spaces.
method Stability theorem development for level sets of functions.
result Quantitative stability theorems for Serrin's problem and Alexandroff's theorem.
C. Gordon conjectured that a connected sum of two Heegaard splittings is stabilized if and only if one of the two factors is stabilized (Problem 3.91 in Kirby's problem list). In this paper, we shall prove this conjecture.
This paper analyzes stability and generalization of Markov chain stochastic gradient methods.
problem Analyzing stability and generalization of Markov chain stochastic gradient methods.
method Algorithmic stability in statistical learning theory.
result Established optimal generalization bounds for both smooth and non-smooth cases.
Stability inequalities for specific solutions in high dimensions.
problem Stability of solutions to the one-phase Bernoulli problem.
method Proving strict stability inequalities for cohomogeneity one solutions with bi-orthogonal symmetry.
result Strict stability for cohomogeneity one solutions in dimensions 7 and above.
Solves Riemann-Hilbert problems on surface triangulations.
problem Riemann-Hilbert problems in Donaldson-Thomas theory.
method Map from stability conditions to cluster variety.
result Constructs solutions to Riemann-Hilbert problems.
Solves Tian's stabilization problem for toric Fano manifolds.
problem Tian's stabilization problem for equivariant global log canonical thresholds.
method Expressed complex singularity exponents in terms of support and gauge functions from convex geometry.
result First general result on Tian's problem.
Novel approach ensures stability of compact schemes for variable PDEs.
problem Ensuring stability of compact schemes for variable coefficient PDEs.
method Difference equation approach to derive stability conditions.
result Derives sufficient condition for unconditional stability.
Paper shows stability of metric reconstruction for orbifolds from spectral data.
problem Determining the metric structure of collapsing orbifolds from spectral data.
method Improved quantitative unique continuation for wave operator on Riemannian manifolds.
result Quantitative stability of inverse problem for Riemannian orbifolds.
The study examines stability of Sobolev inequalities on manifolds with Ricci bounds.
problem Stability of Sobolev inequalities on Riemannian manifolds with Ricci curvature lower bounds.
method Combines techniques from smooth and non-smooth geometry, focusing on direct strategies.
result Effective methods revealed for stability of Sobolev inequalities on manifolds with non-negative Ricci curvature and Euclidean volume growth.
The paper presents a new method to create exotic 4-manifolds and surfaces that remain exotic after stabilization.
problem Stabilization of exotic 4-dimensional phenomena and knotted surfaces.
method Elementary approach to constructing exotic 4-manifolds and surfaces, including examples in closed, simply connected 4-manifolds.
result The construction yields exotic surfaces in the 4-ball that remain exotic after stabilization, detected by Khovanov homology.
Paper proves stability for recovering connections from holonomy traces.
problem Recovering a connection from holonomy traces on Riemannian manifolds.
method Combination of microlocal analysis and non-Abelian approximate Livsic Theorem.
result Hölder type stability estimates for holonomy inverse problem.
The paper presents a model-free method for stabilizing unknown control systems.
problem Stabilizing unknown control systems in engineering.
method Solving discounted LQR problems with increasing discount factors.
result The method efficiently recovers a stabilizing controller for linear and smooth nonlinear systems.
This work explores the trade-offs between stability and accuracy in statistical estimation.
problem Understanding the statistical cost of algorithmic stability.
method Statistical decision-theoretic perspective, focusing on worst-case and average-case stability.
result Optimal stable estimators for mean estimation and regression settings are developed, revealing trade-offs between stability and accuracy.
We define a new notion of "b-stability" for a polarised algebraic variety, adapted to the existence problem for Kahler-Einstein metrics on Fano manifolds.
The paper proves stability of Wulff shapes using anisotropic curvature functionals.
problem Stability of Wulff shapes under anisotropic curvature.
method Estimates distance to Wulff shape using Lp-norm of traceless F-Hessian of a foliating function. result Quantitative stability results for anisotropic inequalities and problems.
