Unified framework for Alexandrov 3-spaces, extending manifold results.
problem Extend manifold topology results to Alexandrov 3-spaces.
method Generalize connected sum, prime decomposition, and Dehn surgery.
result Unified framework for Alexandrov 3-spaces, including non-manifold spaces.
3D spaces without boundaries are found that aren't manifolds.
problem Finding compact aspherical Alexandrov spaces without boundaries.
method Constructing specific examples of 3D spaces.
result Examples of 3D spaces without boundaries that are not topological manifolds.
Unified framework designs LK structures using integer twists on non-manifold meshes.
problem Binary twisting limits topological possibilities and structural behaviors.
method Generalizes twist formulation to arbitrary integer labels for non-manifold meshes.
result Integer twists enable full connectivity and dynamic folding/articulation.
Classifies generalized Seifert fiber spaces and their branched covers.
problem Classifying generalized Seifert fiber spaces and their topological properties.
method Symbolic classification and computation of invariants.
result Canonical double branched cover of non-manifold spaces is a Seifert manifold.
By a construction of Berstein and Edmonds every proper branched cover f between manifolds is a factor of a branched covering orbit map from a locally connected and locally compact Hausdorff space called the monodromy space of f to the target manifold. For proper branched covers between 2-manifolds the monodromy space i…
New map constructed from equivariant spectra for manifold study.
problem Understanding equivariant parametrized h-cobordism in non-manifold settings.
method Constructed a map from suspension G-spectrum to equivariant A-theory spectrum, compatible with tom Dieck splitting formulas.
result Fiber of constructed map is wedge of stable h-cobordism spectra.
We characterize subgroups of the mapping class group that stabilize a Teichmueller disk in terms of ellipses and strips that are immersed in the associated translation surface. In particular, we show that the space of immersed ellipses/strips that meet at least three cone points is naturally a (non-manifold) 2-dimensio…
4-manifolds with Ricci curvature bounds can be approximated by non-manifold spaces.
problem Existence of manifold structure in 4-manifolds with Ricci curvature bounds.
method Constructing 4-rectifiable metric spaces that are limits of manifolds with Ricci curvature bounds.
result 4-manifolds can be approximated by spaces that are nowhere topologically manifolds.
Piecewise Euclidean structures (identified solid Euclidean polyhedra) on topological 3-dimensional manifolds and pseudo-manifolds are constructed so that they admit pseudo-foliations, a generalized type of foliation. The construction of non-manifold point neighborhoods is done to preserve as much of the geometric, and …
We consider certain groups of tree automorphisms as so-called diffeological groups. The notion of diffeology, due to Souriau, allows to endow non-manifold topological spaces, such as regular trees that we look at, with a kind of a differentiable structure that in many ways is close to that of a smooth manifold; a suita…
Detects singularities in complex data to improve machine learning models.
problem Real-world data often contains non-manifold structures (singularities) that can mislead machine learning models.
method Develops a topological framework to quantify local intrinsic dimension and Euclidicity score for multiple scales.
result Identifies singularities and captures local geometric complexity in image data.
We enumerate all spaces obtained by gluing in pairs the faces of the octahedron in an orientation-reversing fashion. Whenever such a gluing gives rise to non-manifold points, we remove small open neighbourhoods of these points, so we actually deal with three-dimensional manifolds with (possibly empty) boundary. There a…
In 1992, David Wright proved a remarkable theorem about which contractible open manifolds are covering spaces. He showed that if a one-ended open manifold M has pro-monomorphic fundamental group at infinity which is not pro-trivial and is not stably Z, then M does not cover any manifold (except itself). In the non-mani…
Given a sample Y from an unknown manifold X embedded in Euclidean space, it is possible to recover the homology groups of X by building a Vietoris--Rips or Čech simplicial complex on top of the vertex set Y. However, these simplicial complexes need not inherit the metric structure of the manifold, in particular…
Deep Gaussian processes on manifolds improve performance on complex data.
problem Complex data on manifolds that shallow models struggle with.
method Residual deep Gaussian processes on Riemannian manifolds.
result Significant improvement in prediction quality and uncertainty calibration.
Study of tangent spaces in diffeological spaces under Lie group actions.
problem Understanding tangent spaces in generalized spaces.
method Generalized tangent space construction and isomorphism proof.
result Internal tangent space isomorphic to stratified tangent space.
