Smoothly isotopic 3-discs in 4-sphere become identical after 5-dimensional push.
problem Smooth isotopy of 3-discs in 4-sphere.
method Pushing 3-discs into 5-dimensional space.
result Isotopic 3-discs in 4-sphere become identical after 5-dimensional push.
New spanning tree model connects knot homology, s-invariant, and exotic discs.
problem Understanding exotic discs in the 4-ball for knots.
method Explicitly defined differential in spanning tree complex, described Rasmussen's s-invariant.
result Identified new infinite family of knots bounding exotic discs.
The paper solves sliceness for odd free knots.
problem Determining if odd free knots have a spanning disc.
method Using the pairing of knot diagram chords.
result An explicit criterion for sliceness of odd free knots.
We define a geometric flow that is designed to change surfaces of cylindrical type spanning two disjoint boundary curves into solutions of the Douglas-Plateau problem of finding minimal surfaces with given boundary curves. We prove that also in this new setting and for arbitrary initial data, solutions of the Teichmüll…
We prove that the supremum of principal curvatures of a minimal embedded disc in hyperbolic three-space spanning a quasicircle in the boundary at infinity is estimated in a sublinear way by the norm of the quasicircle in the sense of universal Teichmüller space, if the quasicircle is sufficiently close to being the bou…
For embedded 2-spheres in a 4-manifold sharing the same embedded transverse sphere homotopy implies isotopy, provided the ambient 4-manifold has no $\BZ_2$-torsion in the fundamental group. This gives a generalization of the classical light bulb trick to 4-dimensions, the uniqueness of spanning discs for a simple close…
Solves area minimizing surface problem in metric spaces with bounded genus.
problem Finding area minimizing surfaces of bounded genus in metric spaces.
method Solves Plateau-Douglas problem in proper metric spaces with local quadratic isoperimetric inequality.
result Generalizes results from Riemannian manifolds to proper metric spaces.
The paper proves conditions for the existence of holomorphic discs in Kähler manifolds.
problem Existence of holomorphic discs for higher A∞ operations. method Showing existence of minimal discs with specific properties implies existence of holomorphic discs.
result Minimal discs in Kähler manifolds with certain boundary conditions are holomorphic.
Study on the topology of ordered disc configurations, revealing nontrivial homotopy classes.
problem Topology of ordered disc configurations and their homotopy types.
method Analysis of ordered configuration spaces of hard discs, focusing on homotopy types and nontrivial classes.
result Exhibit nontrivial classes in π_{n-3} for all n, and their persistence in deformed ambient discs.
Classifies homotopy ribbon discs for certain slice knots.
problem Characterizing homotopy ribbon discs for specific slice knots.
method Classifies Γ-homotopy ribbon slice discs up to topological ambient isotopy. result In the infinite cyclic case, there is a unique equivalence class of such slice discs. For the Baumslag-Solitar group, there are at most two equivalence classes of Γ-homotopy ribbon discs. New knots found with tough, unsliceable discs.
problem Finding tough knots that can't be sliced smoothly.
method Constructed infinitely many knots with non-approximable slice discs.
result Smoothly sliceable knots have non-approximable slice discs.
Classifies ancient flows in a disc with boundary.
problem Ancient convex flows in a disc with boundary.
method Classifies flows using curve shortening.
result Ancient convex flows in a disc are classified.
Holomorphic discs cover a ball in complex space.
problem Covering a ball in complex space with holomorphic discs.
method Showed a nonsingular holomorphic foliation by complete discs.
result The open unit ball in complex space admits a foliation by complete discs.
Circular disc can be tiled with up to 3 congruent pieces, showing symmetry.
problem Tiling a circular disc with congruent pieces.
method Proving the existence of a k-fold rotational symmetry for k≤3. result First nontrivial estimate on minimum number of tiles for certain tiling configurations.
The paper explores isometric models and Busemann functions for Funk and Hilbert discs.
problem Exploring isometric models and Busemann functions for Funk and Hilbert discs.
method Finding and describing isometric models and computing Busemann functions.
result Proving asymptotic harmonicity of the Funk disc and showing its dependence on measure.
Study geodesic discs with boundary length bounds, finding their closure in metric space.
problem Geodesic discs with boundary length constraints in metric spaces.
method Investigate closure in Gromov-Hausdorff space, relate to disc retracts.
result Closure of geodesic discs is related to disc retracts in metric spaces.
