The hyperbolic dodecahedral space is cubulated and covered with special properties.
problem Understanding the cubulation and covering of hyperbolic dodecahedral space.
method Subdividing the dodecahedron into cubes and describing various covers.
result The hyperbolic dodecahedral space has special 60-sheeted covers with a cubulation.
Researchers prove Seifert-Weber dodecahedral space is an L-space.
problem Proving the Seifert-Weber dodecahedral space is an L-space.
method Relating Floer homology and spectral geometry of hyperbolic three-manifolds, exploiting symmetry and arithmetic structure.
result Small eigenvalues on coexact 1-forms must have large multiplicity in Seifert-Weber dodecahedral space.
6 out of 29 dodecahedral 3-spheres are L-spaces, solving a Seiberg-Witten question.
problem Identifying L-spaces among dodecahedral 3-spheres.
method L-space census algorithm and hyperbolic geometry analysis.
result Existence of hyperbolic 4-manifolds with separating L-spaces.
In this article we examine the conjecture of Neumann and Reid that the only hyperbolic knots in the 3-sphere which admit hidden symmetries are the figure-eight knot and the two dodecahedral knots. Knots whose complements cover hyperbolic reflection orbifolds admit hidden symmetries, and we verify the Neumann-Reid con…
We prove that an irreducible 3-manifold whose fundamental group satisfies a certain group-theoretic property called RFRS is virtually fibered. As a corollary, we show that 3-dimensional reflection orbifolds and arithmetic hyperbolic orbifolds defined by a quadratic form virtually fiber. These include the Seifert Weber …
In this paper we settle Thurston's old question of whether the Weber-Seifert dodecahedral space is non-Haken, a problem that has been a benchmark for progress in computational 3-manifold topology over recent decades. We resolve this question by combining recent significant advances in normal surface enumeration, new he…
Observational data hints at a finite universe, with spherical manifolds such as the Poincare dodecahedral space tentatively providing the best fit. Simulating the physics of a model universe requires knowing the eigenmodes of the Laplace operator on the space. The present article provides explicit polynomial eigenmodes…
Minimal Morse functions on Poincaré dodecahedral space are selected via spectral properties.
problem Identifying minimal Morse functions on the Poincaré dodecahedral space.
method Spectral selection property P, obstruction principle, conformal variations, finite dimensional reduction.
result Restoration of minimal Morse selection on the Poincaré dodecahedral space via spectral mechanisms.
Study Froyshov invariants and closed geodesics in hyperbolic 3-manifolds.
problem Understanding Froyshov invariants in hyperbolic 3-manifolds.
method Analytic number theory and hyperbolic geometry.
result Effective upper bounds for Froyshov invariants and algorithm to compute them.
In a flat space, the global topology of comoving space can induce a weak acceleration effect similar to dark energy. Does a similar effect occur in the case of the Poincare dodecahedral space S^3/I^*? Does the effect distinguish the Poincare space from other well-proportioned spaces? The residual acceleration effect in…
We show that for certain arithmetic groups, geometrically finite subgroups are the intersection of finite index subgroups containing them. Examples are the Bianchi groups and the Seifert-Weber dodecahedral space. In particular, for manifolds commensurable with these groups, immersed incompressible surfaces lift to embe…
The only finite nonabelian simple group acting on a homology 3-sphere - necessarily non-freely - is the dodecahedral group A5≅PSL(2,5) (in analogy, the only finite perfect group acting freely on a homology 3-sphere is the binary dodecahedral group A5∗≅SL(2,5)). In the present pa…
Machine learning speeds up the construction of virus assembly fitness landscapes.
problem Constructing realistic evolutionary fitness landscapes for viruses is computationally expensive.
method Developed a neural network to model virus assembly efficiency from a whole genome/phenotype space.
result Machine learning significantly reduces the computational time for constructing fitness landscapes.
The study finds infinitely many counterexamples to a generalized Double Soul Conjecture.
problem The Double Soul Conjecture for non-simply connected manifolds.
method Analysis of double disk bundles and counterexamples.
result Infinitely many counterexamples to the generalized Double Soul Conjecture.
