New theorem: Spaces with coaxial homeomorphisms are 2-equivalent to trees.
problem Understanding spaces with specific homeomorphisms and their topological properties.
method Using a new definition of coaxial homeomorphisms and proving a stronger theorem than previous results.
result Spaces with coaxial homeomorphisms are proper 2-equivalent to the product of a locally finite tree and a line.
Study counts catenoids connecting two coaxial circles in 3D Lorentz-Minkowski space.
problem Determining the number of catenoids connecting two coaxial circles in 3D Lorentz-Minkowski space.
method Separated analysis by circle type and catenoid causal character (spacelike and timelike).
result Number of catenoids is determined for different circle and catenoid types.
Study smoothings of singular intersections of ellipsoids.
problem Smooth singular intersections of ellipsoids.
method Generic singularities of coaxial intersections of ellipsoids studied.
result Special attention to 3D case.
The Brylinski beta function is extended for coaxial layers on submanifolds.
problem Extending the Brylinski beta function to coaxial layers on submanifolds.
method Analytic continuation and computation of residues for the function.
result The Brylinski beta function has an analytic continuation with simple poles.
Method constructs harmonic immersions in R^3 using Enneper-type representation.
problem Constructing harmonic immersions in R^3.
method Enneper-type representation approach.
result Any harmonic immersion in R^3 can be obtained using this approach.
New minimal surfaces in a ball created by merging catenoids and disks.
problem Creating minimal surfaces in a bounded space.
method Desingularizing a critical catenoid and equatorial disk using singular perturbation methods.
result Constructed high genus free boundary minimal surfaces with three boundary components.
Rotational symmetry proven for special self-expanders near cylinders.
problem Proving symmetry of self-expanders near cylinders.
method Analyzing self-expanders to inverse mean curvature flow with specific end conditions.
result Proven rotational symmetry for self-expanders with cylindrical ends.
In this article we give a criterion for the existence of a metric of curvature 1 on a 2-sphere with n conical singularities of prescribed angles 2πϑ1,…,2πϑn and non-coaxial holonomy. Such a necessary and sufficient condition is expressed in terms of linear inequalities in $\vartheta_1,\dot…
Study smoothings of ellipsoid intersections with singularities.
problem Smooth singular intersections of ellipsoids.
method Study 3D manifolds with singularities as small covers of Coxeter polyhedral orbifolds.
result Introduce n-pyramitoid to generalize n-pyramids. Study proves existence and nonexistence for annular surfaces with specific curvature and boundary.
problem Existence and nonexistence of annular surfaces with prescribed mean curvature.
method Proves existence and nonexistence results for normal graphs of unduloids or nodoids.
result Existence and nonexistence results for annular type surfaces with prescribed mean curvature.
This study examines geometric properties and offsets of slant timelike-ruled surfaces.
problem Geometric properties and offsets of slant timelike-ruled surfaces in Minkowski 3-space.
method Derivation of parametric formulation, conditions for coaxial alignment, examination through Blaschke and Darboux frames.
result Conditions ensuring the coaxial alignment of the central normal with the ruling direction of the offset surface.
Study local features of decorated representation spaces for spherical surfaces.
problem Local structure of moduli space of spherical surfaces with conical points.
method Analysis of decorated representation spaces of fundamental groups in SU(2).
result Smooth locus of decorated representation spaces is dense and connected.
New algorithm trains communication systems without a differentiable channel model.
problem Training communication systems with unknown or non-differentiable channel models.
method Iterative training between receiver and transmitter using true and approximated gradients.
result Works as well as model-based training and achieves state-of-the-art performance.
Extends Palais' result on diffeomorphisms to homeomorphisms and bi-Lipschitz mappings.
problem Extending diffeomorphisms to global mappings of manifolds.
method Elementary argument for diffeomorphisms, deep results for homeomorphisms and bi-Lipschitz mappings.
result Extension of Palais' result to homeomorphisms and bi-Lipschitz mappings.
Study on homeomorphism groups of manifolds using set theory.
problem Relationship between set theory and homeomorphism groups of manifolds.
method First-order rigidity, type versus conjugacy, axiom of constructibility, projective determinacy.
result Under V=L, homeomorphism groups of manifolds are first-order rigid and conjugacy class is determined by type.
New proof associates partitions to isotopic pseudo-Anosov homeomorphisms.
problem Stable and unstable foliations for pseudo-Anosov homeomorphisms.
method Geometric realization of Fathi's result for isotopic homeomorphisms.
result Associated stable and unstable partitions for isotopic pseudo-Anosov homeomorphisms.
