We give a short, simple and conceptual proof, based on spin structures, of sphere eversion: an embedded 2-sphere in can be turned inside out by regular homotopy. Ingredients of this eversion are seamlessly connected. We also give the mathematical origins of the proof: the Hopf fibration, and the topological struc…
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Thom announced a homological h-principle, which was later proven by connecting combinatorial techniques with Thurston's jiggling method.
problem The h-principle in immersion theory, particularly the sphere inside-out problem.
method Combining combinatorial techniques with Thurston's jiggling method.
result The announced homological h-principle was proven true.
Forward transition ratesmath.PR
New definition for forward rates in multi-state models.
problem Defining forward rates in multi-state models.
method Established a theoretical framework and provided a novel definition.
result Interchanged transition probabilities and intensities in Kolmogorov forward equations.
Paper introduces IO-NPF for efficient Bayesian experimental design.
problem Efficient Bayesian experimental design in non-exchangeable settings.
method Inside-Out Nested Particle Filter (IO-NPF) for non-Markovian state-space models.
result IO-NPF achieves computational complexity, improving efficiency.
Novel approach to Bayesian experimental design for non-exchangeable data.
problem Optimal experimental design for non-exchangeable data.
method Inside-Out SMC algorithm embedded in particle Markov chain Monte Carlo framework.
result Efficacy demonstrated on a set of dynamical systems.
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.
Smooth homotopy 4-spheremath.GT
Every smooth 4-sphere is the same as the standard one.
problem Identifying smooth 4-spheres.
method Proved diffeomorphism to the standard 4-sphere.
result Smooth homotopy 4-spheres are diffeomorphic to the 4-sphere.
Paper shows maps from m-sphere to n-sphere are trivial cobordant.
problem Understanding cobordism classes of maps and covers for spheres.
method Analyzes cobordism classes of maps from m-sphere to n-sphere.
result For m>n, cobordism classes of maps from m-sphere to n-sphere are trivial.
The study classifies branched Willmore spheres in 3- and 4-spheres.
problem Classifying branched Willmore spheres in 3- and 4-spheres.
method Analyzing variational branched Willmore spheres and their projections onto and .
result Improved regularity of unit normals and integer multiples of width in sphere eversion.
Ribbonness of Kervaire's sphere-link in homotopy 4-sphere and its consequences to 2-complexesmath.GT
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.
Proves a theorem connecting graph theory spheres, reformulating Morse conditions.
problem Defines and connects spheres in graph theory.
method Proves a theorem bridging two graph theory sphere definitions.
result Reformulates Morse conditions using center manifolds and level surface graphs.
The 3-sphere has either 2 minimal 2-spheres or an optimal foliation by 2-spheres.
problem Proving existence of minimal 2-spheres or optimal foliations in arbitrary Riemannian 3-spheres.
method Analyzing the properties of arbitrary Riemannian metrics on 3-spheres.
result The existence of at least two minimal 2-spheres or an optimal foliation in 3-spheres with arbitrary metrics.
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.
The study shows how to construct -spheres from -spheres and -balls without additional vertices.
problem Constructing -spheres from -spheres and -balls without additional vertices.
method Examining specific types of spheres (flag, stacked, join of spheres) and -balls to determine if constructions can be made without extra vertices.
result Affirmative answers to constructing -spheres from -spheres and -balls without additional vertices for certain types of spheres and -balls.
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.
Patterns and Tracksmath.GT
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.
Stability of Convex Spheresmath.DG
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.
Soliton Spheresmath.DG
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 …
Splitting spheres for -links in math.GT
The paper proves infinitely many non-isotopic splitting 3-spheres for split sphere links in 4D.
problem Proving the existence of infinitely many non-isotopic splitting 3-spheres for split sphere links in 4D.
method Establishing a general sufficient condition for infinitely many topologically non-isotopic splitting 3-spheres in connected sums of 4-manifolds.
result Proves infinitely many non-isotopic splitting 3-spheres for split sphere links in 4D.
