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. The paper proves stability of inequalities for nearly spherical sets in various spaces.
problem Stability of geometric inequalities for nearly spherical sets.
method Deriving a quantitative quermassintegral inequality and applying it to derive stability results.
result Stability of geometric inequalities involving weighted curvature integrals and quermassintegrals for nearly spherical sets in Rn+1 and Hn+1. The paper extends geometric inequalities for nearly spherical sets in various space forms.
problem Investigating weighted inequalities for nearly spherical sets in space forms.
method Generalizing and extending inequalities for nearly spherical sets in C1 and W2,∞ settings, with convex weight functions. result Quantitative stability estimates for weighted inequalities in Rn+1 and Hn+1. Quantifies nearly spherical subsets in complex ball geometry.
problem Isoperimetric inequality for nearly spherical domains in Bergman ball.
method Proves a quantitative isoperimetric inequality for nearly spherical subsets of Bergman ball.
result First result on isoperimetric phenomenon in Bergman ball.
New theorem shows nearly spherical manifolds can be mapped from spheres.
problem Generalizing Caffarelli's theorem to nearly spherical manifolds.
method Optimal transport map on the sphere, stability result.
result Every nearly spherical manifold can be mapped from a sphere.
The paper examines the stability of Minkowski inequality for nearly spherical domains.
problem Stability of Minkowski inequality for nearly spherical domains.
method Analyzes stability inequalities for C1 perturbations of a ball and axially symmetric perturbations. result Established stability inequalities for curvature integrals of nearly spherical domains.
Nearly spherical, positively curved surfaces are mapped from a sphere.
problem Mapping nearly spherical, positively curved surfaces from a sphere.
method Combines Ricci flow, Kim-Milman construction, and Bakry-Émery criterion.
result Every nearly spherical, positively curved surface is the contractive image of a round sphere.
We show non-collapsing for the evolution of nearly spherical closed convex curves in \mathbb{R}^2 under power curvature flow using two-point-methods.
For nearly spherical bodies, the unique center is proven under certain conditions.
problem Finding the unique center of nearly spherical bodies.
method Using regularized Riesz potential and asphericity measure.
result The $r^{\an}$-center is unique for sufficiently close bodies to a ball.
Paper proves stability of quermassintegral inequalities using inverse curvature flow.
problem Stability of quermassintegral inequalities for nearly spherical sets.
method Inverse curvature flow with special rescaling to study quermassintegral inequalities.
result Decreasing rate of k-th quermassintegral is faster than Fraenkel asymmetry for nearly spherical sets.
Gradient Descent with Projection learns low-degree polynomials efficiently.
problem Learning low-degree spherical polynomials with neural networks.
method Over-parameterized two-layer neural network with Gradient Descent with Projection.
result Achieves nearly minimax optimal sample complexity and risk bound.
Proves stability of cone-volume measure with nearly constant density.
problem Stability of cone-volume measure with near constant density.
method Proves stability of cone-volume measure with near constant density.
result Homothetic copy of the body is close to the unit ball in the L2-distance. We present a simple, general technique for reducing the sample complexity of matrix and tensor decomposition algorithms applied to distributions. We use the technique to give a polynomial-time algorithm for standard ICA with sample complexity nearly linear in the dimension, thereby improving substantially on previous b…
The c-curvature of a complete surface with Gauss curvature close to 1 in C2 norm is almost-positive (in the sense of Kim--McCann). Our proof goes by a careful case by case analysis combined with perturbation arguments from the constant curvature case, keeping track of an estimate on the closeness curvature conditi…
The study finds regular null hypersurfaces in a perturbed Schwarzschild black hole exterior.
problem Existence of regular null hypersurfaces in a perturbed Schwarzschild black hole.
method Proof of existence for null hypersurfaces in a perturbed Schwarzschild spacetime.
result Existence of many foliations by regular null hypersurfaces in the exterior region of a perturbed Schwarzschild black hole.
