Survey on singularity theory and its relation to minimal model program.
problem Understanding singularities and their role in minimal model program.
method Construction and analysis of dual complexes, proof of ACC conjecture, local stability theory.
result Recent progress on local stability theory of Kawamata log terminal singularities.
The study proves how groups can be split with limited complexity.
problem Understanding the complexity of group splittings.
method Analyzing trees with finite stabilizers and their quotient structures.
result Deformation spaces of trees have maximal complexity.
Groups of importance in group theory have flexible stability properties.
problem Stability and flexibility of groups in geometric and combinatorial group theory.
method Establishing Kirchberg's Local Lifting Property and Lubotzky--Shalom's Property FD for specific groups.
result Groups like 3 3 3 -manifold groups, limit groups, and certain one-relator groups are very flexibly stable. Paper develops a new local convexity condition for non-isolated minima in non-convex optimization.
problem Lack of theory for non-isolated minima in non-convex optimization.
method Formulates a new local convexity condition and studies SGD convergence under this condition.
result Shows SGD can converge locally under the new condition.
Modeling financial systemic risk with optimal control theory for stability.
problem Analyzing and stabilizing systemic risk in interconnected financial entities.
method Developed a theoretical model using optimal control theory, including steps for synthesizing stabilizing controllers.
result The model ensures that the H ∞ H^{\infty} H ∞ norms of the mappings from disturbance to output are less than a predefined constant, stabilizing the system. Paper presents neural network controllers for offset-free setpoint tracking.
problem Offset-free setpoint tracking using neural network controllers.
method Exploiting slope-restricted activation functions, linear matrix inequalities are used to verify stability.
result Global and local stability conditions for neural network controllers are derived.
The EM algorithm's convergence is analyzed using Lyapunov stability theory.
problem Analyzing the convergence of the EM algorithm.
method Reinterpreting the EM algorithm as a dynamical system and applying Lyapunov stability theory.
result Asymptotic stability and convergence of the EM algorithm are established.
The study shows how stabilizing manifolds with projective spaces affects their homotopy structure.
problem Understanding the homotopy of manifolds stabilized by projective spaces.
method Trace the effect of surgery on product manifolds, showing a loop homotopy decomposition after localization.
result A loop homotopy decomposition of a manifold after stabilization by a projective space is provided.
Paper proposes a new method to stabilize noisy gradient algorithms.
problem Stochastic-gradient Langevin algorithms can introduce bias when taming denominators depend on stochastic-gradient realizations.
method Proposes a structure-preserving framework for designing tamed denominators that avoid unnecessary taming and maintain the stabilizing effect of taming.
result The method avoids stationary bias and explains the stationary error split into bias and remaining error.
Local stability of p-Kähler structures studied.
problem Stability of p-Kähler structures under deformations.
method Natural extension map and power series method.
result Local stability theorem for p-Kähler structures.
We develop a Chern-Weil theory for compact Lie group action whose generic stabilizers are finite in the framework of equivariant cohomology. This provides a method of changing an equivariant closed form within its cohomological class to a form more suitable to yield localization results. This work is motivated by our w…
This paper introduces Libra to analyze and optimize generalization in Federated Learning.
problem Inconsistent local optima in Federated Learning lead to poor generalization performance.
method Introduces Libra, a generalization dynamics analysis framework for algorithm-dependent excess risk minimization.
result Libra highlights the trade-offs between model stability and gradient norms in Federated Learning.
Study normal curves in sub-Finsler Lie groups with specific norms, focusing on branching and face stability.
problem Analyzing normal curves in sub-Finsler Lie groups with different norms.
method Using tools from convex analysis, the Pontryagin Maximum Principle is revisited to express the normal equation as a differential inclusion involving the subdifferential of the dual norm.
result Normal curves in polyhedral norms have controls that locally take values in a single face of a sphere with respect to the norm.
FedProx algorithm improved for non-smooth and heterogeneous data.
problem Theoretical understanding of FedProx for non-convex federated optimization.
method Local dissimilarity invariant convergence theory through algorithmic stability.
result Convergence guarantees for non-smooth FL problems and minibatch size.
New knot models analyze local entanglement for robust curve analysis.
problem Lack of local structural information in classical knot theory.
method Proposed multiscale and persistent Jones polynomials.
result Models are stable to small perturbations, robust for real-world applications.
We prove the Poisson geometric version of the Local Reeb Stability (from foliation theory) and of the Slice Theorem (from equivariant geometry). The result is also a generalization of Conn's linearization theorem from one-point leaves to arbitrary symplectic leaves (however, we do not make use of Conn's theorem).
