The paper develops quantitative estimates for holomorphic sections over bounded domains.
problem Establishing precise inequalities for holomorphic sections over bounded domains.
method Develops Sobolev-type inequalities and applies them to holomorphic sections of Hermitian vector bundles.
result Quantitative Carleman-type estimates for holomorphic sections are derived, improving on previous non-quantitative results.
Proves upper bound on filling radius for manifolds with positive scalar curvature.
problem Bounding the filling radius of manifolds with positive scalar curvature.
method Quantitative operator K-theory and index theory.
result Proves a quantitative upper bound on the filling radius.
Quantitative Sobolev extensions lead to Neumann heat kernel bounds.
problem Bounding Neumann heat kernels for domains with integral Ricci curvature.
method Quantitative Sobolev extension operators and Neumann heat kernel estimates.
result Uniform bounds on Neumann heat kernels and eigenvalues.
Study PL bordism theories with quantitative bounds on filling simplices.
problem Understanding PL bordism theories with geometric constraints.
method Quantitative analysis of PL manifolds and exotic theories.
result Bounding the number of simplices in fillings of cycles.
Researchers create integral representations for two-layer ReLU networks with quantitative bounds.
problem Approximating functions with two-layer ReLU networks using explicit integral representations.
method Developed integral representations involving harmonic extension and projection, providing L2 bounds. result Functions can be approximated with L2 errors independent of dimension or degree, depending on coefficients and distribution. Study connects manifold complexity to scalar curvature bounds.
problem Understanding the relationship between manifold complexity and scalar curvature.
method Combining quantitative operator K-theory, Lipschitz topological K-theory, and a vanishing theorem.
result Established a relationship between covering complexity and scalar curvature bounds.
We provide a quantitative obstruction to collapsing surfaces of genus at least 2 under a lower curvature bound and an upper diameter bound. Keywords: curvature; diameter; volume; filling radius; systole; Gromov-Hausdorff distance
Enhanced bounds on rho-invariants for 3-manifolds.
problem Establishing bounds on Cheeger-Gromov rho-invariants for 3-manifolds.
method Constructing chain null-homotopies with linear complexity.
result Linearly bounded complexity of constructed null-homotopies.
Quantitative rigidity theorem for Alexandrov spaces with curvature bounds.
problem Quantifying rigidity in Alexandrov spaces with curvature constraints.
method Using Gromov-Hausdorff distance and properties of Alexandrov spaces.
result Alexandrov spaces with curvature bounds are close to hyperbolic manifolds.
Quantifies scalar curvature under C0 convergence, proving a refined version in all dimensions.
problem Proving a refined quantitative bound for scalar curvature under C0 convergence. method Established the refined quantitative bound in all dimensions using smoothing techniques.
result Established the refined quantitative bound for scalar curvature in all dimensions.
Deep belief networks can approximate any multivariate density with binary hidden units.
problem Approximating multivariate probability densities with binary hidden units.
method Sharp quantitative bounds on approximation error in terms of hidden units.
result Deep belief networks can approximate any multivariate density with binary hidden units under mild integrability requirements.
Quantum neural networks can approximate noisy functions accurately.
problem Approximating noisy functions with quantum neural networks.
method Universal approximation theorem with error bounds for noisy quantum neural networks.
result Quantum neural networks can approximate noisy functions with precise error bounds.
The paper proves a quantitative rigidity result for spaces with specific curvature bounds.
problem Understanding the rigidity of spaces with almost maximal volume entropy.
method Analyzing Riemannian manifolds and RCD-spaces with specific curvature conditions. result Spaces with almost maximal volume entropy are closely related to hyperbolic space forms.
Estimates for Schrödinger operators on manifolds with bounded Ricci curvature.
problem Quantifying unique continuation for Schrödinger operators on manifolds with specific curvature conditions.
method Proving quantitative unique continuation estimates for Schrödinger operators on manifolds with Ricci curvature bounded below.
result Upper bound for energy range and constant in terms of Ricci curvature and parameters of relatively dense set.
Study extends bounds on sample covariance matrices with general dependence.
problem Quantitative bounds on sample covariance matrices with i.i.d. columns.
method Extends previous work on deterministic equivalent to rectangular random matrices with general dependence structure.
result Proves quantitative bounds involving dimensions and spectral parameter, including closer proximity to real positive semi-line.