We assume data sampled from a mixture of d-dimensional linear subspaces with spherically symmetric distributions within each subspace and an additional outlier component with spherically symmetric distribution within the ambient space (for simplicity we may assume that all distributions are uniform on their correspondi…
We consider a variational problem for submanifolds Q ⊂ M with nonempty boundary ∂Q = K. We propose the definition that the boundary K of any critical point Q have constant mean curvature, which seems to be a new perspective when dim Q \textless{} dim M . We then construct small nearly-spherical solutio…
We give a construction of G2 and Spin(7) instantons on exceptional holonomy manifolds constructed by Bryant and Salamon, by using an ansatz of spherical symmetry coming from the manifolds being the total spaces of rank-4 vector bundles. In the G2 case, we show that, in the asymptotically conical model, the conn…
Paper studies generic dynamics of MCFs with spherical singularities.
problem Characterizing the generic behavior of mean curvature flow with spherical singularities.
method Level set formulation of mean curvature flow, analysis of arrival time function.
result Generically, the arrival time function has at most C2 regularity. New algorithms learn multi-index models via harmonic analysis, achieving statistical and computational trade-offs.
problem Learning multi-index models with unknown projections of input data.
method Exploiting the equivariance of the problem under the orthogonal group, we derive lower bounds and construct spectral algorithms based on harmonic tensor unfolding.
result Achieve statistical and computational trade-offs between sample and runtime complexity.
The reductivity of a spherical curve is the minimal number of a local transformation called an inverse-half-twisted splice required to obtain a reducible spherical curve from the spherical curve. It is unknown if there exists a spherical curve whose reductivity is four. In this paper, an unavoidable set of configuratio…
The reductivity of a spherical curve represents how reduced the spherical curve is. It is unknown if there exists a spherical curve whose reductivity is four. In this paper we give an unavoidable set for spherical curves with reductivity four by considering 4-gons.
We show that we can obtain a reducible spherical curve from any non-trivial spherical curve by four or less inverse-half-twisted splices, i.e., the reductivity, which represents how reduced a spherical curve is, is four or less. We also discuss unavoidable sets of tangles for spherical curves.
Convexity properties are preserved under radial transformations in hyperbolic and spherical geometries.
problem Preserving convexity in hyperbolic and spherical geometries under radial transformations.
method Used Poincaré disk model for hyperbolic geometry and stereographic projection for spherical geometry to prove preservation of convexity under radial expansion and contraction.
result Radial expansion and contraction preserve hyperbolic and spherical convexity, respectively.
Global existence and geometry of constant mass aspect function foliation in perturbed Schwarzschild spacetime studied.
problem Null Penrose inequality on a null hypersurface.
method Global existence of constant mass aspect function foliation on a nearly spherically symmetric incoming null hypersurface in a vacuum perturbed Schwarzschild spacetime.
result Geometry of the constant mass aspect function foliation compared to the spherically symmetric foliation in the Schwarzschild spacetime.
The abstract proves spherical surface decompositions with conical singularities.
problem Decomposing surfaces with spherical metrics and conical singularities.
method Geometric triangulations and irreducible components of standard shapes.
result Spherical polygons, including half-spherical concave polygons, can be arbitrarily complicated.
Study integrability of generalized almost complex structures on S^6.
problem Integrability of generalized almost complex structures on the 6-dimensional sphere.
method Local coordinate criteria for integrability with respect to brackets and Courant integrability for strong structures.
result No nontrivial spherical combinations of the canonical structures are integrable with respect to the Levi-Civita connection.
Study on evolutes and focal surfaces of pseudo-spherical framed immersions in anti-de Sitter space.
problem Investigating singularities of evolutes and focal surfaces of pseudo-spherical framed immersions.
method Introduced pseudo-spherical non-null framed curves, defined moving frames, and analyzed evolutes and focal surfaces.
result Evolutes of pseudo-spherical framed immersions are the sets of singular points of their focal surfaces.
We explore the limit set of a particular spherical CR uniformization of a cusped hyperbolic manifold. We prove that the limit set is the closure of a countable union of R-circles, is connected, and contains a Hopf link with three components; we also show that the fundamental group of its complement in S3 …
Study proposes initial data sets for solving gravitational equations, proving energy estimates.
problem Solving the constraint equations in the evolutionary form.
method Proposes a family of initial data sets, proving Penrose-like energy estimates.
result Established existence of solutions for specific cases.
The paper defines plat closures for spherical braids and shows links in RP3 can be realized this way.
problem Defining and analyzing plat closures for spherical braids in RP3. method Defining plat closures, associating residual permutations, and presenting moves on spherical braids.
result The number of components of the plat closure link of a spherical braid is equal to the number of disjoint cycles in its residual permutation.
We study high-dimensional distribution learning in an agnostic setting where an adversary is allowed to arbitrarily corrupt an ε-fraction of the samples. Such questions have a rich history spanning statistics, machine learning and theoretical computer science. Even in the most basic settings, the only known…
A new overlapping space solves the configuration search problem for graph embeddings.
problem Configuring product spaces for graph embeddings is resource-intensive and impractical.
method Introducing overlapping spaces that share subsets of coordinates between different types of spaces (Euclidean, hyperbolic, spherical).
result Overlapping spaces achieve nearly optimal results without configuration tuning, reducing training time.
