Lower bounds on average normal curvature for submanifolds in Riemannian domains.
problem Finding bounds on the average normal curvature of submanifolds in Riemannian domains.
method Using an invariant measuring optimal n-trace convexity under unit-gradient normalization. result Lower bounds for the average normal curvature expressed in terms of an invariant.
Lower bounds on curvature integral for manifolds with curvature constraints.
problem Bounding curvature integrals under curvature constraints.
method Proving a lower bound for the curvature integral using dimension, upper curvature bounds, and injectivity radius.
result Uniformly bounded below integral of scalar curvature.
Metric measure boundary vanishes on certain spaces without boundary.
problem Existence of infinite geodesics on Alexandrov spaces without boundary.
method Solving conjecture by showing metric measure boundary vanishes on mRCD(K,N) spaces. result Metric measure boundary vanishes on mRCD(K,N) spaces without boundary. Paper shows equivalence of two curvature notions on singular surfaces.
problem Equivalence of two curvature notions on singular surfaces.
method Demonstrates equivalence between two curvature definitions.
result Inequalities of curvature measure imply Alexandrov curvature bounds.
New shapes enclose less volume than the sphere, surprising in 3D.
problem Finding the minimal volume enclosed by smooth spheres with bounded curvatures.
method Produced a family of bodies parameterized by ε, each bounded by a smooth topological sphere with principal curvatures in [-1, 1].
result The unit sphere does not enclose the minimal volume among all smooth spheres in R^3 with principal curvatures in [-1, 1].
We prove the generalized Margulis lemma with a uniform index bound on an Alexandrov n-space X with curvature bounded below, i.e., small loops at p∈X generate a subgroup of the fundamental group of unit ball B1(p) that contains a nilpotent subgroup of index ≤w(n), where w(n) is a constant depending on…
Proves non-existence of certain flat manifolds.
problem Non-existence of asymptotically flat 4-manifolds.
method Analyzes conjecture of Petrunin and Tuschmann.
result Proves conjecture on non-existence of asymptotically flat 4-manifolds.
Integral of scalar curvature over a manifold is bounded by a constant depending on dimension and curvature threshold.
problem Bounding curvature integral over open manifolds and smooth manifolds with boundary.
method Induced Riemannian metric on tangent cones.
result Integral of scalar curvature over a smooth manifold is bounded.
Smooth surface encloses less volume than a ball.
problem Can a smooth surface enclose less volume than a ball?
method Example of a smooth closed surface in R3 with specific curvature constraints. result No, a supersqueezed sphere encloses less volume than a unit ball.
Study on extremal subsets in geodesically complete spaces with curvature constraints.
problem Characterizing extremal subsets in GCBA spaces.
method Introduced and analyzed extremal subsets in GCBA spaces, proving their properties.
result Set of topological singularities forms an extremal subset under additional assumptions.
Proves spheres with bounded curvatures must contain a unit ball.
problem Proving spheres with bounded curvatures enclose a unit ball.
method Analyzing topological spheres in R^3 with bounded normal curvatures.
result Spheres with normal curvatures bounded by 1 must contain a unit ball.
We show that on every RCD spaces it is possible to introduce, by a distributional-like approach, a Riemann curvature tensor. Since after the works of Petrunin and Zhang-Zhu we know that finite dimensional Alexandrov spaces are RCD spaces, our construction applies in particular to the Alexandrov setting.…
Random 3-manifolds have no totally geodesic submanifolds.
problem Existence of totally geodesic submanifolds in random 3-manifolds.
method Analysis of metrics on compact 3-manifolds in Cq-topology. result The set of such metrics contains an open and dense set in the Cq-topology for any q≥3. Stability of tori under curvature conditions is proven.
problem Stability of tori under curvature conditions.
method Gromov-Hausdorff convergence and Alexandrov spaces.
result Stability of tori under curvature conditions is proven.
Convex hypersurfaces in curved spaces bound convex regions.
problem Characterizing convex hypersurfaces in curved spaces.
method Gauss-Codazzi equations, Schur comparison theorem, Alexandrov geometry.
result Closed convex hypersurfaces bound convex regions in curved spaces.
Alexandrov spaces have a special stratification that maps to spheres.
problem Characterizing the structure of Alexandrov spaces.
method Extremal stratification and space of directions analysis.
result Alexandrov spaces are homeomorphic to spheres in their space of directions.