The paper develops algorithms to detect stability and Morse properties in various groups.
problem Detecting stability and Morse properties in finitely generated groups.
method Various detection and decidability algorithms for stability and Morse properties in specific types of groups.
result The algorithms provide a way to determine if a subgroup is stable or Morse in specific group types.
This work uses Lyapunov theory to improve the robustness of deep neural networks against adversarial attacks.
problem Vulnerability of deep neural networks to subtle adversarial perturbations.
method Treated each layer as a nonlinear dynamical system and used Lyapunov theory for stability and robustness.
result Developed empirically tight bounds on adversarial perturbations and proved stability and robustness globally.
The paper proves stability of curvature bounds in geometric analysis.
problem Stability of local Riemannian Ricci curvature bounds under convergence.
method Gromov-Hausdorff convergence, Lagrangian approach, heat flow, weak gradients, Evolution Variational Inequality.
result Almost everywhere existence of Euclidean weak tangents.
Paper proves stability of positive mass theorem for specific types of manifolds.
problem Stability of positive mass theorem for compact graphical manifolds.
method Used Federer--Fleming flat distance and static quasi-local Brown-York energy.
result Proved stability of positive mass theorem for compact (locally) hyperbolic graphical manifolds.
The paper establishes curvature inequalities and rigidity results for surfaces in Riemannian and Lorentzian geometry.
problem Curvature and rigidity of surfaces in Riemannian and Lorentzian geometries.
method Establishes curvature inequalities and rigidity results using stability conditions and extrinsic curvature sign conditions.
result Sharp inequality ∣ H ⃗ ∣ 2 ≤ 16 π / ∣ Σ ∣ |\vec{H}|^2\leq 16π/ |Σ| ∣ H ∣ 2 ≤ 16 π /∣Σ∣ for spacetime constant mean curvature surfaces under the dominant energy condition. New method stabilizes quantum ergodicity for mixed quantization and partial hyperbolicity.
problem Stabilizing quantum ergodicity for complex systems.
method Combines mixed quantization techniques with stable ergodicity results for partially hyperbolic systems.
result Establishes stable quantum ergodicity for spin Hamiltonians.
Geometric invariant theory introduces stability conditions mirroring abelian category theory.
problem Stability conditions in geometric invariant theory.
method Axiomatic notion of central charge and stability condition on schemes and stacks.
result Introduction of stability conditions for polarized schemes and smooth projective varieties.
We investigate rigidity and stability properties of critical points of quadratic curvature functionals on the space of Riemannian metrics. We show it is possible to "gauge" the Euler-Lagrange equations, in a self-adjoint fashion, to become elliptic. Fredholm theory may then be used to describe local properties of the m…
Geometric perspective on unique solution in matrix completion with a deterministic pattern.
problem Identifying unique solutions in matrix completion with a specific pattern of observed entries.
method Geometric and algebraic analysis, focusing on the well-posedness condition and local stability.
result A sufficient condition for local uniqueness of matrix completion solutions, called the well-posedness condition.
New stabilizing number defined for knots, linking bounds in 4D.
problem Defining a new measure for knot boundaries in 4D.
method Defining stabilizing number sn ( K ) \operatorname{sn}(K) sn ( K ) , bounding it by signatures, Casson-Gordon invariants, and 4-genus. result Found examples where stabilizing number is less than 4-genus.
Unified theory explains GAN convergence, highlighting interaction term's dual roles.
problem Understanding and accelerating convergence of GANs.
method Unified non-asymptotic local convergence theory for smooth two-player games.
result Interaction term explains slow-down and exponential convergence for GAN training.
Study on birational rigidity and stability of hypersurfaces and complete intersections, proving non-locally closed property.
problem Birational rigidity and stability of hypersurfaces and complete intersections.
method Optimal results on birational rigidity and K-stability, proving non-locally closed property.
result Birational superrigidity is not a locally closed property.
Study local properties of Chern-scalar curvature through linearization stability.
problem Local properties of Chern-scalar curvature function.
method Linearization analysis of the Chern-scalar curvature function.
result Stability of linearization and structure of metrics with prescribed curvature.
New stabilization found in planar elasticae with degenerate diffusion.
problem Existence of local minimizers in degenerate p p p -elasticae. method Analysis of pinned planar p p p -elasticae with degenerate diffusion. result Uncountably many local minimizers with diverging energy in degenerate regime.
Proves stability of lcK spaces under holomorphic mappings.
problem Stability of locally conformally Kähler spaces with singularities.
method Analyzes sufficient conditions for proper open morphisms of lcK spaces.
result Extends Varouchas' result to lcK spaces with singularities.