On a Riemannian manifold with a positive lower bound on the Ricci tensor, the distance of isoperimetric sets from geodesic balls is quantitatively controlled in terms of the gap between the isoperimetric profile of the manifold and that of a round sphere of suitable radius. The deficit between the diameters of the mani…
Study shows how close functions are to optimal in Riemannian manifolds.
problem Understanding how close functions are to optimal in Riemannian manifolds.
method Analyzes quantitative stability of Sobolev inequalities on compact Riemannian manifolds.
result Functions that nearly saturate a critical Sobolev inequality are quantitatively close to extremal functions.
We prove the following quantitative version of the celebrated Soap Bubble Theorem of Alexandrov. Let S be a C2 closed embedded hypersurface of Rn+1, n≥1, and denote by osc(H) the oscillation of its mean curvature. We prove that there exists a positive ε, depending on n and upper …
Quantitative CLTs show neural network distributions converge to Gaussian as width increases.
problem Understanding the distribution of fully connected neural networks with random weights and biases.
method Analyzing the distribution of a fully connected neural network with random Gaussian weights and biases, proving quantitative bounds on normal approximations.
result The distance between a random fully connected network and the corresponding infinite width Gaussian process scales like n−γ for γ>0. The paper improves inequalities for nearly spherical sets using quermassintegrals.
problem Improving inequalities for nearly spherical sets.
method Establishing quantitative Alexandrov-Fenchel inequalities for quermassintegrals.
result Lower bounds on the (k,m)-isoperimetric deficit found using spherical deviation and asymmetry. We prove a quantitative version of Obata's Theorem involving the shape of functions with null mean value when compared with the cosine of distance functions from single points. The deficit between the diameters of the manifold and of the corresponding sphere is bounded likewise. These results are obtained in the genera…
We establish a quantitative lower bound on the reach of flat norm minimizers for boundaries in R2.
We obtain a quantitative estimate on the generalised index of translators for the mean curvature flow with bounded norm of the second fundamental form. The estimate involves the dimension of the space of weighted square integrable f-harmonic 1-forms. By the adaptation to the weighted setting of Li-Tam theory developed …
Proves a quantitative index theorem for positive scalar curvature metrics.
problem Studying conjectures and open questions on positive scalar curvature.
method Quantitative relative index theorem and λ-Lipschitz rigidity theorem. result Positive answers to Gromov's open questions on scalar curvature.
A sharp quantitative polygonal isoperimetric inequality is obtained.
Study geodesic Lie groups' convergence to limits with quantitative estimates.
problem Quantifying convergence rates of geodesic Lie groups to their limits.
method Estimates on the difference between original metrics and asymptotic/tangent metrics.
result Sharpens existing bounds on convergence rates.
New principles prove precompactness of domains with lower Ricci curvature bound.
problem Proving precompactness of domains with lower Ricci curvature bound.
method Quantitative Hopf-Rinow theorem and doubling property.
result New precompactness principles applicable to incomplete Riemannian manifolds.
Bounds neural network output distribution to Gaussian for random initialization.
problem Quantifying the distribution of randomly initialized deep neural networks.
method Quantitative Gaussian approximation using quadratic Wasserstein distance.
result Explicit inequalities show how network sizes affect Gaussian behavior.
We introduce a "qualitative property" for Bott-Chern cohomology of complex non-Kähler manifolds, which is motivated in view of the study of the algebraic structure of Bott-Chern cohomology. We prove that such a property characterizes the validity of the ∂∂-Lemma. This follows from a quantitativ…
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.
Quantifies tightness in 3D contact manifolds using sub-Riemannian metrics.
problem Estimating maximal tight neighbourhoods of Reeb orbits on 3D contact manifolds.
method Sub-Riemannian metrics and geometric constructions.
result Sharp estimates of tightness radius in terms of Schwarzian derivative bounds.