Alexandrov spaces with non-negative curvature are characterized by the matrix displacement convexity of an entropy tensor.
problem Characterizing non-negative curvature in Alexandrov spaces
method Constructing a parallel trivialization of the entropy tensor
result The entropy tensor is matrix displacement convex on Alexandrov spaces
Theory of parallel transport on non-collapsed RCD spaces established.
problem Parallel transport on non-collapsed RCD spaces.
method General theory developed for parallel transport on non-collapsed RCD spaces, including geodesics and curves via time-dependent vector fields.
result Existence and uniqueness of parallel transport results obtained.
We prove a Lipschitz-Volume rigidity theorem in Alexandrov geometry, that is, if a 1-Lipschitz map f:X=⨿Xℓ→Y between Alexandrov spaces preserves volume, then it is a path isometry and an isometry when restricted to the interior of X. We furthermore characterize the metric structure on Y with re…
Characterizes submanifolds with minimum ratio of diameter to focal radius.
problem Finding submanifolds with the minimum ratio of extrinsic diameter to focal radius.
method Combining K. Sakamoto's classification of submanifolds with planar geodesics and A. Schur's Bow Lemma for space curves.
result Essentially round spheres or Veronese embeddings of projective spaces achieve the minimum ratio.
In this paper we discuss an extension of Perelman's comparison for quadrangles. Among applications of this new comparison theorem, we study the equidistance evolution of hypersurfaces in Alexandrov spaces with non-negative curvature. We show that, in certain cases, the equidistance evolution of hypersurfaces become tot…
We study, from the extrinsic point of view, the structure at infinity of open submanifolds isometrically immersed in the real space forms of constant sectional curvature κ≤0. We shall use the decay of the second fundamental form of the the so-called tamed immersions to obtain a description at infinity of the subm…
Uniform bounds for eigenvalues of Hodge Laplacian on manifolds with lower Ricci curvature.
problem Establishing bounds for eigenvalues of Hodge Laplacian under lower Ricci curvature.
method Using geometric assumptions including lower Ricci curvature, injectivity radius, and diameter bounds.
result Uniform eigenvalue bounds for the Hodge Laplacian and connection Laplacian.
Lower bounds on ribbon distance using Bar-Natan and α-Homology.
problem Calculating the minimum number of ribbon operations to unknot a knot.
method Bar-Natan Homology and α-Homology approaches.
result Lower bounds on ribbon distance via both Bar-Natan and α-Homology.
New study on regret lower bounds for multi-agent multi-armed bandit problems.
problem Understanding the limits of performance in multi-agent multi-armed bandit problems.
method Comprehensive study on different settings, establishing tight lower bounds.
result First comprehensive study on regret lower bounds across various settings.
Along the line of the Yang Conjecture, we give a new estimate on the lower bound of the first non-zero eigenvalue of a closed Riemannian manifold with negative lower bound of Ricci curvature in terms of the in-diameter and the lower bound of Ricci curvature.
There has been renewed recent interest in developing effective lower bounds for Dynamic Time Warping (DTW) distance between time series. These have many applications in time series indexing, clustering, forecasting, regression and classification. One of the key time series classification algorithms, the nearest neighbo…
In this paper we present a self-contained combinatorial proof of the lower bound theorem for normal pseudomanifolds, including a treatment of the cases of equality in this theorem. We also discuss McMullen and Walkup's generalised lower bound conjecture for triangulated spheres in the context of the lower bound theorem…
Lower bound found for Kähler manifold eigenvalues.
problem Finding bounds for eigenvalues on Kähler manifolds.
method Comparison results of Li and Wang applied to Kähler manifolds.
result Explicit lower bound of the first eigenvalue determined.
Paper presents a reduction-based framework for conservative bandits and RL with improved lower and upper bounds.
problem Conservative bandits and reinforcement learning problems.
method Reduction technique to calculate necessary and sufficient budget from baseline policy.
result Improved lower and upper bounds for various conservative settings.
Survey on gluing constructions under lower curvature bounds.
problem Understanding lower curvature bounds in various geometric contexts.
method Analyzes gluing constructions in smooth and non-smooth settings.
result Provides conjectures and theorems on synthetic lower Ricci curvature bounds.
The paper establishes bounds on scalar curvature on asymptotically flat manifolds.
problem Establishing scalar curvature bounds on asymptotically flat manifolds.
method Using Ricci-DeTurck flow and distributional scalar curvature, the paper derives bounds on scalar curvature.
result The scalar curvature lower bound under Ricci-DeTurck flow depends on the scalar curvature lower bound in the β-weak sense and time.
