Constructs index for elliptic operators using rapidly decaying kernels.
problem Index of elliptic operators in Fréchet algebra.
method Uses heat operators and heat kernel asymptotics.
result Index can be represented by an idempotent involving heat operators.
Study on Kähler manifolds with nonnegative Ricci curvature, focusing on rigidity.
problem Rigidity of Kähler manifolds with nonnegative Ricci curvature.
method Analysis of Kähler manifolds with specific properties.
result Complete noncompact Kähler surface with nonnegative Ricci curvature, Euclidean volume growth, and quadratic curvature decay is biholomorphic to the resolution of an affine algebraic variety.
A new flow method reduces Lorentz contraction to a simple algebraic decay.
problem Reducing Lorentz contraction in geometric models.
method Variational scalar conformal flow with algebraic decay.
result Explicit algebraic decay law for energy functional.
We consider any pseudo holomorphic integral 2-cycle in an arbitrary almost complex manifold and perform a blow up analysis at an arbitrary point. Building upon a pseudo algebraic blow up (previously introduced by the author) we prove a geometric rate of decay for the mass ratio towards the limiting density, with an exp…
Study shows rapid decay of Hitchin metric from semi-flat metric on Higgs bundles.
problem Analyzing the asymptotic behavior of the Hitchin metric on moduli spaces of Higgs bundles.
method Examined the decay rate of the difference between Hitchin and semi-flat metrics on smooth spectral curves.
result Exponential decay of the difference between Hitchin and semi-flat metrics as t approaches infinity.
We study the geometry of infinitely presented groups satisfying the small cancelation condition C'(1/8), and define a standard decomposition (called the criss-cross decomposition) for the elements of such groups. We use it to prove the Rapid Decay property for groups with the stronger small cancelation property C'(1/10…
We compare systematically several classes of stochastic volatility models of stock market fluctuations. We show that the long-time return distribution is either Gaussian or develops a power-law tail, while the short-time return distribution has generically a stretched-exponential form, but can assume also an algebraic …
The paper discusses methods to compute Green's function on algebraic surfaces using Schottky uniformization.
problem Computing Green's function on algebraic surfaces using Schottky uniformization.
method Investigates convergence of deformations of a formula related to Green's function.
result Provides insights into the geometric interpretation of the formula for Green's function.
This paper speeds up K-FAC for deep learning by focusing on only a few eigen-modes.
problem Time-consuming computation of Kronecker factors in K-FAC for large layers.
method Theoretical analysis and randomized numerical linear algebra to approximate eigen-spectrum decay.
result Reduces time complexity from cubic to quadratic in layer width, improving efficiency.
We started from computer experiments with simple one-dimensional ergodic dynamical systems called interval exchange transformations. Correlators in these systems decay as a power of time. In the simplest non-trivial case the exponent is equal to 1/3. We found a formula connecting characteristic exponents with explicit …
In this paper, by combining techniques from Ricci flow and algebraic geometry, we prove the following generalization of the classical uniformization theorem of Riemann surfaces. Given a complete noncompact complex two dimensional Kähler manifold M of positive and bounded holomorphic bisectional curvature, suppose its…
Proves K-polystability for Kähler-Ricci shrinkers with decaying curvature.
problem K-stability of Kähler-Ricci shrinkers with decaying curvature.
method Developed algebraic theory for Kähler-Ricci shrinkers and proved K-polystability.
result Existence of Kähler-Ricci shrinker metric implies K-polystability in decaying curvature case.
Study efficient neural operator learning using variation spaces.
problem Operator learning using encoder-decoder neural networks.
method Introduce variation space for nonlinear operators, establish approximation bounds.
result Algebraic approximation and learning rates for polynomially decaying input and output encoding errors.
We prove compactification theorems for some complete Kähler manifolds with nonnegative Ricci curvature. Among other things, we prove that a complete noncompact Kähler Ricci flat manifold with maximal volume growth and quadratic curvature decay is a crepant resolution of a normal affine algebraic variety. Furthermore, s…
We consider the SL(2,R) action on moduli spaces of quadratic differentials. If μ is an SL(2,R)-invariant probability measure, crucial information about the associated representation on L2(μ) (and in particular, fine asymptotics for decay of correlations of the diagonal action, the Teichmüller flow) is encoded …
Global stability proved for Navier-Stokes equations on hyperbolic space.
problem Stability of the Navier-Stokes equations on hyperbolic space.
method Proved global stability with exponential decay rate for small initial data.
result Exponential decay rate of $μλ_\Def^{(3)}$ for Navier-Stokes equations on hyperbolic space.