Uniform interpretation of group theory in manifold homeomorphisms.
problem Understanding group properties in manifold homeomorphisms.
method First order theory interpretation of second order group theory.
result Many group theory problems encoded in homeomorphism groups.
Proves existence of sentences to identify homeomorphic manifolds.
problem Identifying homeomorphic manifolds using group properties.
method Defines sentences in group language to match homeomorphic manifolds.
result Existence of sentences to distinguish homeomorphic manifolds.
New homeomorphism found in Klein bottle group.
problem Understanding homeomorphisms of Klein bottle.
method Using recent results on commutator length.
result Existence of homeomorphism with positive stable commutator length.
Study weak conjugacy in surface homeomorphisms.
problem Understanding weak conjugacy in homeomorphisms of surfaces.
method Exploring the group of homeomorphisms isotopic to the identity.
result New insights into weak conjugacy relations.
Paper explains dynamics of homeomorphisms to mapping tori geometry.
problem Understanding dynamics of end-periodic homeomorphisms.
method Illustration-driven overview of recent results.
result Analogue of Brock's theorem for infinite-type surfaces.
Characterizes sphere covers with lifted homeomorphisms.
problem Characterizing sphere covers with lifted homeomorphisms.
method Characterization through cyclic branched covers and homeomorphisms.
result Every homeomorphism of the sphere lifts to a homeomorphism of the covering surface in these cases.
Unified framework recovers and improves classical Brouwer homeomorphism results.
problem Classical Brouwer homeomorphism theory and its dynamics.
method Unified foliated framework combining Le Calvez's and Handel's methods.
result Recovery and improvement of classical results in Brouwer homeomorphism theory.
Every homeomorphism of Euclidean space is a commutator of two homeomorphisms.
problem Understanding the structure of homeomorphisms in Euclidean space.
method Proving every orientation-preserving homeomorphism can be written as a commutator of two such homeomorphisms.
result Every orientation-preserving homeomorphism of Euclidean space is a commutator of two homeomorphisms.
Paper improves proof of theorem on convex 2-disk homeomorphisms.
problem Space of SL homeomorphisms of a convex 2-disk. method Improved proof of main lemma.
result Major improvement in understanding homeomorphisms of convex 2-disk.
The study of periodic subgroups in homeomorphism groups of manifolds.
problem Burnside problem for homeomorphism groups of manifolds.
method Analyzing surface and circle homeomorphism groups, extending Tits alternative.
result Every finitely generated periodic subgroup is finite for most manifolds.
Adapts pivoting technique to circle homeomorphisms for proofs.
problem Probabilistic Tits alternative and exponential synchronization.
method Adapts Gou{ë}zel's pivoting technique.
result Different proofs of probabilistic Tits alternative and exponential synchronization.
This paper explores when homeomorphisms between Morse boundaries of spaces induce quasi-isometries.
problem When does a homeomorphism between Morse boundaries of spaces imply a quasi-isometry?
method Investigates quasi-mobius homeomorphisms and 2-stability conditions.
result A homeomorphism between Morse boundaries of proper, cocompact spaces is induced by a quasi-isometry if and only if it is quasi-mobius and 2-stable.
No algorithm exists to decide 4-manifold homeomorphism.
problem Deciding homeomorphism for 4-manifolds.
method Demonstrated through a specific example of a connected sum of 12 copies of S^2 × S^2.
result No algorithm exists for deciding homeomorphism of 4-manifolds.
Proof shows homeomorphism problem is unsolvable.
problem Unsolvable homeomorphism problem in topology.
method Detailed proof of Markov's theorem.
result Existence of unrecognizable manifolds in higher dimensions.
Study covers of sphere with homeomorphisms lifting property.
problem Finite abelian covers of sphere with lifting homeomorphisms.
method Completely determined covers with specific lifting property.
result Properties of finite abelian covers with lifting homeomorphisms.
Goldman bracket distinguishes surface homeomorphisms.
problem Characterizing homeomorphisms between non-compact surfaces.
method Using the Goldman bracket to distinguish homeomorphisms.
result A homotopy equivalence is a homeomorphism if it preserves the Goldman bracket.
Study shows similar result to Margulis for Cantor set homeomorphisms.
problem Understanding groups of homeomorphisms of Cantor sets.
method Analogous to Margulis's proof for linear groups.
result Groups of homeomorphisms either preserve a measure or contain a free subgroup.