We prove non-asymptotic lower bounds on the expectation of the maximum of d independent Gaussian variables and the expectation of the maximum of d independent symmetric random walks. Both lower bounds recover the optimal leading constant in the limit. A simple application of the lower bound for random walks is an (…
We give a lower bound on the number of non-simple closed curves on a hyperbolic surface, given upper bounds on both length and self-intersection number. In particular, we carefully show how to construct closed geodesics on pairs of pants, and give a lower bound on the number of curves in this case. The lower bound for …
Study sets lower bounds for Kähler manifolds' Laplacian eigenvalues.
problem Finding bounds for eigenvalues on Kähler manifolds.
method Establishes lower bounds using geometric data like dimension, diameter, and curvature.
result Proves bounds for Laplacian eigenvalues on Kähler manifolds.
New algorithms for private GLM estimation with minimax lower bounds.
problem Privacy in generalized linear models.
method Differentially private algorithms using projected gradient descent.
result Nearly rate-optimal performance with privacy-constrained minimax lower bounds.
Lower bounds for eigenvalues on Bakry-Emery manifolds proven.
problem Eigenvalue estimates on Bakry-Emery manifolds.
method Generalised maximum principle and heat kernel estimates.
result Lower bounds for all eigenvalues proven.
Estimates lower bounds for isoperimetric profiles and improves on previous estimates for specific manifolds.
problem Estimating lower bounds for isoperimetric profiles of specific Riemannian manifolds.
method Explicit lower bounds for isoperimetric profiles of Riemannian product manifolds.
result Improved lower bounds for isoperimetric profiles and Yamabe constants.
Sharp lower bound for Hodge Laplacian on Kähler hyperbolic manifolds.
problem Finding a sharp lower bound for the spectrum of the Hodge Laplacian.
method Explicitly expressed in terms of the supremum norm of the 1-form.
result Explicit spectral lower bounds for bounded symmetric domains.
Lower bound found for volatility swap in SABR model.
problem Finding a lower bound for volatility swap in SABR model.
method Short time to maturity limit analysis of conditionally lognormal SABR model.
result Zero vanna implied volatility is a lower bound for volatility swap strike.
Lower bounds for geodesically convex optimization show curvature negatively impacts complexity.
problem Understanding the impact of curvature on the query complexity of geodesically convex optimization.
method Building on recent lower bounds, the study proposes and proves new lower bounds for various settings of geodesically convex optimization.
result Negative curvature is detrimental to the complexity of geodesically convex optimization.
Paper proves tight lower bounds for online multicalibration, separating it from marginal calibration.
problem Proving lower bounds for online multicalibration in relation to marginal calibration.
method Information-theoretic approach, constructing group families from orthonormal bases.
result Establishes tight lower bounds for online multicalibration, matching upper bounds up to logarithmic factors.
Paper establishes new lower bounds for MDPs with changing transition kernels.
problem Minimizing sample complexity and regret in non-stationary MDPs.
method Developed novel lower bounds and constructed hard MDPs.
result Proved Ω((H3SA/ε2)log(1/δ)) sample complexity lower bound. Lower bounds found for nonconvex-strongly-concave min-max optimization problems.
problem Finding stationary points in nonconvex-strongly-concave min-max optimization.
method Provided lower bounds for first-order oracle complexity.
result Lower bounds of Ω(√κε⁻²) for deterministic oracles and Ω(√κε⁻² + κ¹/₃ε⁻⁴) for stochastic oracles.
The paper proves inequalities under Bakry-Émery-Ricci curvature bounds.
problem Proving functional inequalities under lower Bakry-Émery-Ricci curvature bounds.
method Lower m-Bakry-Émery-Ricci curvature bounds with ε-range. result Proves Cheng type inequality and local Sobolev inequality.
New lower bounds for combinatorial multi-armed bandits for general reward functions.
problem Maximizing reward in sequential decisions with sets of arms.
method Proved tight regret lower bounds for all smooth reward functions under mild assumptions.
result Lower bounds are tight up to log-factors for monotone reward functions.
Surveying Ricci flow for weak lower scalar curvature bounds.
problem Creating local definitions for weak lower scalar curvature bounds for C0 metrics. method Using Ricci flow to define and analyze weak lower scalar curvature bounds.
result Properties and applications of Ricci flow in defining weak lower scalar curvature bounds.
Abstract reviews known and open questions on spaces with lower Ricci bounds.
problem Understanding the structure and regularity of spaces with lower Ricci curvature bounds.
method Review of known results and presentation of open questions.
result Presentation of new open questions in the field.