Maps approximable by homeomorphisms under certain conditions.
problem Finding when a map is approximable by homeomorphisms.
method Analyzing the cell-like point-inverses and using the disjoint disc property.
result Maps are approximable by homeomorphisms if M has dimension ≥ 5 and Q has the disjoint disc property.
A manifold is T-embedded into an affine space if its tangent spaces at distinct points are disjoint. We prove that an n-dimensional disc cannot be T-embedded into 2n-dimensional space.
Study on non-orientable surfaces for maximal disc packings.
problem Maximizing disc packings in non-orientable surfaces.
method Analyzing compact non-orientable surfaces of genus g≥3 for maximal k-packings. result Characterization of maximal disc packings in non-orientable surfaces.
The paper proves properties of surfaces with holes and ends.
problem Properties of Riemann surfaces with holes and ends.
method Analyzes Riemann surfaces with countably many ends and discs removed.
result Completes the complex structure of a minimal surface in 3D space.
Maximizes filling systems on surfaces with given boundary components.
problem Finding the maximum size of filling systems on surfaces with specific boundary conditions.
method Analyzing the structure of filling systems and their complements.
result The maximum size of a filling system on a surface of genus g with 1 ≤ b ≤ 2g-2 boundary components is 2g + b - 1.
The paper proves conditions for converting homotopies to monotone homotopies in Riemannian discs and spheres.
problem Conditions for converting homotopies to monotone homotopies in Riemannian discs and spheres.
method Analyzing the boundary of Riemannian discs and spheres to determine if they can be contracted monotonously.
result A monotone homotopy can be constructed for a Riemannian disc and sphere under certain length constraints.
We prove a "gluing" theorem for monotone homotopies; a monotone homotopy is a homotopy through simple contractible closed curves which themselves are pairwise disjoint. We show that two monotone homotopies which have appropriate overlap can be replaced by a single monotone homotopy. The ideas used to prove this theorem…
Let Fg denote a closed oriented surface of genus g. A set of simple closed curves is called a filling of Fg if its complement is a disjoint union of discs. The mapping class group Mod(Fg) of genus g acts on the set of fillings of Fg. The union of the curves in a filling forms a graph on the surfa…
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.
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.
Study minimal discs in metric spaces with a quadratic isoperimetric inequality.
problem Understanding the geometry of minimal discs in metric spaces.
method Associate a compact metric space to each minimal disc, controlling its properties by the isoperimetric inequality.
result The geometry of the associated space can control the shapes of curves and the original space's topology.
We show that all finite-dimensional resolvable generalized manifolds with the piecewise disjoint arc-disk property are codimension one manifold factors. We then show how the piecewise disjoint arc-disk property and other general position properties that detect codimension one manifold factors are related. We also note …
A Heegaard splitting of a closed, orientable three-manifold satisfies the disjoint curve property if the splitting surface contains an essential simple closed curve and each handlebody contains an essential disk disjoint from this curve [Thompson, 1999]. A splitting is full if it does not have the disjoint curve proper…
Paper equates torsions on wedge singularities.
problem Analyzing spaces with wedge singularities.
method Equating analytic and intersection Reidemeister torsions.
result Equation of torsions in specific wedge singularities.
We present a new property, the Disjoint Path Concordances Property, of an ENR homology manifold X which precisely characterizes when X times R has the Disjoint Disks Property. As a consequence, X times R is a manifold if and only if X is resolvable and it possesses this Disjoint Path Concordances Property.
Paper finds a disc with finite area and dense boundary in a ball.
problem Constructing a holomorphic disc with finite area and dense boundary.
method Constructs a holomorphic disc in the unit ball with specific properties.
result Disc has finite area and dense boundary curve.
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 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 present and discuss several old and new methods for mapping a circular disc to a square. In particular, we present analytical expressions for mapping each point (u,v) inside the circular disc to a point (x,y) inside a square region. Ideally, we want the mapping to be smooth and invertible. In addition, we put emphas…
We prove that the Teichmueller disc stabilized by the Arnoux-Yoccoz pseudo-Anosov diffeomorphism contains at least two closed Teichmueller geodesics. This proves that the corresponding flat surface does not have a cyclic Veech group. In addition, we prove that this Teichmueller disc is dense inside the hyperelliptic lo…
We study regularity properties of solutions to the Dirichlet problem for the complex Homogeneous Monge-Ampère equation. We show that for certain boundary data on P1 the solution Φ to this Dirichlet problem is connected via a Legendre transform to an associated flow in the complex plane called the Hele-Shaw…
The paper proves bounded cohomology properties of Euclidean space groups.
problem Understanding bounded cohomology of transformation groups of Euclidean spaces and discs.
method Analyzes groups of homeomorphisms and diffeomorphisms of Euclidean spaces and discs, proving boundedly acyclic properties.
result Groups of orientation-preserving homeomorphisms and diffeomorphisms of Rn are boundedly acyclic, with implications for characteristic classes of bundles. Paper constructs graphs with girth four and distinct properties.
problem Characterize Ricci-flat graphs with specific girth.
method Constructs and characterizes graphs with girth four, focusing on edge-disjoint and vertex-disjoint 4-cycles.
result Characterizes all Ricci-flat graphs of girth four with vertex-disjoint 4-cycles.
