The study improves bounds on space waists using advanced geometry techniques.
problem Understanding the waist of different geometric spaces.
method Advanced geometric techniques, including Borsuk-Ulam-Crofton method, Hausdorff measure.
result Established new waist bounds for various spaces.
Study incompressible surfaces to find cable knots' representativity and waist.
problem Computing the representativity and waist of cable knots.
method Study incompressible surfaces in cable knots' exteriors.
result Compute representativity and waist for most cable knots.
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.
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 studies the waist size of cusps in hyperbolic 3-manifolds, proving unique smallest sizes for specific manifolds.
problem Determining the smallest waist sizes of cusps in hyperbolic 3-manifolds.
method Analyzing the shortest nontrivial curves generated by parabolic isometries in maximal cusp boundaries.
result The next two smallest waist sizes are realized uniquely for specific manifolds.
The paper proves waist inequalities for convex bodies and their linear images.
problem Understanding geometric characteristics of convex bodies through waist inequalities.
method Connections between Gromov's and Milman's work, proving waist inequalities for convex bodies and their linear images.
result Any convex body has a linear image satisfying a waist inequality with a universal constant.
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.
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.
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 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.
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.
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…
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.
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.
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)
Tune simplifies hyperparameter search for distributed computing.
problem Adapting hyperparameter search to distributed environments.
method Unified framework for model selection and training.
result Simplifies implementation of state-of-the-art search algorithms.
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.
The study finds infinitely many periodic orbits just above a critical value on a 2-sphere.
problem Finding periodic orbits just above a critical value on a 2-sphere.
method Introduced a new critical value c∞(L) and showed its strict inequality to the Mañé critical value c(L), proving the existence of infinitely many periodic orbits on energy levels e∈(c(L),c∞(L)). result Infinitely many periodic orbits exist on energy levels just above the Mañé critical value.
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. 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.
Paper compares ML models for a wall-following robot, achieving high accuracy.
problem Improving prediction accuracy of a wall-following robot's direction.
method Trained various machine learning models on a dataset of ultrasound sensor readings.
result Presented machine learning models with higher accuracy than previous work.