The rotation angle of a rolling disc is shown to be a geometric phase related to the Hopf fibration.
problem Understanding the geometric nature of rotation angles in kinematic models.
method Using the Hopf fibration and Gauss map, the geometric phase is decomposed into dynamical and geometric components.
result The geometric phase of rotation is described as the holonomy of the Hopf fibration.
Study smooth manifolds using disc-presheaves.
problem Understanding smooth manifolds.
method Using presheaves on a category of discs.
result Disc-presheaves have desirable properties and strong applications.
This note characterizes monohedral tilings of regular polygons with up to three tiles.
problem Characterizing monohedral tilings of regular polygons with up to three tiles.
method Connecting the results for squares and circles to generalize for any regular n-gon. result Characterization of monohedral tilings of any regular n-gon with up to three tiles. Study horizontal discs in fat distributions, proving their existence.
problem Existence of embedded horizontal discs in fat distributions.
method Analyzing nonlinear PDEs and proving local invertibility.
result Existence of germs of embedded horizontal discs.
Study on invariants of complex hyperbolic disc bundles over surfaces, proving a conjecture.
problem Investigating relationships between three invariants of complex hyperbolic disc orbibundles.
method Analyzing Euler characteristic, Euler number, and Toledo invariant of disc orbibundles over 2-orbifolds.
result Proved that -3|τ| = 2e + 2χ holds for certain complex hyperbolic disc orbibundles.
We calculate the asymptotic average rate at which a generic geodesic on a finite area hyperbolic 2-orbifold returns to an embedded disc on the surface, as well as the average amount of time it spends in the disc during each visit. This includes the case where the center of the disc is a cone point.
The paper classifies homotopy ribbon discs with specific groups.
problem Classifying homotopy ribbon discs with given fundamental groups.
method Using geometric and algebraic properties of groups, particularly Farrell-Jones conjecture.
result Classification of homotopy ribbon discs for specific knot groups and Baumslag-Solitar groups.
Study of rotation angles in a rotating disc model.
problem Understanding geometric phase in rotating systems.
method Analyzes a simple kinematic model of rotating discs.
result Explicit form of geometric phase Δg found using Baumkuchen lemma. Study calculates homotopy groups and derivatives for disc diffeomorphisms.
problem Understanding the homotopy groups of diffeomorphisms of discs.
method Computes rational homotopy groups and uses Weiss' orthogonal calculus.
result Determines optimal rational concordance stable range for high-dimensional discs.
It is well-known that Teichmuller discs that pass through "integer points'' of the moduli space of abelian differentials are very special: they are closed complex geodesics. However, the structure of these special Teichmuller discs is mostly unexplored: their number, genus, area, cusps, etc. We prove that in genus two …
The study finds conditions for free boundary Hamiltonian stationary discs in complex 2-space.
problem Conditions for free boundary Hamiltonian stationary Lagrangian discs in complex 2-space.
method Established conditions for weakly conformal, branched Ω-free boundary Hamiltonian stationary Lagrangian immersions of discs. result If conditions are met, a disc is a free boundary minimal immersion.
We study the intrinsic structure of parametric minimal discs in metric spaces admitting a quadratic isoperimetric inequality. We associate to each minimal disc a compact, geodesic metric space whose geometric, topological, and analytic properties are controlled by the isoperimetric inequality. Its geometry can be used …
Algebraic treatment of connection reduction over a special disc.
problem Reduction theory for connections over a specific geometric structure.
method Purely algebraic approach for arbitrary groups, with quantitative results.
result New quantitative results in reduction theory.
Holomorphic discs converge to maximal surfaces under specific flows.
problem Understanding the evolution of holomorphic discs under mean curvature flow.
method Mean curvature flow with boundary conditions in the space of oriented lines.
result Holomorphic discs converge to Bishop filling by holomorphic discs under certain conditions.
Modified Engel structures allow complete h-principle for overtwisted discs.
problem Engel structures and their overtwisted discs.
method Engel twist modification and h-principle proof.
result Complete h-principle for overtwisted Engel structures.
