The paper extends vector bundle theory to non-Hausdorff manifolds.
problem Generalizing vector bundle theory to non-Hausdorff manifolds.
method Using Čech cohomology to classify real non-Hausdorff line bundles.
result Vector bundles over non-Hausdorff manifolds can be constructed as colimits of standard vector bundles.
The paper classifies differentiable structures on a line with two origins.
problem Classifying differentiable structures on a non-Hausdorff line with two origins.
method Using homeomorphisms and diffeomorphisms, the paper establishes a bijection between structures and coset classes.
result The line with two origins admits uncountably many non-diffeomorphic structures for each differentiability class.
The study classifies differentiable structures on a 'Y' shape manifold.
problem Classifying differentiable structures on a 'Y' shape manifold.
method Similar to real line classification, but with gluing positive reals.
result Both manifolds admit uncountably many non-diffeomorphic structures.
The paper recasts Penrose-Sparling's non-Hausdorff twistor space using noncommutative geometry.
problem Reinterpreting Penrose-Sparling's non-Hausdorff twistor space.
method Introduces noncommutative geometry techniques to reinterpret the space, using explicit etale gluing groupoid and convolution algebra.
result The source-adapted cyclic pairing recovers the Coulomb charge, demonstrating the effectiveness of the new algebraic model.
Study of de Rham cohomology on non-Hausdorff manifolds.
problem De Rham cohomology on non-Hausdorff manifolds.
method Careful discussion of non-Hausdorff differential forms, Mayer-Vietoris sequences.
result Proved de Rham's Theorem and Gauss-Bonnet theorem for non-Hausdorff manifolds, including counterterms.
Study of one-dimensional non-Hausdorff manifolds and their quotient to CW complexes.
problem Understanding and characterizing one-dimensional non-Hausdorff manifolds.
method Analyzing properties of connected non-Hausdorff manifolds and their quotient spaces to CW complexes.
result Existence of a quotient map from a connected non-Hausdorff manifold to an open one-dimensional CW complex.
The paper classifies coverings and non-Hausdorff extensions of Misner spacetime.
problem Classifying coverings and non-Hausdorff extensions of Misner spacetime.
method Covering constructions of the punctured Minkowski plane, quotient spacetimes, and explicit embeddings.
result A natural family of spacetimes including the Hawking--Ellis extension and its universal-cover analogue.
In this paper, using similar idea as in Fukaya-Oh's work ([9]), we devise a method to compute the Fukaya category of certain exact symplectic manifolds by reducing it to the corresponding Morse category of non-Hausdorff manifold as perturbation of the Lagrangian skeleton of the exact symplectic manifold.
Investigates Künneth formula for foliated de Rham cohomology, overcoming non-Hausdorff issues.
problem Computing cohomology for foliated manifolds, especially when non-Hausdorff.
method Develops a Künneth formula for specific cases of Hausdorff foliated cohomology and finite-dimensional cohomology.
result Valid Künneth formula for certain foliated cohomology spaces, with counterexamples for others.
This paper connects foliations of the plane to non-Hausdorff 1-manifolds.
problem Connecting foliations of the plane to non-Hausdorff 1-manifolds.
method Establishes a connection between foliations of the plane and non-Hausdorff 1-dimensional manifolds.
result Establishes a beautiful connection between foliations of the plane and non-Hausdorff 1-dimensional manifolds.
The paper classifies submanifolds in pseudo-Riemannian space forms.
problem Characterizing and classifying totally umbilical submanifolds.
method Classification of congruent classes of totally umbilical submanifolds in non-flat pseudo-Riemannian space forms.
result Some moduli spaces of isometric immersions between space forms are non-Hausdorff.