We study the behavior of hyperbolic affine automorphisms of a translation surface which is infinite in area and genus that is obtained as a limit of surfaces built from regular polygons studied by Veech. We find that hyperbolic affine automorphisms are not recurrent and yet their action restricted to cylinders satisfie…
The study establishes minimax bounds for estimating operators from noisy samples.
problem Estimating unknown operators between Hilbert spaces from noisy data.
method Developed a minimax theory for uniformly bounded Lipschitz operators, proving lower and upper bounds.
result Sharp characterizations of minimax risk for generic Lipschitz operators, showing a curse of sample complexity.
We prove the existence of a Hermitian-Einstein metric on holomorphic vector bundles with a Hermitian metric satisfying the analytic stability condition, under some assumption for the underlying Kähler manifolds. We also study the curvature decay of the Hermitian-Einstein metrics. It is useful for the study of the class…
Extends Minkowski stability proof to minimal decay assumptions.
problem Global stability of Minkowski spacetime with minimal decay.
method Extends Christodoulou-Klainerman's proof to minimal decay assumptions.
result Exterior stability of Minkowski holds with borderline decay.
In this paper, we discuss the Cramér-Lundberg model with investments, where the price of the invested risk asset follows a geometric Brownian motion with drift a and volatility σ>0. By assuming there is a cap on the claim sizes, we prove that the probability of ruin has at least an algebraic decay rate if $2a/σ^2 …
New method creates vacuum data at minimal and borderline decay thresholds.
problem Creating vacuum initial data at specific decay thresholds.
method Conical solution-operator method applied to vacuum asymptotically flat initial data.
result Demonstrates global and exterior stability of Minkowski spacetime.
Study shows mixing of flows on specific geometric spaces.
problem Mixing of one-parameter diagonal flows on Anosov homogeneous spaces.
method Proves local mixing for flows on $Γackslash G$ with deviations in transverse subspaces.
result Local mixing of flows on $Γackslash G$ for various directions.
Unique solutions found for wave-like decaying null infinity equations.
problem Wave-like decaying null infinity equations with spherically symmetric Einstein-scalar-field.
method Local and global unique solutions for small initial data.
result Sharp decaying condition for unique solutions.
New findings on flatness of certain metrics with fast decay.
problem Rigidity of positive mass theorem under fast metric decay.
method Considered metrics with nonnegative scalar curvature and rapid decay at infinity.
result Any such metric is necessarily flat in dimensions 4 and higher if decay rate exceeds Schwarzschild metric.
Study on curvature decay in steady Ricci solitons, proving dichotomy.
problem Curvature decay in steady Ricci solitons.
method Established a dichotomy for curvature decay in specific types of solitons.
result Proved a dichotomy on curvature decay for certain steady Ricci solitons.
Open problem: Establishing bounds for Cayley-table completion to discover discrete algorithmic axioms.
problem Discovering discrete algorithmic axioms missing in deep learning.
method Cayley-table completion as a testbed for algorithmic complexity minimization.
result Formal exact recovery bounds for Cayley-table completion.
Cautious Weight Decay modifies weight decay for better optimization.
problem Improving optimization in deep learning models.
method Applies weight decay selectively based on parameter sign alignment.
result Consistently improves model performance across various tasks and scales.
Study on scalar curvature decay on non-compact manifolds linked at infinity.
problem Understanding scalar curvature decay on non-compact manifolds with topological linking at infinity.
method Analyzing polynomial decay, developing obstruction theory, using μ--bubble exhaustions, and index theory. result Topological linking at infinity forces polynomial decay of scalar curvature on manifolds of weakly bounded geometry.
Study on massless Vlasov equation on Reissner-Nordström spacetimes, showing decay rates and non-decay phenomena.
problem Analyzing decay and non-decay rates of solutions to the massless Vlasov equation on Reissner-Nordström spacetimes.
method Quantitative analysis of geodesic flow and comparison to wave equation instability results.
result Exponential decay rates in subextremal cases and polynomial rates in extremal cases, with non-decay of transversal derivatives in extremal cases.
We first show that a Kähler cone appears as the tangent cone of a complete expanding gradient Kähler-Ricci soliton with quadratic curvature decay with derivatives if and only if it has a smooth canonical model (on which the soliton lives). This allows us to classify two-dimensional complete expanding gradient Kähler-Ri…
Introduces gradient decay in Softmax for better generalization.
problem Improving generalization performance in neural networks.
method Gradient decay hyperparameter in Softmax for varying gradient rates based on probability.
result Gradient decay rate affects generalization performance and can be tuned for better optimization.
