Existence proved for static vacuum extensions near Schwarzschild spheres.
problem Proving existence of static vacuum extensions near Schwarzschild spheres.
method Existence and local uniqueness of static vacuum extensions for Bartnik data on a sphere near a Schwarzschild sphere.
result Existence of static vacuum extensions near Schwarzschild spheres.
Spherical T-duality for iterated sphere bundles
problem T-duality for iterated sphere bundles
method Repackaging cohomological data into Massey products
result Found T-dual iterated sphere bundles associated to Massey products
Study spherical T-duality and Massey products in iterated sphere bundles.
problem Understanding spherical T-duality and Massey products in iterated sphere bundles.
method Analyzing Gysin sequences and Massey products to find T-dual iterated sphere bundles.
result For certain iterated sphere bundles, spherical T-duality can be represented by Massey products.
Reconstructing Finsler manifolds from sphere data.
problem Recovering a Finsler manifold from sphere data.
method Solving the geometrical inverse problem locally along geodesics.
result Local reconstruction of Finsler manifolds.
Prove realizability of genus-0 branch data for tetrahedral coverings of the sphere.
problem Prove realizability of genus-0 branch data for tetrahedral coverings of the sphere.
method Use an explicit combinatorial description of coverings via dessins d'enfants.
result Prove realizability for a broader class of branch data with more critical values.
Estimates mass of static vacuum metrics with small Bartnik data.
problem Estimating mass of static vacuum metrics with small perturbations.
method Second-order mass estimation using Bartnik data.
result New upper bound on Bartnik mass to fifth order.
The Weierstrass representation for spheres in R3 and, in particular, effective construction of immersions from data of spectral theory origin is discussed. These data are related to Dirac operators on a plane and on an infinite cylinder and these operators are just representations of Dirac operators acting in spino…
The Willmore energy for Frenet curves in quaternionic projective space is the generalization of the Willmore functional for immersions into the 4-sphere. Critical points of the Willmore energy are called Willmore curves in quaternionic projective space. Using a Baecklund transformation on Willmore curves, we generalize…
New criterion for branched covers between 2-spheres.
problem Existence of specific types of branched covers between 2-spheres.
method Combining complex analysis and topology techniques.
result Established a complete criterion for the existence of branched covers.
New method explains high-dimensional sphere data with latent factors.
problem Understanding intricate dependence structure in high-dimensional sphere data.
method Exploratory factor analysis of the projected normal distribution with a fast alternating expectation profile conditional maximization algorithm.
result Uniformly excellent results on various data types, including tweets, brain imaging, and cancer gene expression.
New findings about twists in 4-sphere diffeomorphisms.
problem Understanding isotopy classes of diffeomorphisms in 4-sphere.
method Using Cerf theory and twists along Montesinos twins.
result The subgroup of twists along Montesinos twins is trivial or cyclic of order two.
The study classifies tilings of the sphere by congruent quadrilaterals.
problem Classifying edge-to-edge tilings of the sphere by congruent quadrilaterals.
method Classification of tilings into three classes based on geometric data and parameters.
result Three classes of tilings are identified: 2-layer earth map tilings, quadrilateral subdivisions of the octahedron, and 3-layer earth map tilings.
New discrete cmc surfaces defined from sphere packings and combinatorics.
problem Creating constant mean curvature surfaces from discrete data.
method Discrete cmc surfaces defined via sphere packings and combinatorial patterns.
result Construction of discrete cmc surfaces from orthogonal ring patterns.
New instanton invariants for rational homology spheres defined and shown to be functorial.
problem Defining and proving invariance of instanton homology groups for rational homology spheres.
method Novel suspended flow category technique to handle obstructed cobordisms and prove wall-crossing formula.
result Instanton invariant λI(Y) conjecturally equals Casson-Walker invariant for rational homology spheres. Motivated by the foliation by stable spheres with constant mean curvature constructed by Huisken-Yau, Metzger proved that every initial data set can be foliated by spheres with constant expansion (CE) if the manifold is asymptotically equal to the standard [t=0]-timeslice of the Schwarzschild solution. In this paper, w…
A new method estimates the number of clusters on spherical data.
problem Estimating the number of clusters in spherical data.
method Spherical X-means (SX-means) method assuming von Mises-Fisher distributions.
result Shows the performance of SX-means in estimating the number of clusters.