When each data point is a large graph, graph statistics such as densities of certain subgraphs (motifs) can be used as feature vectors for machine learning. While intuitive, motif counts are expensive to compute and difficult to work with theoretically. Via graphon theory, we give an explicit quantitative bound for the…
We study generalizations of Reifenberg's Theorem for measures in Rn under assumptions on the Jones' β-numbers, which appropriately measure how close the support is to being contained in a subspace. Our main results, which holds for general measures without density assumptions, give effective measure bounds…
Study Gaussian-process limits of neural networks using tensor programs.
problem Understanding the behavior of neural networks as they approach infinite width.
method Quantitative analysis through tensor programs and Wasserstein distance.
result Explicit finite-width error bounds, showing convergence to Gaussian-process limits.
Sharp estimates for mean curvature flow confirm bounded diameter conjecture.
problem Bounding the diameter of mean curvature flow in 3D.
method Quantitative estimates on second fundamental form and singular set.
result Uniform boundedness of intrinsic diameter and sharp estimates on flow properties.
Sharp 3D Alexandrov inequality applied to volume-preserving flows.
problem Volume-preserving geometric flows in 3D space.
method Sharp quantitative Alexandrov inequality for C2-regular sets. result Established a 3D sharp quantitative version of the Alexandrov inequality.
In this paper we prove quantitative regularity results for stationary and minimizing extrinsic biharmonic maps. As an application, we determine sharp, dimension independent Lp bounds for ∇kf that do not require a small energy hypothesis. In particular, every minimizing biharmonic map is in W4,p for all…
This paper proves explicit bilipschitz bounds on the change in metric between the thick part of a cusped hyperbolic 3-manifold N and the thick part of any of its long Dehn fillings. Given a bilipschitz constant J > 1 and a thickness constant epsilon > 0, we quantify how long a Dehn filling suffices to guarantee a J-bil…
We establish a quantitative isoperimetric inequality for weighted Riemannian manifolds with Ric∞≥1. Precisely, we give an upper bound of the volume of the symmetric difference between a Borel set and a sub-level (or super-level) set of the associated guiding function (arising from the needle deco…
This article is about inverse spectral problems for hyperbolic surfaces and in particular how length spectra relate to the geometry of the underlying surface. A quantitative answer is given to the following: how many questions do you need to ask a length spectrum to determine it completely? In answering this, a quantit…
Paper proves a noncompact version of Gromov's band-width estimate.
problem Proving a precise upper bound for noncompact Riemannian bands.
method Developed a quantitative partitioned manifold index theory.
result Proved a version of Gromov's band-width estimate for noncompact Riemannian bands.
This paper proves properties of uniformly hyperbolic sets and constructs Markov partitions.
problem Establishing properties of uniformly hyperbolic sets and constructing Markov partitions.
method Backward graph transform, spectral decomposition, shadowing lemma, Markov partitions construction.
result Explicit bounds and Hölder continuity for the coding map.
Estimates for harmonic functions in curved spaces.
problem Quantifying harmonic functions in curved spaces.
method Quantitative Sobolev estimates for p-harmonic functions in manifolds with curvature conditions. result Established a quantitative second order Sobolev estimate for p-harmonic functions. Solves linearity problem for acyclic groups, bounds Cheeger-Gromov ρ-invariants.
problem Linearity problem for acyclic groups and Cheeger-Gromov ρ-invariants.
method Quantitative algebraic and geometric techniques over simplicial classifying spaces.
result Universal linear bound for Cheeger-Gromov ρ-invariants of PL (4k-1)-manifolds.
The paper proves a bound on the length of the shortest geodesic flower on certain manifolds.
problem Finding the shortest geodesic flower on a specific class of manifolds.
method Analyzing a non-compact Riemannian manifold with locally convex ends and finite volume, proving the existence of a geodesic net with constraints on its length.
result The existence of a non-trivial geodesic flower with a bounded total length on the manifold.
We show that any closed n-dimensional Riemannian manifold can be embedded by a map constructed from heat kernels at a certain time from a finite number of points. Both this time and this number can be bounded in terms of the dimension, a lower bound on the Ricci curvature, the injectivity radius and the volume. It foll…
Similarity algebra extends algebraic structures with quantitative bounds.
problem Exact algebraic structures with strict axioms.
method Framework for approximate algebraic and Lie structures with ε-estimates. result Similarity structures converge to classical algebraic objects as εightarrow0. New bounds show transformers need longer training for length generalization.
problem Understanding when transformers can generalize to longer inputs.
method Analyzing different settings of transformers, providing quantitative bounds.
result Transformers need training data longer than previously thought for length generalization.