Uniform waist inequalities proven for manifolds with Kazhdan groups in codimension two.
problem Proving uniform waist inequalities for manifolds with specific group properties.
method Using finite covers and Cheeger inequality for manifolds with Kazhdan fundamental groups.
result Finite covers of manifolds with Kazhdan groups satisfy uniform waist inequalities in codimension two.
Constructs foliations for 3-manifolds with positive scalar curvature.
problem Finding surfaces in 3-manifolds with positive scalar curvature.
method Singular foliations of compact three-manifolds with controlled properties.
result Extends Urysohn and Gromov-Lawson waist inequalities.
The abstract applies waist inequality to dynamical systems and entropy.
problem Understanding the relationship between waist inequality and dynamical systems.
method Applying waist inequality to entropy and mean dimension of dynamical systems.
result Maps between dynamical systems have positive conditional metric mean dimension under certain conditions.
We introduce two numerical invariants, the waist and the trunk of knots. The waist of a closed incompressible surface in the complement of a knot is defined as the minimal intersection number of all compressing disks for the surface in the 3-sphere and the knot. Then the waist of a knot is defined as the maximal waist …
This paper presents connections between Gromov's work on isoperimetry of waists and Milman's work on the M-ellipsoid of a convex body. It is proven that any convex body K⊆Rn has a linear image K~⊆Rn of volume one satisfying the following waist inequality: Any continu…
The waist inequality states that for a continuous map from S^n to R^q, not all fibers can have small (n-q)-dimensional volume. We construct maps for which most fibers have small (n-q)-dimensional volume and all fibers have bounded (n-q)-dimensional volume.
We prove several new results around Gromov's waist theorem. We give a simple proof of Vaaler's theorem on sections of the unit cube using the Borsuk--Ulam--Crofton technique. We consider waists of real and complex projective spaces, flat tori, convex bodies in Euclidean space. We establish waist-type results in terms o…
3-manifolds with positive scalar curvature have controlled foliations.
problem Understanding foliations in 3-manifolds with positive scalar curvature.
method Showed a singular foliation by surfaces with controlled area and diameter.
result 3-manifolds with positive scalar curvature admit controlled foliations.
We study the incompressible surfaces in the exterior of a cable knot and use this to compute the representativity and waist of most cable knots.
The waist size of a cusp in an orientable hyperbolic 3-manifold is the length of the shortest nontrivial curve generated by a parabolic isometry in the maximal cusp boundary. Previously, it was shown that the smallest possible waist size, which is 1, is realized only by the cusp in the figure-eight knot complement. In …
Continuous sweepouts cover manifolds with bounded curve lengths.
problem Covering closed Riemannian manifolds with bounded curve lengths.
method Continuous family of 1-cycles parametrized by a sphere, with length bounds in terms of volume and dimension.
result Polyhedral 1-waist equals filling radius up to a constant factor.
Study geodesics on neck-degenerate manifolds, focusing and winding behavior observed.
problem Geodesics behavior on neck-degenerate manifolds with cuspidal singularities.
method Detailed multiscale analysis, blow-up techniques.
result Geodesics exhibit focussing and winding behavior as the neck degenerates.
Minimal submanifolds in octonionic hyperbolic spaces have large volume.
problem Characterizing minimal submanifolds in locally symmetric spaces.
method Analyzing higher expansion properties and volume constraints.
result Codimension two minimal submanifolds have at least linear volume in the ambient space.
Study minimal annuli in a slab, estimating their area.
problem Estimating the area of minimal annuli in a slab.
method Organized minimal annuli based on winding number, deduced convexity of length function, compared to catenoid waist area.
result Deduced convexity of length function and estimated area of minimal annuli.
The study quantifies topological expansion properties of complexes and their embeddings.
problem Understanding topological expansion properties of simplicial complexes.
method Quantifying topological expansion through sublinear functions and proving monotonicity under regular maps.
result Proves topological expanders contain graphical expanders and gives lower bounds for specific embeddings.