We prove that the mod Z reduction of the torsion of a rational homology 3-sphere is completely determined by three data: a certain canonical spin^c structure, the linking form and a Q/Z-valued constant c. This constant is a new topological invariant of the rational homology sphere. Experimentations with lens spaces sug…
Our team won the second prize of the Safe Aging with SPHERE Challenge organized by SPHERE, in conjunction with ECML-PKDD and Driven Data. The goal of the competition was to recognize activities performed by humans, using sensor data. This paper presents our solution. It is based on a rich pre-processing and state of th…
New approach for principal curves on spherical data.
problem Dimension reduction of spherical data.
method Projection of data onto a continuous curve on a sphere.
result Stationary principal curves on a sphere.
Study proves inequality linking black hole properties and angular momentum.
problem Establishing a Penrose-type inequality for black holes with 3-sphere horizons.
method Analyzing biaxially symmetric, maximal, asymptotically flat initial data sets for the Einstein equations.
result Equality holds only for stationary Myers-Perry black holes.
Study of Randers metrics on spheres with simple cut loci.
problem Understanding Randers metrics on spheres and their cut loci.
method Analyzing geodesics, conjugate, and cut loci of Finsler metrics of Randers type.
result Found new families of Randers metrics with simple cut loci.
In a matter-filled spacetime, perhaps with positive cosmological constant, a stable marginally outer trapped 2-sphere must satisfy a certain area inequality. Namely, as discussed in the paper, its area must be bounded above by 4π/c, where c>0 is a lower bound on a natural energy-momentum term. We then consider th…
Estimates Manolescu's κ-invariant using spin 4-orbifolds.
problem Estimating κ-invariant of rational homology 3-spheres.
method Using spin 4-orbifolds bounded by rational homology 3-spheres.
result Restricts and determines the value of κ for specific cases.
Fast and efficient homology algorithms are in demand in the applied sciences for analyzing solid materials and proteins, processing digital imaging data, or pattern classification among others. Recent advances employ discrete Morse theory as a preprocessor. Research in this area has lead to the need to find complicated…
We propose and evaluate alternative ensemble schemes for a new instance based learning classifier, the Randomised Sphere Cover (RSC) classifier. RSC fuses instances into spheres, then bases classification on distance to spheres rather than distance to instances. The randomised nature of RSC makes it ideal for use in en…
We solve Bartnik's stationary extension problem near Schwarzschild spheres.
problem Existence and uniqueness of asymptotically flat stationary vacuum spacetimes.
method Developed a double geodesic gauge, reducing equations to elliptic and transport-type problems.
result Local well-posedness for Bartnik stationary metric extension problem near Schwarzschild spheres.
Improved weather forecasting using deep CNN on cubed-sphere grid.
problem Global weather prediction accuracy and speed.
method Deep convolutional neural network (CNN) on cubed-sphere grid, offline mapping, loss minimization.
result Significantly improved weather forecasts, indefinitely stable, realistic patterns at long lead times.
We give a local representation for the pseudoholomorphic surfaces in Euclidean spheres in terms of holomorphic data. Similar to the case of the generalized Weierstrass representation of Hoffman and Osserman, we assign such a surface in $\Sf^{2n}$ to a given set of n holomorphic functions defined on a simply-connected…
In this note, we compute the limit of the Wang-Yau quasi-local mass on unit spheres at spatial infinity of an asymptotically flat initial data set. Similar to the small sphere limit of the Wang-Yau quasi-local mass, we prove that the leading order term of the quasi-local mass recovers the stress-energy tensor. For a va…
Paper solves a conjecture about minimal surfaces using sphere intersections and Weierstrass data.
problem Solving the Fraser-Li conjecture for minimal surfaces.
method Using the Weierstrass representation formula and sphere intersections.
result The conjecture can be translated into problems about the Gauss map.
New research finds 145 infinite families of CS spheres are standard.
problem Determining which Cappell-Shaneson spheres are diffeomorphic to the standard 4-sphere.
method Using Kirby calculus and new families of CS spheres.
result Proves 145 new infinite families of CS spheres are standard.
Given a sphere with Bartnik data close to that of a round sphere in Euclidean 3-space, we compute its Bartnik-Bray outer mass to first order in the data's deviation from the standard sphere. The Hawking mass gives a well-known lower bound, and an upper bound is obtained by estimating the mass of a static vacuum extensi…
Method estimates densities on manifolds using dequantization.
problem Estimating densities on non-Euclidean manifolds.
method Inspired by dequantization, coordinate transformation, and normalizing flows.
result Successfully models densities on spheres, tori, and orthogonal groups.