Study examines wave equation decay and Strichartz estimates on conic manifolds.
problem Analyzing wave equation behavior on conic spaces with critical electromagnetic potentials.
method Established decay and Strichartz estimates through localized spectral measure construction.
result Extended and improved previous results on wave equation behavior with critical potentials.
Study shows uniform decay rate for singular mean curvature flows.
problem Understanding singularities in mean curvature flows.
method Rescaled flow analysis near compact singularities.
result Uniform decay order bound for the rescaled flow.
Weight decay stabilizes training dynamics by slowing progressive sharpening.
problem Understanding how weight decay affects training stability in deep learning models.
method Analyzing weight decay effects at the Edge of Stability, developing a mathematical framework.
result Weight decay dampens oscillations and stabilizes sharpness in CNNs, causing a phase transition in MLPs.
We provide a full characterisation of the large-maturity forward implied volatility smile in the Heston model. Although the leading decay is provided by a fairly classical large deviations behaviour, the algebraic expansion providing the higher-order terms highly depends on the parameters, and different powers of the m…
Polynomial decay of correlations shown for curved surfaces.
problem Analyzing geodesic flows on curved surfaces.
method Proving polynomial decay of correlations for geodesic flows on nonpositively curved surfaces.
result Polynomial decay of correlations for geodesic flows on nonpositively curved surfaces.
Last SGD iterate bounds for overparameterized linear regression.
problem Analyzing the last iterate risk bounds of SGD with decaying stepsize for overparameterized linear regression.
method Problem-dependent analysis of last iterate risk bounds of SGD with geometrically decaying stepsize.
result Proved nearly matching upper and lower bounds on the excess risk for last iterate SGD with geometrically decaying stepsize.
Optimal learning rates decay to zero in easy tasks and maintain a warmup phase in hard tasks.
problem Optimizing learning rates under functional scaling laws for model training.
method Deriving optimal learning-rate schedules based on exponents s and β. result Sharp phase transition between easy and hard tasks, with different decay behaviors.
We found that factors decay over time, with momentum fitting best.
problem Understanding how factors decay over time and their impact on performance.
method Derived a hyperbolic decay model for factors, tested against linear and exponential alternatives.
result Momentum exhibits hyperbolic decay, outperforming linear and exponential models.
The paper predicts and explains the decay of stock anomaly performance over time.
problem Predicting and explaining the drop in risk-adjusted performance of stock anomalies.
method The authors propose ex-ante characteristics based on hypotheses of out-of-sample decay and in-sample overfitting.
result The year of publication explains 30% of the variance in Sharpe decay across factors.
Sharp decay found for solutions of a specific equation in Lie groups.
problem Asymptotic decay of solutions to a Yamabe type equation.
method Analysis of a specific pseudodifferential operator in a homogeneous Lie group.
result Established sharp asymptotic decay of positive solutions.
We give conditions which imply that a complete noncompact manifold with quadratic curvature decay has finite topological type. In particular, we find links between the topology of a manifold with quadractic curvature decay and some properties of the asymptotic cones of such a manifold.
The paper extends decay estimates to graphs with positive spectrum.
problem Proving decay estimates for nonnegative functions on graphs.
method Sharp ℓ2 decay estimates for nonnegative generalized subharmonic functions. result Extends Li and Wang's result to graphs with positive Laplacian spectrum.
The paper studies harmonic map heat flow stability and decay rates.
problem Analyzing stability and decay rates of harmonic map heat flow solutions.
method Use of homogeneous Besov space B˙p,∞pd(Rd) for small initial data and self-similar decay assumption. result Decay rates for solutions of the harmonic map flow of the form ∥ablau(t)∥L∞(Rd)≤Ct−21 and self-similar decay under stronger initial conditions. In this paper, we give a description for steady Ricci solitons with a linear decay of sectional curvature. In particular, we classify all 3-dimensional steady Ricci solitons and 4-dimensional κ-noncollpased steady Ricci solitons with nonnegative sectional curvature under the linear curvature decay.
Extends global stability of Minkowski spacetime to minimal decay assumptions.
problem Global stability of Minkowski spacetime under minimal decay assumptions.
method Uses rp-weighted estimates instead of vectorfield method. result Proves global stability of Minkowski spacetime under minimal decay assumptions.
SHIFT framework identifies subgroups with large ML model performance decay.
problem Large model performance decay in subgroups when deployed.
method Subgroup-scanning Hierarchical Inference Framework (SHIFT) for performance drift.
result SHIFT identifies interpretable subgroups with large performance decay and suggests targeted actions to mitigate it.