Computer proof verifies key differential properties in sphere classifications.
problem Verifying holomorphic properties of meromorphic differentials in sphere classifications.
method Computer-assisted proof using Sage software.
result Holomorphicity of quartic and octic differentials confirmed.
Study on sphere immersions and their stability indices.
problem Analyzing the stability of sphere immersions.
method Calculation of Morse indices and stability indices for specific sphere immersions.
result Bounds on stability index of associative cone in .
Classification of constant curvature surfaces in Berger spheres.
problem Identifying complete rotationally invariant surfaces with constant Gauss curvature in Berger spheres.
method Complete classification through detailed analysis of Clifford tori and spheres.
result Rotationally invariant spheres with constant Gauss curvature are the only topological spheres in Berger spheres for .
The paper constructs biharmonic maps between spheres using polynomial maps.
problem Creating biharmonic maps between spheres.
method Using harmonic homogeneous polynomial maps of different degrees to generate proper biharmonic maps.
result Established a method for constructing proper biharmonic product maps.
Reduces connectivity problem for genus-4 Heegaard surface in 3-sphere.
problem Connectivity problem in reducing sphere complex for genus-4 Heegaard surface.
method Presented a sufficient condition for a non-separating weak reducing pair to be separated by a reducing sphere.
result Reduced connectivity problem to showing disjointness of representative reducing spheres from a fixed disk.
Characterizes a specific type of convex curves on a 3-sphere.
problem Understanding convex curves on a 3-sphere.
method Decomposes curves on 3-sphere into 2-sphere curves, characterizes locally convex ones.
result Completely characterized a class of convex curves on the 3-sphere.
Paper studies inscribed sphere and lines through centers of Apollonius spheres in n dimensions.
problem Tangency of spheres and lines through their centers.
method Lie sphere geometry and two-step construction of Apollonius spheres.
result Center of inscribed sphere coincides with point .
Author provides an alternate proof of the free ribbon lemma.
problem Proving that every free sphere-link in the 4-sphere is a ribbon sphere-link.
method An alternate proof of the free ribbon lemma.
result Provides an alternate proof of the free ribbon lemma.
Study confirms conjecture linking 4-spheres, finds new spheres.
problem Determine if all Cappell-Shaneson homotopy 4-spheres are standard.
method Algebraic number theory techniques, symmetry analysis.
result Gompf conjecture confirmed for trace and cases.
The study characterizes surface singularities in Lie sphere geometry.
problem Understanding singularities of surfaces in Lie sphere geometry.
method Analyzing conditions for cuspidal edges, swallowtails, and Lie sphere transformations.
result Conditions for various surface singularities in Lie sphere geometry.
Sharp convergence theorem for sphere submanifolds proved.
problem Sphere submanifolds in spheres.
method Proved a sharp convergence theorem.
result New differentiable sphere theorem for submanifolds in spheres.
In the 1-parameter family of Berger spheres S^3(a), a > 0 (S^3(1) is the round 3-sphere of radius 1) we classify the stable constant mean curvature spheres, showing that in some Berger spheres (a close to 0) there are unstable constant mean curvature spheres. Also, we classify the orientable compact stable constant mea…
We introduce and study a new class of homotopy spheres called Farrell-Jones spheres. Using Farrell-Jones sphere we construct examples of closed negatively curved manifolds , where or , which are homeomorphic but not diffeomorphic to complex hyperbolic manifolds, thereby giving a partial answer to a que…
Characterizes 3-punctured spheres in non-hyperbolic link exteriors.
problem Characterizing 3-punctured spheres in non-hyperbolic link exteriors.
method Analyzes non-hyperbolic 3-component links and multibranched surfaces.
result Existence of embeddings of multibranched surfaces in the 3-sphere.