Consider a non-planar orientable minimal surface S in a slab which is possibly with genus or with more than two boundary components. We show that there exists a catenoidal waist W in the slab whose flux has the same vertical component as S such that Area(S)>= Area(W), provided the intersections of S with horizontal pla…
Hyperbolic spaces have many cells in their fibers.
problem Finding lower bounds on the topological complexity of fibered spaces
method Using a freedom theorem for ideals in group rings of hyperbolic groups
result For large injectivity radii, there are many cells of dimension k in the fiber p^{-1}(z)
Modern machine learning algorithms are increasingly computationally demanding, requiring specialized hardware and distributed computation to achieve high performance in a reasonable time frame. Many hyperparameter search algorithms have been proposed for improving the efficiency of model selection, however their adapta…
We introduce a new critical value c∞(L) for Tonelli Lagrangians L on the tangent bundle of the 2-sphere without minimizing measures supported on a point. We show that c∞(L) is strictly larger than the Mañé critical value c(L), and on every energy level e∈(c(L),c∞(L)) there exist infinitely…
New maxfaces with catenoid or planar ends constructed using node-opening technique.
problem Lack of examples of maxfaces with catenoid or planar ends.
method Adapted node-opening technique to construct maxfaces of high genus.
result Singularities on constructed maxfaces form curves around the waists of the necks, with most singularities being cuspidal edges and the rest swallowtails.
Machine learning predicts obesity causes using genetic and imaging data.
problem Predicting causes of obesity in children and adults.
method Use ML techniques like decision trees, SVM, RF, GBM, LASSO, BN, and ANN on genetic and imaging data.
result ML models accurately predict obesity causes and chronic diseases.
Study of minimal surfaces in 4D with specific ends.
problem Characterize minimal surfaces in R4 with specific ends. method Modification of Costa and Hoffman-Meeks method, generalized Weierstrass representation.
result Minimal surfaces with specific ends are J-holomorphic under certain conditions. A comparison of the performance of various machine learning models to predict the direction of a wall following robot is presented in this paper. The models were trained using an open-source dataset that contains 24 ultrasound sensors readings and the corresponding direction for each sample. This dataset was captured u…
Method predicts NAFLD risk with high accuracy and distribution-free coverage guarantees.
problem Insufficient population-level screening tools for NAFLD.
method Gradient-boosted decision trees with conformal prediction.
result Method achieves AUROC of 0.912 internally and 0.891 externally, superior to other models.
The isoperimetric inequality and related inequalities are explored.
problem Proving the isoperimetric inequality and related inequalities.
method Discussing classical and recent proofs.
result Various proofs of the isoperimetric inequality and Sobolev inequality.
New proof of Willmore inequality using geometric divergence inequality.
problem Proving the Willmore inequality for bounded domains.
method Using a parametric geometric inequality derived from a divergence form geometric differential inequality.
result New proofs of quantitative Willmore-type and weighted Minkowski inequalities.
Lorentz-Finsler geometry reveals new and old inequalities.
problem Finding new inequalities using Lorentz-Finsler geometry.
method Applying reverse Cauchy-Schwarz and reverse triangle inequalities in Lorentz-Finsler geometry.
result Proved new and refined inequalities, including refinements of Aczél's inequality.
The paper derives new inequalities on manifolds and applies them to convex hypersurfaces.
problem Deriving new inequalities on manifolds and convex hypersurfaces.
method Using Fourier theory and geometric implications of Poincare-type inequalities.
result Sharp Minkowski-type inequalities, including stability and Alexandrov-Fenchel inequalities.
The paper proves inequalities on Finsler manifolds under Ricci curvature bounds.
problem Proving (p,q)-Sobolev and Nash inequalities on Finsler metric measure manifolds. method Global p-Poincaré inequality, (p,q)-Sobolev inequality, Nash inequality derivation. result Established global optimal (p,q)-Sobolev inequality with a sharp constant. New inequality on sphere generalizes circle inequality.
problem Generalizing circle inequality to sphere.
method Develops a new inequality on the sphere that incorporates mass center deviation.
result Improves Aubin's inequality and Onofri's inequality.
Paper proves anisotropic Minkowski inequality and related inequalities.
problem Proving anisotropic Minkowski inequality and related inequalities.
method Utilizes a nonlinear potential theoretic approach.
result Sharp anisotropic Minkowski inequality and related inequalities proved.
Explains geometric inequalities for minimal hypersurfaces.
problem Geometric inequalities for minimal hypersurfaces.
method Expository discussion of known inequalities.
result Discussion of classical inequalities for minimal hypersurfaces.
The paper finds new inequalities for convex polygons.
problem Finding precise inequalities for convex polygons.
method Analytic isoperimetric inequalities based on Schur convex functions, followed by Bonnesen-style and inverse Bonnesen-style inequalities.
result Sharp discrete isoperimetric inequalities for planar convex polygons.