A new spherical Sliced-Wasserstein distance for data on spheres.
problem Defining Wasserstein distance on manifolds, especially spheres.
method Closed-form solutions of the Wasserstein distance on the circle and a new spherical Radon transform.
result A novel spherical Sliced-Wasserstein (SW) discrepancy for data on spheres.
The paper explores how data geometry influences generalization in neural networks.
problem Understanding generalization in overparameterized neural networks.
method Theoretical exploration of overparametrized two-layer ReLU networks trained below the edge of stability.
result Generalization bounds adapt to the intrinsic dimension of data distributions and deteriorate as data concentrates towards the unit sphere.
A conformal metric on a 4-ball induces on the boundary 3-sphere a conformal metric and a trace-free second fundamental form. Conversely, such a data on the 3-sphere is the boundary of a unique selfdual conformal metric, defined in a neighborhood of the sphere. In this paper we characterize the conformal metrics and tra…
We study a functional on the boundary of a compact Riemannian 3-manifold of nonnegative scalar curvature. The functional arises as the second variation of the Wang-Yau quasi-local energy in general relativity. We prove that the functional is positive definite on large coordinate spheres, and more general on nearly roun…
Kervaire's sphere-link is equivalent to a ribbon sphere-link, simplifying complex 2-complexes.
problem Understanding the structure of 2-complexes and their asphericity.
method Using Kervaire's sphere-link and ribbon sphere-link equivalence, analyzing the compact complement of ribbon disk-links.
result Every connected subcomplex of a contractible finite 2-complex is aspherical.
New theory proves infinite homology 3-spheres in homology 4-spheres.
problem Existence of homology 3-spheres in homology 4-spheres.
method Diagrammatics of surface cross sections, Taubes' work.
result Infinite number of homology 3-spheres in homology 4-spheres.
Infinitely many splitting spheres found for unlinked 2-spheres in 4-space.
problem Existence of pairwise non-isotopic splitting spheres for unlinked 2-spheres in 4-space.
method Analytical proof showing non-isotopic spheres.
result Infinitely many non-isotopic splitting spheres found.
Reduces weak reducing pairs to spheres in 3-sphere Heegaard surfaces.
problem Finding reducing spheres for weak reducing pairs in Heegaard surfaces.
method Proves existence of reducing spheres for weak reducing pairs in 3-sphere Heegaard surfaces.
result Reduction of weak reducing pairs to spheres if genus is at most 3.
The study shows how to construct d-spheres from (d−1)-spheres and d-balls without additional vertices.
problem Constructing d-spheres from (d−1)-spheres and d-balls without additional vertices. method Examining specific types of spheres (flag, stacked, join of spheres) and d-balls to determine if constructions can be made without extra vertices. result Affirmative answers to constructing d-spheres from (d−1)-spheres and d-balls without additional vertices for certain types of spheres and d-balls. New proof for sphere recognition algorithm.
problem Sphere recognition algorithm proof.
method New proof of a lemma in Abigail Thompson's algorithm.
result New proof of a lemma in Abigail Thompson's proof of the Recognition Algorithm for 3-spheres.
We give examples of asymptotically flat three-manifolds (M,g) which admit arbitrarily large constant mean curvature spheres that are far away from the center of the manifold. This resolves a question raised by G. Huisken and S.-T. Yau in 1996. On the other hand, we show that such surfaces cannot exist when (M,g) ha…
Proves stability of convex spheres with similar geodesic lengths.
problem Stability of convex spheres with specific geodesic properties.
method Proves C^0 Cheeger-Gromov closeness to the round sphere.
result Strictly convex 2-spheres are close to the round sphere.
For a given branched covering between closed connected surfaces, there are several easy relations one can establish between the Euler characteristics of the surfaces, their orientability, the total degree, and the local degrees at the branching points, including the classical Riemann-Hurwitz formula. These necessary re…
Soliton spheres are immersed 2-spheres in the conformal 4-sphere S^4=HP^1 that allow rational, conformal parametrizations f:CP^1->HP^1 obtained via twistor projection and dualization from rational curves in CP^{2n+1}. Soliton spheres can be characterized as the case of equality in the quaternionic Pluecker estimate. A …
The paper explores rational functions with 3 branching points on the Riemann sphere.
problem Existence of rational functions with specific branching points.
method Utilizes complex analysis to establish properties of rational functions.
result Identifies new types of exceptional branching data.