If a closed smooth n-manifold M admits a finite cover whose Z/2Z-cohomology has the maximal cup-length, then for any riemannian metric g on M, we show that the systole Sys(M,g) and the volume Vol(M,g) of the riemannian manifold (M,g) are related by the following isosystolic inequality: Sys(M,g)^n \leq n! Vol(M,g). The …
We consider orientation-preserving actions of a finite group G on the 3-sphere S^3 (and also on Euclidean space R^3). By the geometrization of finite group actions on 3-manifolds, if such an action is smooth then it is conjugate to an orthogonal action, and in particular G is isomorphic to a subgroup of the orthogonal …
We present a new, practical algorithm to test whether a knot complement contains a closed essential surface. This property has important theoretical and algorithmic consequences; however, systematically testing it has until now been infeasibly slow, and current techniques only apply to specific families of knots. As a …
Computer program constrains foam evaluation dimensions.
problem Determining dimensions of foam evaluation for nonreducible webs.
method Developed a computer program to analyze the dimension and graded dimension of J♭(K) for webs K. result Suggests dimJ♭(W1)=58 for the dodecahedral web W1 with Tait(W1)=60. Paper finds inequalities for convex domains in hyperbolic space.
problem Finding inequalities for convex domains in hyperbolic space.
method Introducing hyperbolic ellipsoids and using orthogonal projection to establish inequalities.
result Affine isoperimetric inequalities for static convex domains in hyperbolic space characterized by hyperbolic ellipsoids.
Introduces Möbius structures and hyperbolic ends for k-surfaces in hyperbolic space.
problem Understanding k-surfaces in hyperbolic space. method Studies Möbius structures and hyperbolic ends.
result Applications to k-surfaces in hyperbolic space. New criteria for relative hyperbolicity in hierarchically hyperbolic spaces.
problem Characterizing relative hyperbolicity in hierarchically hyperbolic spaces.
method New formulation of relative hyperbolicity in terms of hierarchy structures, applied to graphs associated to surfaces.
result The separating curve graph of a surface is relatively hyperbolic when the surface has zero or two punctures.
We show the equivalence of several characterizations of relative hyperbolicity for metric spaces, and obtain extra information about geodesics in a relatively hyperbolic space. We apply this to characterize hyperbolically embedded subgroups in terms of nice actions on (relatively) hyperbolic spaces. We also study the d…
Classifies minimal submanifolds in complex hyperbolic spaces.
problem Identifying minimal submanifolds in complex hyperbolic spaces.
method Classification based on extrinsic homogeneity.
result Classification of minimal extrinsically homogeneous submanifolds.
Survey solves curvature problems with hyperbolic spaces.
problem Singularities in hypersurface geometry.
method Hyperbolic unfolding correspondence linking hypersurfaces to Gromov hyperbolic spaces.
result Eliminates hypersurface singularities in scalar curvature geometry.
New formulas for hyperbolic mass using horospheres.
problem Mass calculation of asymptotically hyperbolic manifolds.
method Geometric formulas derived using coordinate horospheres.
result Improved rigidity results of hyperbolic space.
The article proves inequalities for capillary hypersurfaces in hyperbolic space.
problem Proving inequalities for capillary hypersurfaces in hyperbolic space.
method Constructing a new locally constrained inverse curvature flow.
result Obtained Alexandrov-Fenchel inequalities for convex capillary hypersurfaces in hyperbolic space.
Hyperbolic SVM improves classification in complex networks.
problem Accurately classifying points in hyperbolic space with hierarchical relationships.
method Introducing hyperbolic SVM, a hyperbolic formulation of SVM classifiers.
result Hyperbolic SVM outperforms Euclidean SVM in multi-class prediction tasks.
Introduces a new metric to measure deviation from hyperbolicity.
problem Measuring how much a metric space deviates from being hyperbolic.
method Defining the quasi-hyperbolicity constant and analyzing its properties.
result The quasi-hyperbolicity constant provides a measure of deviation from hyperbolicity.
Proves heat kernel superconvexity in hyperbolic space.
problem Heat kernel superconvexity in hyperbolic space.
method Proves conjecture by Bernstein in all dimensions.
result Analog of Huisken's monotonicity formula for mean curvature flow.
New condition prevents hyperbolic spaces from matching curve complexes.
problem Identifying when hyperbolic spaces cannot match curve complexes.
method Analyzing specific hyperbolic complexes and identifying a condition.
result Identified a condition preventing quasi-isometry between hyperbolic spaces and curve complexes.
Characterizes specific harmonic manifolds using Laplacian eigenfunctions.
problem Characterizing harmonic manifolds.
method Using radial eigenfunctions of the Laplacian.
result Characterizes space forms, complex hyperbolic spaces, and quaternionic hyperbolic spaces as harmonic manifolds.
Geometric inequalities for static convex domains in hyperbolic space proved.
problem Proving geometric inequalities for static convex domains in hyperbolic space.
method Using static convexity of flow hypersurfaces, new inequalities are derived.
result New family of geometric inequalities for static convex domains in hyperbolic space.