(1,1) non-L-space knots are foliar in 3D space.
problem Proving (1,1) non-L-space knots are foliar.
method Analyzing (1,1) non-L-space knots in S3 and lens spaces. result (1,1) non-L-space knots are persistently foliar.
Universal spaces for finite topological spaces simplify shape descriptions.
problem Describing shape properties of compact metric spaces.
method Inverse limits of finite spaces and Alexandroff extensions.
result Universal spaces simplify shape descriptions of compact metric spaces.
The paper extends Stone duality to topological convexity spaces.
problem Understanding the relationship between topological convexity spaces and sup-lattices.
method Extending Stone duality to topological convexity spaces using preconvexity spaces.
result An adjunction between topological convexity spaces and sup-lattices.
No Einstein hypersurfaces found in Damek-Ricci spaces.
problem Existence of Einstein hypersurfaces in symmetric spaces.
method Analyzing properties of Damek-Ricci spaces and proving no Einstein hypersurface exists.
result No Einstein hypersurfaces in Damek-Ricci spaces.
The distance function ϱ(p,q) (or d(p,q)) of a distance space (general metric space) is not differentiable in general. We investigate such distance spaces over Rn, whose distance functions are differentiable like in case of Finsler spaces. These spaces have several good properties, yet they are no F…
New kernels defined for various spaces, including measures.
problem Defining kernels on non-standard spaces like measures.
method Integrally strictly positive definite and characteristic kernels on Hilbert, Banach, and metric spaces.
result Explicit classes of kernels on Lp spaces and sets of measures. Metric spaces uniquely split into Hilbert and non-line-split parts.
problem Understanding the structure of metric spaces.
method Proved unique decomposition into Hilbert and non-line-split parts.
result Metric spaces have a unique decomposition into a Hilbert space and a non-line-split part.
In a paper (math.DG/0403528) we obtained explicit examples of Moishezon twistor spaces of some compact self-dual four-manifolds admitting a non-trivial Killing field, and also determined their moduli space. In this note we investigate minitwistor spaces associated to these twistor spaces. We determine their structure, …
Study geometry of tetrahedra in complex hyperbolic space and Hilbert spaces.
problem Understanding geometric relationships between complex hyperbolic spaces and Hilbert spaces.
method Use a complex analog of the cosine of a vertex angle as a novel technical tool.
result Describe possible triangular faces of tetrahedra in hyperbolic space and three-dimensional subspaces in Hilbert spaces with Pick kernels.
New curvature positivity helps classify spherical spaces and complex projective spaces.
problem Classifying spherical space forms and complex projective spaces.
method Introducing a new positivity notion for curvature.
result Characterizations for spherical space forms and complex projective spaces.
Study of complete space-like self-expanders in Minkovski space.
problem Characterize complete space-like self-expanders in Minkovski space.
method Use of maximum principle of Omori-Yau type to prove rigidity theorems.
result Classification of 2-dimensional complete space-like self-expanders with constant squared norm of the second fundamental form.
The abstract discusses the linear and smooth structures of mapping spaces.
problem The structure of mapping spaces in differential geometry.
method Proving diffeomorphisms and fibre bundle properties.
result Path spaces and base point preserving mapping spaces are Fréchet spaces.
Study on convergence of transformed metric spaces as dimensions grow.
problem Conditions for convergence of transformed metric spaces.
method Clarifying conditions for convergence of transformed spaces from original sequence and vice versa.
result Spheres and projective spaces converge to Gaussian space and its quotient as dimensions increase.
Characterizes when almost smooth spaces become RCD spaces.
problem Understanding conditions for almost smooth spaces to be RCD spaces.
method Characterizations via local volume doubling and Poincaré inequality.
result Characterizes Einstein 4-orbifolds.
Stability of Wasserstein spaces under various convergence types.
problem Stability and finiteness of Wasserstein spaces over singular and non-singular spaces.
method Gromov--Hausdorff convergence and equivariant Gromov--Hausdorff convergence.
result Analogue of Perelman's stability theorem on Wasserstein spaces.
New infinite families of flat spaces found from symmetric spaces.
problem Finding new flat homogeneous spaces.
method Starting from compact symmetric spaces, constructing infinite families of compact homogeneous spaces with invariant Bismut connections.
result Infinite families of compact homogeneous spaces with vanishing Ricci tensor.