The study improves Bochner inequality on Finsler manifolds to derive important inequalities.
problem Improving Bochner inequality on Finsler manifolds to derive new inequalities.
method Using improved Bochner inequality and its integrated form, the study derives a sharp Poincaré-Lichnerowicz inequality, a new proof for logarithmic Sobolev inequality, and an estimate of geodesic ball volumes.
result Derivation of new inequalities and estimates on Finsler manifolds.
The paper proves various inequalities on gradient shrinking Ricci solitons.
problem Understanding geometric inequalities on gradient shrinking Ricci solitons.
method Proving multiple inequalities equivalent on complete gradient shrinking Ricci solitons.
result Various inequalities (Sobolev, logarithmic Sobolev, Schrödinger, etc.) are equivalent on gradient shrinking Ricci solitons.
Sharp inequality found on three-balls for fourth order Sobolev traces.
problem Fourth order Sobolev trace inequality on three-balls.
method Established through equivalence to a third order Sobolev inequality on two-spheres.
result Sharp fourth order Sobolev trace inequality on three-balls.
Sharp inequalities for star bodies in 2D space.
problem Understanding star bodies in 2D space.
method Sharp inequalities for star bodies in R2. result New inequalities and proofs for star bodies.
The paper develops inequalities for log-concave functions and related surface areas.
problem Understanding log-concave functions and their inequalities.
method Establishing new inequalities through f-divergences and functional affine surface areas.
result New inequalities on functional affine surface area and bounds for Kullback-Leibler divergence.
Study on functional inequalities on simple edge spaces.
problem Whether classical functional inequalities hold in simple edge spaces.
method Analyzing Sobolev and Poincaré inequalities, proving optimality of Sobolev constant.
result Optimality result concerning the B-constant of the Sobolev inequality.
Proves inequalities on curved spaces with positive curvature.
problem Proving inequalities on manifolds with nonnegative Ricci curvature.
method Analyzes manifolds with nonnegative Ricci curvature and Euclidean volume growth.
result Proves Heisenberg-Pauli-Weyl, Hardy-Sobolev, and Caffarelli-Kohn-Nirenberg inequalities.
The paper derives inequalities on Finsler manifolds, influenced by their curvatures.
problem Deriving inequalities on Finsler manifolds.
method Local and global geometric inequalities on Riemannian and Finsler manifolds.
result Generalized Caffarelli-Kohn-Nirenberg and Hardy type inequalities on Finsler manifolds.
Paper refines Talagrand inequality on Euclidean spaces.
problem Improving Talagrand inequality for Euclidean spaces.
method Symmetrization and alternative proof methods.
result Several refined functional inequalities derived.
New inequalities for convex curves with multiple geometric factors.
problem Establishing inequalities for convex curves with multiple geometric factors.
method Parametric isoperimetric-type inequalities for closed convex curves with parameter conditions and equality conditions.
result Derived new inequalities and improved versions of existing inequalities.
Alternative proofs for various inequalities on Riemannian manifolds.
problem Various functional inequalities on Riemannian manifolds.
method Generic functional inequality, Riccati pairs, solving Riccati-type ODE.
result Alternative proofs for multiple inequalities, including Hardy-type and Caccioppoli inequalities.
The paper establishes inequalities and gradient estimates for harmonic functions on Finsler measure spaces.
problem Functional and geometric inequalities on Finsler measure spaces.
method Local uniform Poincaré and Sobolev inequalities, mean value inequality, Harnack inequalities, and gradient estimates.
result Global gradient estimates for positive harmonic functions on Finsler measure spaces.
The paper proves a Harnack inequality for heat equations on Finsler metric measure manifolds.
problem Proving a Harnack inequality for positive solutions to heat equations on Finsler metric measure manifolds.
method Volume comparison theorem, weighted Poincaré inequality, local uniform Sobolev inequality, mean value inequalities.
result Derives a Harnack inequality for positive solutions to heat equations.
Extends Riemannian geometry inequalities with sharper estimates.
problem Deriving new inequalities on Riemannian manifolds.
method Investigates advanced Hardy and Rellich-type inequalities on complete noncompact manifolds with weight functions.
result Provides sharper estimates conforming to the geometry and structure of the manifold.
The paper explores how information geometry impacts classical CR inequalities.
problem Deriving and generalizing CR inequalities using information geometry.
method Examining Eguchi's theory and applying Amari-Nagoaka's theory to KL-divergence, and then extending to other divergences.
result Generalized CR inequalities derived from various divergences.