The paper finds the Finsler structure of Apollonian weak metric on unit disc.
problem Understanding the Finsler structure of Apollonian weak metric on the unit disc.
method Analyzing the deformation of hyperbolic Poincaré metric by a closed 1-form.
result The Apollonian weak-Finsler structure has bounded below S-curvature and flag curvature K satisfying −∞<K<−1. This paper compares two methods for training neural ODEs in time-series regression and CNFs.
problem Training neural ODEs for time-series regression and CNFs efficiently.
method Discretize-Optimize (Disc-Opt) vs. Optimize-Discretize (Opt-Disc) approaches.
result Disc-Opt methods can achieve similar performance as Opt-Disc at inference with drastically reduced training costs.
We show that a relatively hyperbolic graph with uniformly hyperbolic peripheral subgraphs is hyperbolic. As an application, we show that the disc graph and the electrified disc graph of a handlebody H of genus g>1 are hyperbolic, and we determine their Gromov boundaries.
Constructs infinite-dimensional Siegel disc as symplectic and Kaehler quotient.
problem No specific problem stated; focuses on mathematical construction.
method Symplectic and Kaehler quotient construction.
result Infinite-dimensional Siegel disc constructed as symplectic and Kaehler quotient.
New phenomena in 4-manifolds show discs with special properties.
problem Exploring special properties of discs in 4-manifolds.
method Construction of discs with geometrically dual spheres and analysis of isotopy.
result Discs with common geometrically dual spheres are not properly isotopic but are otherwise related.
We reduce a question of Eskin-Kontsevich-Zorich and Forni-Matheus-Zorich, which asks for a classification of all SL2(R)-invariant ergodic probability measures with completely degenerate Kontsevich-Zorich spectrum, to a conjecture of Möller's. Let Dg(1) be the subset of the moduli space …
Curvature bounds preserved in length-minimizing disks.
problem Preserving curvature bounds in length-minimizing disks.
method Analyzing length-minimizing disks and their curvature properties.
result Length-minimizing disks inherit curvature bounds from the target.
In this paper we describe the homology and cohomology of some natural bimodules over the little discs operad, whose components are configurations of non-k-overlapping discs. At the end we briefly explain how this algebraic structure intervenes in the study of spaces of non-k-equal immersions.
We construct knotted proper holomorphic embeddings of the unit disc in C^2.
Novel approach localizes optic disc and fovea centers efficiently.
problem Localizing optic disc and fovea centers in retinal images.
method Simultaneously process optic disc and fovea, modeling their relative geometry and appearance.
result Improves localization and recognition by incorporating object-object relations.
Study on holomorphic discs in bundles over compact surfaces, proving Fredholm regularity under certain conditions.
problem Analyzing holomorphic discs with boundary on surfaces in vector bundles over compact manifolds.
method Proves Fredholm regularity for sections with a single complex point under specific conditions.
result Holomorphic discs are Fredholm regular under certain conditions, including neutral Kähler and symplectic actions.
The paper proves properties of discs bounded by 2-links.
problem Whether all 2-links are slice.
method Link cobordism and intersection properties of slice discs.
result For any 2-component 2-link, there exists a disc intersecting transversely with itself trivially.
Modified proof constructs dual spheres for 4-manifolds.
problem Prove dual spheres for 4-manifolds.
method Modified proof of disc embedding theorem, geometric construction.
result Constructs geometrically dual spheres.
New minimal discs and annuli found in ellipsoids.
problem Constructing minimal surfaces in ellipsoids.
method Equivariant variational methods.
result At least three distinct embedded free boundary minimal annuli in ellipsoids.
Hyperbolic geometry reveals financial network structure and systemic importance.
problem Understanding the structure and importance of financial networks.
method Data from European banking stress tests, hyperbolic geometry analysis.
result Latent dimensions of `popularity' and `similarity' are strongly associated with systemic importance.
The article proves Randers Poincaré disc satisfies isoperimetric equality.
problem Extending Riemannian isoperimetric equality to Finslerian case.
method Analyzes Randers Poincaré disc with different volume forms.
result Osserman's result cannot be extended to Finslerian case.
We prove a general contractibility criterion for Riemannian metrics on a disc.
problem Contractibility of subsets of Riemannian metrics on a disc.
method General contractibility criterion for Riemannian metrics.
result The space of metrics with positive Gauss curvature and convex boundary is contractible.