The Pachner graph of 2-spheres is studied, focusing on subgraphs of flag and stacked 2-spheres.
problem Characterize subgraphs of the Pachner graph of 2-spheres.
method Analyzes various induced subgraphs of the Pachner graph of -vertex triangulated 2-spheres, proving connectivity and providing bounds on the number of connected components.
result The subgraph of -vertex flag 2-spheres is connected, while the subgraph of -vertex stacked 2-spheres has at least as many connected components as trees with specific properties.
The paper proves sphere theorems for submanifolds in Kähler manifolds.
problem Sphere theorems for submanifolds in Kähler manifolds.
method Differentiable and topological sphere theorems for submanifolds in Kähler manifolds, especially in complex space forms.
result Proves sphere theorems for submanifolds in Kähler manifolds.
Study finds infinite families of Sasaki-Einstein metrics on spheres.
problem Finding Sasaki-Einstein metrics on spheres and exotic spheres.
method Analyzing odd-dimensional spheres and exotic spheres that bound parallelizable manifolds.
result Infinitely many families of Sasaki-Einstein metrics on spheres and exotic spheres.
We show that among Seifert fibered integer homology spheres, Poincare sphere (with either orientation) is the only non-trivial example which has trivial Heegaard Floer homology. Together with an earlier result, this shows that if an integer homology sphere has trivial Heegaard Floer homology, then it is a connected sum…
The paper extends geodesic orbit sphere classification to Finsler geometry.
problem Classifying geodesic orbit spheres in Finsler geometry.
method Generalized from Riemannian to Finsler geometry, proving constant curvature conditions.
result Geodesic orbit Finsler spheres with constant flag curvature are Randers.
Study biharmonic hypersurfaces in spheres, proving unique continuation theorem.
problem Characterize biharmonic hypersurfaces in spheres.
method Prove CMC Unique Continuation Theorem for biharmonic hypersurfaces of spheres.
result Supports the conjecture that biharmonic submanifolds of Euclidean spheres must be of constant mean curvature.
On a homotopy -spheremath.GT
Standard proved to be diffeomorphic to a curious homotopy sphere.
problem Determining the diffeomorphism of a curious homotopy sphere to the standard .
method Proof based on properties of homotopy spheres and loose corks.
result The curious homotopy sphere is diffeomorphic to the standard .
Formula derived for discrete improper affine spheres.
problem Constructing discrete improper affine spheres.
method Loop group factorizations and Birkhoff decomposition.
result Representation formula for discrete indefinite affine spheres.
Study on stochastic flows on 7-dimensional spheres.
problem Stochastic processes on 7-dimensional spheres.
method Isometric stochastic flows of Stratonovich SDE on spheres.
result Properties of stochastic processes on Gromoll-Meyer exotic sphere.
The paper proves spheres in certain 3-manifolds are always rotational.
problem Characterizing spheres in homogeneous 3-manifolds with specific curvature relations.
method Analyzing surfaces with elliptic Weingarten equations and proving rotational symmetry.
result Spheres in the specified 3-manifolds are always rotational.
Sphere theorems for submanifolds in Kähler and Sasaki spaces.
problem Proving sphere theorems for Lagrangian and Legendrian submanifolds.
method Differentiable and topological sphere theorems for submanifolds in Kähler and Sasaki spaces.
result Proved sphere theorems for Lagrangian and Legendrian submanifolds.
New actions found on exotic spheres using group theory.
problem Understanding smooth transformations on exotic spheres.
method Recent progress in stable homotopy groups of spheres and group theory.
result Smooth circle and cyclic group actions on exotic spheres produced.
Here we generalize the Gromoll-Meyer construction of an exotic 7-sphere by producing geometric models of exotic 8, 10 and Kervaire spheres as quotients of sphere bundles over spheres by free isometric actions. We give a geometric application at the end.
New bounds and examples for sphere unknotting numbers.
problem Comparing unknotting numbers for 2-spheres in 4-space.
method Algebraic and geometric techniques.
result Stabilization number is bounded above by one more than Casson-Whitney number.