Proves stability of Einstein metrics on specific symmetric spaces.
problem Stability of Einstein metrics on symmetric spaces of compact type.
method Linear stability analysis of the Einstein-Hilbert action.
result Resolves stability problem for irreducible symmetric spaces of compact type.
Study on stability of free boundary Willmore problem using new gradient inequality.
problem Stability of free boundary Willmore problem.
method New Łojasiewicz-Simon gradient inequality for functionals on infinite dimensional manifolds.
result Existence and convergence of solutions for the free boundary Willmore flow.
The paper analyzes stability and convergence rates of entropic and Sinkhorn potentials.
problem Stability and convergence rates of entropic and Sinkhorn potentials.
method Semiconcavity properties of entropic potentials and Schrödinger bridges.
result Exponential convergence rates for gradient and Hessian of Sinkhorn iterates.
Study stability of Einstein manifolds with boundary.
problem Stability of Einstein manifolds with geometric boundary conditions.
method Using Ricci flow and calculus of variations, analyze stability with respect to the Einstein-Hilbert action.
result Introduce a new subspace of tensors (TVg tensors) for stability condition due to boundary constraints.
Derives stability for curvature measure near constant density, proving dual Minkowski problem solutions.
problem Stability of curvature measure near constant density
method Derives stability result for curvature measure, proves existence and uniqueness of solutions to dual Minkowski problem.
result Existence and uniqueness of solutions to dual Minkowski problem for positive indices, stability result for curvature measure.
Investigates stability of Epstein-Zin problem under market distortions.
problem Stability of Epstein-Zin problem in incomplete markets.
method Analyzes perturbations in returns and volatility, and interest rate; proves convergence of optimal solutions.
result Proves convergence of optimal consumption streams and value functions in the limit of model perturbations.
Paper proves stability and Dirichlet problem for translating hypersurfaces.
problem Stability and Dirichlet problem for translating hypersurfaces.
method Analyzes translating solitons in en+k, proves stability conditions, and studies Dirichlet problem. result Proves the infimum of mean curvature is zero for translating solitons and conditions for stability.
Paper analyzes stability and generalization of SCO algorithms.
problem Understanding how SCO algorithms perform on unseen data.
method Algorithmic stability analysis in statistical learning theory.
result Derives dimension-independent excess risk bounds for SCGD and SCSC.
Paper explores stability, regularization, and gradient flows for stochastic inverse problems.
problem Recovering random probability distributions from measurements.
method Direct inversion, variational formulation with regularization, and optimization via gradient flows.
result The choice of metric impacts stability and properties of the optimizer.
The matter of the stability for multi-asset American option pricing problems is a present remaining challenge. In this paper a general transformation of variables allows to remove cross derivative terms reducing the stencil of the proposed numerical scheme and underlying computational cost. Solution of a such problem i…
Stability result for a popular algorithm in optimal transport.
problem Stability of the Iterative Proportional Fitting Procedure in time and metric.
method Uniform stability analysis in the 1-Wasserstein metric.
result Quantitative stability result for entropy-regularized Optimal Transport and Schrödinger bridges.
We introduce the idea of *representation stability* (and several variations) for a sequence of representations V_n of groups G_n. A central application of the new viewpoint we introduce here is the importation of representation theory into the study of homological stability. This makes it possible to extend classical t…
Study stability of contingent claim solutions under probabilistic perturbations.
problem Stability of solutions to discrete-time contingent-claim problems under uncertainty.
method Use Rockafellian perturbations to analyze stability of solutions.
result Establishes convergence of dual problems and shadow prices.
New method uses generative models to improve phase retrieval stability.
problem Improving stability of solutions in phase retrieval problems.
method Unified reconstruction approach using generative models to mitigate overfitting.
result Mitigates overfitting to generative model for varying noise levels.
New methods provide stable ranking without assumptions on data distributions.
problem Stability issues in ranking problems with noisy data.
method Developed a stability framework and two ranking operators.
result Guaranteed stability without assumptions on data distributions.