Positive factorization for pseudoperiodic homeomorphisms on surfaces.
problem Factorization of pseudoperiodic homeomorphisms on surfaces.
method Generalization of classical results on smooth germs of surfaces, topological characterization of monodromies, and use of positive factorization criteria.
result Pseudoperiodic homeomorphisms on surfaces with positive fractional Dehn twist coefficients and screw numbers admit a positive factorization.
Proves non-properness set of 3D polynomial homeos can't be a line.
problem Non-properness set of 3D polynomial homeomorphisms.
method Proof by contradiction, using topological properties.
result Non-properness set cannot be homeomorphic to the real line.
The study explores homeomorphism groups of self-similar 2-manifolds, including the 2-sphere and Cantor set.
problem Understanding the structure and properties of homeomorphism groups of self-similar 2-manifolds.
method Survey of recent results, exposition of classical results, treatment of stable sets, and proof of new theorems.
result Characterization of homeomorphisms of perfectly self-similar 2-manifolds and extensions of existing results.
Let X be a path-connected topological space admitting a universal cover. Let Homeo(X,a) denote the group of homeomorphisms of X preserving degree one cohomology class a. We investigate the distortion in Homeo(X,a). Let g be an element of Homeo(X,a). We define a Nielsen-type equivalence relation on the space of g-invari…
Study shows certain surface homeomorphisms groups can't be precompact.
problem Understanding precompactness of groups of surface homeomorphisms.
method Analyzing subgroup properties and transitivity of homeomorphisms.
result No subgroup of surface homeomorphisms can be Roelcke precompact.
For each n>0 there is a one complex parameter family of homeomorphisms of the circle consisting of linear fractional transformations `conjugated by z→zn'. We show that these families are free of relations, which determines the structure of `the group of homeomorphisms of finite type'. We also discuss a numbe…
Extends continuity proof to non-compact manifolds and groups.
problem Automatic continuity for homeomorphism groups of non-compact manifolds.
method Proof extension to non-compact manifolds and groups.
result Homomorphisms from these groups to separable topological groups are continuous.
Study chaotic behavior in homeomorphism groups of countable products of spaces.
problem Investigate chaotic behavior in homeomorphism groups of countable products of various metrizable topological spaces.
method Construct numerous examples of chaotic groups of homeomorphisms of countable products of spaces.
result New chaotic groups of homeomorphisms of countable products of various metrizable topological spaces are discovered.
Study classifies 2-manifolds with special homeomorphism groups.
problem Classifying 2-manifolds with specific homeomorphism groups.
method Introduced virtual Rokhlin property; classified homeomorphism groups of 2-manifolds.
result 2-manifolds with the virtual Rokhlin property have specific homeomorphism groups.
Homeomorphisms of hyperbolic 3-manifolds have invariant sets under certain conditions.
problem Understanding invariant sets for homeomorphisms on hyperbolic 3-manifolds.
method Shadowing techniques and analysis of invariant sets under positive speed with respect to a uniform foliation.
result Homeomorphisms of hyperbolic 3-manifolds are forced to have invariant sets, preventing them from being minimal.
Paper disproves a Farb question about pseudo-Anosov homeomorphisms.
problem Relation between pseudo-Anosov homeomorphisms and surface genus.
method Analyzes algebraic degree of stretch factor vs. surface genus.
result Negative answer to Farb's question about pseudo-Anosov homeomorphisms.
The study examines the realizability of a 4-manifold invariant for homeomorphisms.
problem Realizing the Casson-Sullivan invariant for homeomorphisms of 4-manifolds.
method Investigation of the invariant's realizability and application to surface isotopy.
result The invariant can be realized fully after stabilizing with a single S2imesS2 for all orientable pairs of homeomorphic 4-manifolds. Generates special homeomorphisms for complex surfaces.
problem Creating specific homeomorphisms for infinite-type surfaces.
method General conditions for producing endperiodic loxodromics.
result Produces homeomorphisms acting loxodromically on arc graphs.
Abstract: Lipschitz homeomorphisms are deformed using Perelman's methods.
problem Deformation of Lipschitz homeomorphisms
method Lipschitz analogues of Siebenmann's and Perelman's homeomorphism theory
result Lipschitz stability theorem and gluing theorem
Paper shows isomorphic curve graphs imply homeomorphic surfaces.
problem Identifying when curve graphs of infinite-type surfaces are isomorphic.
method Proved that a simplicial isomorphism between curve graphs implies homeomorphic surfaces and induced homeomorphisms.
result Isomorphic curve graphs of infinite-type surfaces imply homeomorphic surfaces and induced homeomorphisms.