New quasi space forms solve Thurston's geometrical space form problem.
problem Solving Thurston's geometrical space form problem.
method Introducing quasi space forms as non-real space forms with specific geometric properties.
result Quasi space forms offer a metrical, local geometrical solution to Thurston's problem.
Classifies compact spaces by shape, finite spaces by weak homotopy.
problem Classifying compact Hausdorff spaces and finite topological spaces.
method Constructs a category that classifies spaces by shape and weak homotopy.
result Classifies compact spaces by shape, finite spaces by weak homotopy.
Estimates eigenvalues of Jacobi operator for harmonic spaces.
problem Estimating eigenvalues of Jacobi operator.
method Using density function of a harmonic space.
result Sharp estimates imply symmetric Osserman space.
Introduces new types of homogeneous spaces and their properties.
problem Defining and understanding new types of homogeneous spaces.
method Introducing and analyzing (strongly) (Θ-)discrete homogeneous spaces. result Discovers relationships between new and existing homogeneous space types.
This work describes compactifications of metric spaces and vector spaces using asymmetric norms.
problem Compactifying metric spaces and vector spaces using asymmetric norms.
method Nonstandard methods, ultrapowers of the spaces at hand.
result Polyhedral compactifications of vector spaces with stratified structure.
In this paper, we present a unified study of the moduli space of tropical curves and Outer space which we link via period maps to the moduli space of tropical abelian varieties and the space of positive definite quadratic forms. Our work is a first step towards exhibiting Outer space and the space of positive definite …
Survey Bernstein-type theorems for graphical surfaces in Euclidean and Lorentz-Minkowski spaces.
problem Proving theorems for minimal and constant mean curvature graphs in Euclidean and Lorentz-Minkowski spaces.
method Explains several proofs and provides mean curvature estimates for graphs in Euclidean and Lorentz-Minkowski spaces.
result Bernstein-type theorems for constant mean curvature graphs in Euclidean 3-space and space-like graphs in Lorentz-Minkowski 3-space.
CAT(0) spaces with small volume growth are homeomorphic to Euclidean space.
problem Characterizing CAT(0) spaces with specific volume growth properties.
method Analyzing asymptotic topological regularity and volume growth.
result CAT(0) spaces with small volume growth are homeomorphic to Euclidean space.
Cantor Riemannium is a new type of space from holomorphic germs.
problem Defining a new type of space from holomorphic germs.
method Constructing the Cantor Riemannium by Borel monogenic continuation.
result The Cantor Riemannium is a metric, path connected, Gromov length space.
We construct a metric on the moduli space of bodies in Euclidean space. The moduli space is defined as the quotient space with respect to the action of integral affine transformations. This moduli space contains a subspace, the moduli space of Delzant polytopes, which can be identified with the moduli space of symplect…
Study static Einstein-Maxwell space invariant by translation.
problem Classify static Einstein-Maxwell space invariant under translation.
method Analyze static Einstein-Maxwell space conformal to pseudo-Euclidean space.
result Complete classification of static Einstein-Maxwell space invariant under translation.
A "Chen space" is a set X equipped with a collection of "plots" - maps from convex sets to X - satisfying three simple axioms. While an individual Chen space can be much worse than a smooth manifold, the category of all Chen spaces is much better behaved than the category of smooth manifolds. For example, any subspace …
The classification of G-spaces by Palais is refined for the case where the orbit space satisfies certain mild topological hypotheses. It is shown that when a sequence of such orbit spaces is "close" to a limit orbit space, in some suitable sense, within a larger ambient orbit space, the G-spaces in the tail of the sequ…
New spaces help connect manifold structures on equivariant Poincaré spaces.
problem Creating manifold structures on equivariant Poincaré spaces.
method Introducing semifree isovariant G-Poincaré spaces and gap conditions. result Space of isovariant structures on semifree G-Poincaré spaces is highly connected. Study shows how maps from certain geometric spaces behave near their edges.
problem Boundary regularity of harmonic maps in specific geometric spaces.
method Analysis of RCD(K,N) and CAT(0) spaces. result Established boundary regularity of harmonic maps.
Research explores the space-like embeddings in pseudo-hyperbolic space, finding geometric frames and actions.
problem Flat maximal space-like embeddings in pseudo-hyperbolic space.
method Description of Codazzi tensors, introduction of pseudo-Kähler metrics, Hamiltonian actions, moment maps, and geometric frames.
result Existence of two Hamiltonian actions with moment maps and geometric global Darboux frame.