Sharp stability in Almgren problem solved in any dimension.
problem Quantitative stability in the radial isotropic Almgren problem.
method Developed a theory for estimating the sharp modulus under minimal assumptions.
result Sharp ε2 in any dimension, solving the critical mass problem. Study stabilizes adversarial training in neural networks over infinite-dimensional spaces.
problem Stability issues in adversarial training of neural networks.
method Functional analysis of minimax optimization over infinite-dimensional spaces of continuous functions and probability measures.
result Convergence property of minimax problems under certain conditions, interpreted as stabilization techniques.
The paper examines stability of Yamabe boundary problem under perturbations.
problem Stability of Yamabe boundary problem under perturbations of mean curvature and scalar curvature.
method Analyzes stability of the Yamabe boundary problem with respect to perturbations of mean curvature and scalar curvature.
result The stability of the Yamabe boundary problem is proven under perturbations from below, but not from above.
We prove a sharp stability estimate for the problem of reconstructing a symmetric 2-tensor from its integrals along all maximal geodesics on a simple manifold.
The paper simplifies K-stability conditions for spherical varieties.
problem K-stability of polarized spherical varieties.
method Expressed K-stability in combinatorial terms, provided sufficient conditions.
result G-uniform K-stability provides a checkable condition for existence of constant scalar curvature metrics.
We investigate a semi-continuity property for stability conditions for sheaves that is important for the problem of variation of the moduli spaces as the stability condition changes. We place this in the context of a notion of stability previously considered by the authors, called multi-Gieseker-stability, that general…
Proves stability of convex disks close to round caps.
problem Stability of convex disks with positive curvature and strictly convex boundary.
method Compactness result for a Liouville-type PDE problem.
result Proves stability for a theorem of F. Hang and X. Wang.
Paper addresses travel time tomography stability and statistical inversion.
problem Determining conformal factors of metrics from geodesic lengths.
method Established forward and inverse stability estimates; applied to Bayesian statistical inversion.
result Consistency of statistical inversion technique for travel time tomography.
An introduction is provided to some current research trends in stability in geometric invariant theory and the problem of Kaehler metrics of constant scalar curvature. Besides classical notions such as Chow-Mumford stability, the emphasis is on several new stability conditions, such as K-stability, Donaldson's infinite…
Study on stability of geodesic maps in non-isotropic manifolds.
problem Stability of totally geodesic wave maps in non-isotropic manifolds.
method Factorization property, PDE system in geodesic normal coordinates, global existence result via hyperboloidal foliation.
result Established global existence for small initial data, leading to geometric stability.
We extend classical Euclidean stability theorems corresponding to the nonrelativistic Hamiltonians of ions with one electron to the setting of non parabolic Riemannian 3-manifolds.
Study on combustion theory solutions, proving nondegeneracy and stability in limit.
problem One-phase singular perturbation problem in combustion theory.
method Introduce density condition to preserve nondegeneracy, classify stable solutions.
result Global stable solutions have flat level sets in dimensions ≤ 4.
Study tackles inverse problems on low-dimensional manifolds, proving stability and proposing a reconstruction algorithm.
problem Inverse problems in infinite-dimensional spaces with nonlinear and ill-posed nature.
method Assumption of low-dimensional manifold, proving stability, proposing Landweber-type algorithm.
result Global convergence of the proposed algorithm, Lipschitz stability for specific inverse problems.
Study examines stability of image-reconstruction algorithms using variational regularization.
problem Stability and robustness of image-reconstruction algorithms in medical imaging.
method Review and novel stability results for ℓp-regularized linear inverse problems, focusing on p∈(1,∞). result Guarantees Lipschitz continuity for small p and Hölder continuity for larger p in Lp(Ω) function spaces. The paper studies K-stability of spherical varieties and their degenerations.
problem Understanding K-stability and degenerations of polarized spherical varieties.
method Reduction to a variational problem on the moment polytope, convexity constraint, and solving the HMA equation.
result Determines strict semistability and polystable degenerations for Fano spherical varieties of rank two.
Study shows stability of Schrödinger operator spectral data on a manifold.
problem Determining a manifold and potential function from spectral data.
method Approximation of spectral data on a subset to determine manifold and potential.
result Quantitative stability estimate for Schrödinger operator inverse problem.