Global existence of Yamabe flows on hyperbolic space proved without curvature bounds.
problem Global existence of Yamabe flows on hyperbolic space without completeness or curvature bounds.
method Instantaneously complete initial metrics, no curvature bounds required.
result Global existence of Yamabe flows on hyperbolic space of arbitrary dimension m≥3. Paper introduces a hyperbolic Gaussian distribution for better learning in hierarchical data.
problem Learning hierarchical data in hyperbolic space.
method Developed a novel hyperbolic distribution for gradient-based learning.
result Demonstrated improved learning on various datasets.
Study on relative entropy for hypersurfaces in hyperbolic space.
problem Understanding relative entropy for hypersurfaces in hyperbolic space.
method Relate relative entropy to renormalized area and apply monotonicity formula to mean curvature flows.
result Obtained a monotonicity formula for relative entropy in hyperbolic space.
Maps and embeddings between hyperbolic spaces and their boundaries studied.
problem Understanding relations between maps and embeddings between relatively hyperbolic spaces and their boundaries.
method Establishing correspondences between quasi-isometric embeddings and quasisymmetric embeddings, using polynomial distortion.
result Characterization of hyperbolic relative groups with polynomial distortion embeddings.
Unique median structures found in hyperbolic spaces.
problem Uniqueness of median structures in hyperbolic spaces.
method Analyzing product of hyperbolic spaces and properties of relative hyperbolicity.
result Non-hyperbolic pants graphs can have unique median structures.
Harmonic maps from hyperbolic planes to hyperbolic space exist with given boundary data.
problem Existence of harmonic maps from product of hyperbolic planes to hyperbolic space.
method Existence result for asymptotic Dirichlet problem.
result Existence of harmonic maps with given boundary data.
We introduce curvature-adapted foliations of complex hyperbolic space and study some of their properties. Generalized pseudo-Einstein hypersurfaces of complex hyperbolic space are classified. Analogous results for curvature-adapted hypersurfaces of quaternionic hyperbolic space are also obtained.
Introduces hyperbolic generalized framed surfaces and their properties.
problem None explicitly stated; focuses on introducing new geometric objects.
method Generalization of hyperbolic framed surfaces and curves.
result Established conditions for a surface to be a hyperbolic generalized framed base surface and explored their singularities.
The Willmore inequality is extended to star-shaped and mean-convex hypersurfaces in hyperbolic space.
problem Proving a geometric inequality for specific hypersurfaces in hyperbolic space.
method Inverse mean curvature flow to prove the inequality.
result The Willmore inequality is generalized to star-shaped and mean-convex hypersurfaces in hyperbolic space.
Geometric argument proves projection theorems in hyperbolic space.
problem Proving projection theorems for hyperbolic space.
method Geometric argument for orthogonal projections.
result Characterization of purely unrectifiable sets in hyperbolic space.
Classifies CR submanifolds in complex hyperbolic spaces.
problem Understanding CR submanifolds in complex hyperbolic spaces.
method Classifying orbits of a subgroup of the solvable part of the Iwasawa decomposition.
result Classification of homogeneous CR submanifolds.
Lectures on complex hyperbolic spaces and their groups.
problem Understanding interactions between complex hyperbolic spaces and discrete groups.
method Discussion of function theory and discrete group theory.
result Interactions between complex hyperbolic spaces and discrete groups.
Paper provides first theoretical guarantees for hyperbolic space learning.
problem Learning a classifier in hyperbolic space for hierarchical data.
method Efficient algorithm for large-margin hyperplane learning in hyperbolic space.
result The low embedding dimension in hyperbolic space leads to superior classifier learning guarantees.
Proof of mass positivity for hyperbolic space near Euclidean space.
problem Positivity of mass in asymptotically hyperbolic manifolds.
method Elementary proof generalizing Bartnik's Euclidean case.
result Positivity of hyperbolic mass near hyperbolic space.
Boundary rigidity defined for hyperbolic spaces, tied to geometric properties.
problem Understanding boundary rigidity in Gromov hyperbolic spaces.
method Analyzing properties of Gromov hyperbolic spaces and their boundaries.
result Boundary rigidity is equivalent to positive Cheeger isoperimetric constant and non-amenability.
Proves new inequality for hyperbolic space hypersurfaces.
problem Finding inequalities for hypersurfaces in hyperbolic space.
method Proves a Heintze-Karcher type inequality for shifted mean convex hypersurfaces.
result Proves Alexandrov type theorem and uniqueness result for hypersurfaces.