Study inequalities involving torsional rigidity and spherical deficit on Riemannian manifolds.
problem Inequalities involving torsional rigidity and spherical deficit on Riemannian manifolds.
method Establish inequalities and derive an integral identity for a Dirichlet problem.
result Characterize metric balls and measure spherical deficit on Riemannian manifolds.
The paper proves and analyzes Minkowski inequalities for nearly spherical domains.
problem Validating and stabilizing Minkowski inequalities for perturbed balls.
method Analyzing C1-perturbations of the ball, proving sharp and almost sharp inequalities. result Sharp geometric and almost sharp Minkowski inequalities for nearly spherical domains.
In early 1930s Seifert and Threlfall classified up to conjugacy the finite subgroups of SO(4), this gives an algebraic classification of orientable spherical 3-orbifolds. For the most part, spherical 3-orbifolds are Seifert fibered. The underlying topological space and singular set of non-fibered spherical 3…
Study of marginally trapped surfaces in a perturbed Schwarzschild spacetime.
problem Understanding marginally trapped surfaces in perturbed Schwarzschild spacetime.
method Developed a method to study spacelike surfaces in a double null coordinate system.
result For every incoming null hypersurface nearly spherically symmetric, there exists a unique embedded marginally trapped surface.
Study shows spherical embedding and immersion components are related to homotopy groups.
problem Understanding the connected components of spherical embeddings and immersions.
method Analyzing the spaces of spherical embeddings and immersions modulo immersions, and relating them to homotopy groups.
result The set of connected components of spherical embeddings and immersions modulo immersions is isomorphic to π_{n+1}(SG,SG_q).
Lower bounds show learning mixtures of linear classifiers is nearly impossible.
problem Learning mixtures of linear classifiers under Gaussian covariates.
method Statistical Query (SQ) lower bounds and new spherical designs.
result Complexity of any SQ algorithm is \( n^{\mathrm{poly}(1/Δ) \log(r)} \), where Δ is the pairwise \(\ell_2\)-separation.
Developed a diffusion model on spherical data, addressing geometric and stochastic challenges.
problem Diffusion models on spherical data face unique geometric and stochastic issues.
method Extended spectral diffusion to spherical harmonics, introducing modified stochastic differential equations.
result Introduced a geometry-dependent inductive bias in spectral diffusion models.
In an ordinary feature selection procedure, a set of important features is obtained by solving an optimization problem such as the Lasso regression problem, and we expect that the obtained features explain the data well. In this study, instead of the single optimal solution, we consider finding a set of diverse yet nea…
The paper compares spectral geometry in hyperbolic and spherical manifolds.
problem Understanding spectral geometry in spherical manifolds.
method Survey of known results and open problems.
result Analogous results hold in hyperbolic manifolds but not necessarily in spherical manifolds.
Unique metric found for discrete curvature on spherical cone-metrics.
problem Finding a unique metric with prescribed curvature on spherical cone-metrics.
method Discrete conformal approach to spherical cone-metrics.
result Existence of a unique metric realizing prescribed curvature in each conformal class.
A spherical set is called convex if for every pair of its points there is at least one minimal geodesic segment that joins these points and lies in the set. We prove that for n >= 3 a complete locally-convex (topological) immersion of a connected (n-1)-manifold into the n-sphere is a surjection onto the boundary of a c…
In this paper, we study the problem of learning a mixture of Gaussians with streaming data: given a stream of N points in d dimensions generated by an unknown mixture of k spherical Gaussians, the goal is to estimate the model parameters using a single pass over the data stream. We analyze a streaming version of …
New homogeneous special Lagrangian submanifolds discovered in nearly Kähler CP3.
problem Exploring special Lagrangian submanifolds in nearly Kähler CP3.
method Intrinsically and extrinsically using moving frame and moment-type maps.
result Classification of totally geodesic special Lagrangian submanifolds and homogeneity of special Lagrangians.
Study on spherical conical metrics and their reducibility on compact Riemann surfaces.
problem Existence and geometric structure of reducible spherical conical metrics.
method Analysis of monodromy groups and geometric cutting of surfaces.
result Existence of reducible spherical conical metrics with saddle points on the same geodesic.
Paper explores neural network approximations on sphere domains.
problem Approximating functionals on sphere domains using neural networks.
method Encoder-decoder framework with spherical harmonics for infinite-dimensional domain.
result Approximation rates of neural networks with different encoder structures.