Stability of Einstein metrics under Ricci iteration studied.
problem Stability of Einstein metrics under Ricci iteration.
method Sufficient condition based on Lichnerowicz Laplacian spectrum.
result Stability of several Einstein manifolds including symmetric spaces.
The study proves the stability of smooth embeddings of Riemannian metrics into Euclidean space.
problem Stability of smooth embeddings of Riemannian metrics into Euclidean space.
method Local perturbation method to derive a time-dependent local perturbation method.
result Construction of a smooth parametrized family of isometric embeddings for a short time.
New method solves ∂ ˉ \bar{\partial} ∂ ˉ -equations for logarithmic forms on Kahler manifolds.
problem Solving ∂ ˉ \bar{\partial} ∂ ˉ -equations for logarithmic forms on Kahler manifolds. method Using harmonic integral theory for currents on Kahler manifolds.
result Constructs the extension for logarithmic ( n , q ) (n,q) ( n , q ) -forms on the central fiber. Introduces stability conditions for polarized varieties, linking to K-stability.
problem Stability conditions for polarized varieties.
method Analogue of Bridgeland's stability for polarized varieties, Z-stability, Z-critical Kähler metrics.
result Polarized varieties with certain stability conditions admit Z-critical Kähler metrics.
Quadratic model surprisingly predicts optimization dynamics in large neural networks.
problem Complexity of neural network loss landscapes and optimization dynamics.
method Stress testing the quadratic model, Taylor expansion, Lanczos quadrature, and local linear stability analysis.
result The quadratic model can accurately predict optimization dynamics over long windows in large neural networks.
The paper proves stability of a quasi-local positive mass theorem for graphical hypersurfaces.
problem Stability of a quasi-local positive mass theorem for graphical hypersurfaces.
method Worked with the Brown--York quasi-local mass, considering compact n-manifolds with boundary as graphs in R^(n+1).
result If the Brown--York mass of the boundary of a compact manifold is small, then the manifold is close to a Euclidean hyperplane.
Surveying stability of klt singularities with new solutions.
problem Stability of klt singularities.
method Survey and solution of the stable degeneration conjecture.
result Solution to the stable degeneration conjecture.
We define a notion of stability for chiral ring of four dimensional N=1 theory by introducing test chiral rings and generalized a maximization. We conjecture that a chiral ring is the chiral ring of a superconformal field theory if and only if it is stable. We then study N=1 field theory derived from D3 branes probing …
Paper proves C 0 C^0 C 0 -semi-rigidity of meandering-hyperbolic actions.
problem Stability of meandering-hyperbolic actions in C 0 C^0 C 0 topology. method Proves semi-rigidity using local C 0 C^0 C 0 -semi-rigidity. result Every meandering-hyperbolic action is locally semi-rigid in C 0 C^0 C 0 topology. The study establishes conditions for orientability in spaces with lower Ricci curvature bounds.
problem Conditions for orientability in spaces with lower Ricci curvature bounds.
method Equivalent characterizations of orientability using Ricci limit and RCD spaces.
result Four-manifolds with Ricci curvature bounded below and volume non-collapsing are uniformly locally orientable.
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 shows stability of locally conformally balanced condition under modifications but not under small deformations.
problem Stability of locally conformally balanced condition under small deformations and modifications.
method Proved stability under proper modifications and instability under small deformations using examples and Hilbert-Chow map.
result Stability of locally conformally balanced condition under proper modifications and instability under small deformations.
New proof for stability estimates in complex equations without pluripotential theory.
problem Stability estimates for complex Monge-Ampère and Hessian equations.
method New proof using general degenerations of background metrics.
result Uniform stability estimates for both equations under various degenerations.
Cubic fourfolds have K-stability and admit Kähler-Einstein metrics.
problem Understanding K-stability and Kähler-Einstein metrics for cubic fourfolds.
method Local volume estimates and Ambro-Kawamata's non-vanishing theorem.
result All smooth cubic fourfolds admit Kähler-Einstein metrics.
Stability of capillary hypersurfaces with higher order mean curvature.
problem Stability of capillary hypersurfaces with constant higher order mean curvature.
method Generalization of classical stability theory for capillary hypersurfaces.
result Results on stability for capillary hypersurfaces with higher order mean curvature.
New findings show privacy affects generalization error in a non-monotonic way.
problem Privacy and robustness in distributed learning.
method Theoretical analysis and matching lower/upper bounds on algorithmic stability.
result Generalization error is non-monotonically affected by privacy, depending on noise level.
We prove a version the local Reeb-Thurston stability theorem for symplectic foliations.