We give several new positive finite presentations for the pure braid group that are easy to remember and simple in form. All of our presentations involve a metric on the punctured disc so that the punctures are arranged "convexly", which is why we describe them as geometric presentaitons. Motivated by a presentation fo…
Approximates cycles in planar and bounded-genus graphs.
problem Finding many disjoint cycles in planar and bounded-genus graphs.
method Constant-factor approximation algorithms for vertex-disjoint and edge-disjoint cycles.
result First algorithms for vertex-disjoint paths in fully planar and bounded-genus instances.
The paper describes p-adic analogs of hyperbolic discs and their geometric properties.
problem Exploring p-adic analogs of hyperbolic discs and their geometric properties.
method Geometric construction in the projective p-adic plane, using p-adic convexity and transitive action of PGL.
result Found three (or seven if p=2) p-adic analogs of hyperbolic discs, unlike the real case where only one exists.
Loops in surfaces and chord diagrams are studied with graph factorizations and grammars.
problem Understanding loops in surfaces and their properties.
method Factorization of filoops into spheric and toric sums, and grammars generating chordiagraphs.
result Minimal genus of filoops and stability properties under factorizations.
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.
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.
This paper gives a construction for all minimal immersions f of the Poincaré disc into the complex hyperbolic plane CH2 which are equivariant with respect to an irreducible representation ρ of a hyperbolic surface group into PU(2,1). We exploit the fact that each such immersion is a twisted conformal …
The paper solves an embedding problem for discs with specific curvature properties.
problem Embedding discs with positive Gauss curvature into Euclidean 3-space.
method Analyzes the free boundary isometric embedding problem with specific curvature conditions.
result Discs with specified curvature can be isometrically embedded into R3 orthogonally to the unit sphere. Metric spaces with upper curvature bounds have controlled Dehn functions.
problem Understanding the relationship between curvature bounds and Dehn functions in metric spaces.
method Proving equivalence between upper curvature bounds and bounded Dehn functions using ultralimits and minimal discs.
result A length space has curvature bounded above by κ if and only if its Dehn function is bounded by the model surface of constant curvature κ.
We give an algorithm to decide which elements of pi_2(S^2\times S^1#...#S^2\times S^1) can be represented by embedded spheres. Such spheres correspond to splittings of the free group on k generators. Equivalently our algorithm decides whether, for a handlebody N, an element in pi_2(N,\partial N) can be represented by a…
The study proves a geometric inequality for surfaces with genus G.
problem Proving a geometric inequality for surfaces with genus G.
method Analyzing the relationship between a genus G surface and a disc with specific properties.
result A new geometric inequality involving the area of surfaces and discs.
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.
Let D2 be the open unit disc in the Euclidean plane and let G:=Diff(D2;area) be the group of smooth compactly supported area-preserving diffeomorphisms of D2. We investigate the properties of G endowed with the autonomous metric. In particular, we construct a bi-Lipschitz homomorphism Zk→G of a…
Investigates properties of a pseudometric on domains in Euclidean space, linking it to hyperbolic geometry.
problem Defines and analyzes a pseudometric on domains in Rn to understand their hyperbolic properties. method Introduces a pseudometric based on conformal harmonic discs and studies its properties and conditions for hyperbolicity.
result Characterizes domains as hyperbolic based on their geometric properties and provides sufficient conditions for hyperbolicity.
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.
We show some generic (robust) properties of smooth surfaces immersed in the real 3-space (Euclidean, affine or projective), in the neighbourhood of a {\em godron} (term due to R.Thom): an isolated parabolic point at which the (unique) asymptotic direction is tangent to the parabolic curve. With the help of these proper…
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.
We study quasi-morphisms on the groups Pn of pure braids on n strings and on the group D of compactly supported area-preserving diffeomorphisms of an open two-dimensional disc. We show that it is possible to build quasi-morphisms on Pn by using knot invariants which satisfy some special properties. In particular, we st…
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.
A Margulis spacetime is a complete affine 3-manifold M with nonsolvable fundamental group. Associated to every Margulis spacetime is a noncompact complete hyperbolic surface S. We show that every Margulis spacetime is orientable, even though S may be nonorientable. We classify Margulis spacetimes when S is homeomorphic…