Measure homology was introduced by Thurston in order to compute the simplicial volume of hyperbolic manifolds. Berlanga endowed measure homology with a structure of graded locally convex (possibly non-Hausdorff) topological vector space. In this note we completely characterize Berlanga's topology on measure homology of…
We analyze the obstruction to metrics of positive scalar curvature within a given bounded distortion class of metrics. This obstruction lives in a non-Hausdorff cohomology group Poincare dual to the uniformly finite homology studied by Block and Weinberger. One of the applications is a converse to their theorem on infi…
We study isometric cohomogeneity one actions on the (n+1)-dimensional Minkowski space up to orbit-equivalence. We give examples of isometric cohomogeneity one actions on the Minkowski space whose orbit spaces are non-Hausdorff. We show that there exist isometric cohomogeneity one actions on the Minkowski space which ar…
The paper proves a conjecture about spacetimes and singularities.
problem Understanding naked singularities and causally simple spacetimes.
method Analyzes null geodesics and spacetime properties to prove conjectures.
result Proves a conjecture about spacetimes and singularities, including implications for two-dimensional spacetimes.
Milnor-Thurston homology theory is a construction of homology theory that is based on measures. It is known that it is equivalent to singular homology theory in case of manifolds and complexes. Its behaviour for non-tame spaces is still unknown. This paper provides results in this direction. We prove that Milnor-Thurst…
This paper compares two invariants of foliated manifolds which seem to measure the non-Hausdorffness of the leaf space: the transversal length on the fundamental group and the foliated Gromov norm on the homology. We consider foliations with the property that the set of singular simplices transverse to the foliation sa…
We introduce a category of rigid geometries on singular spaces which are leaf spaces of foliations and are considered as leaf manifolds. We single out a special category F0 of leaf manifolds containing the orbifold category as a full subcategory. Objects of F0 may have non-Hausdorff topology u…
The paper describes and analyzes maximal Lorentzian surfaces with a Killing field, proving completeness criteria.
problem Characterizing and understanding completeness of Lorentzian surfaces with a Killing field.
method Global description and analysis of surfaces, completeness criteria involving topology and geometry.
result Completeness of surfaces is equivalent to null completeness under bounded curvature hypothesis.
We describe explicitly the moduli spaces Mgpst(S,E) of polystable holomorphic structures E with detE≅K on a rank 2 vector bundle E with c1(E)=c1(K) and c2(E)=0 for all minimal class VII surfaces S with b2(S)=1 and with respect to all possible Gauduchon metrics g. These surfaces S are …
Noncommutative geometry is used to study the local geometry of ultrametric spaces and the geometry of trees at infinity. Connes's example of the noncommutative space of Penrose tilings is interpreted as a non-Hausdorff orbit space of a compact, ultrametric space under the action of its local isometry group. This is gen…
We construct the geometric Baum-Connes assembly map for twisted Lie groupoids, that means for Lie groupoids together with a given groupoid equivariant PU(H)−principle bundle. The construction is based on the use of geometric deformation groupoids, these objects allow in particular to give a geometric construction of …
For G a topological group, existence theorems by Milnor (1956), Gelfand-Fuks (1968), and Segal (1975) of classifying spaces for principal G-bundles are generalized to G-spaces with torsion. Namely, any G-space approximately covered by tubes (a generalization of local trivialization) is the pullback of a univers…
We study how the notion of tangent space can be extended from smooth manifolds to diffeological spaces, which are generalizations of smooth manifolds that include singular spaces and infinite-dimensional spaces. We focus on two definitions. The internal tangent space of a diffeological space is defined using smooth cur…
The paper describes the geometric properties of line congruences' singularities.
problem Understanding the singularities of generic line congruences.
method Use of an equiaffine pair to define generic line congruences.
result Geometric description of folds, cusps, and swallowtails as singularities of generic line congruences.
This paper generalizes the envelope of mid-lines to intermediate lines for a plane curve.
problem Understanding the envelope of intermediate lines for a plane curve.
method Using singularity theory techniques to analyze the local behavior of the envelope of intermediate lines.
result The envelope of intermediate lines (EIL) is formed by three disconnected sets: AEIL, the curve itself, and IPTL. Study restricts line arrangements with odd points using topological arguments.
problem Restrictions on line arrangements with singular points of odd multiplicity.
method Topological arguments on locally-flat spheres in 4-manifolds.
result No line arrangement with 13 lines and only triple points exists.
A special group of transformations of the real line cannot act effectively on it.
problem Understanding the limitations of transformations on the real line.
method Analyzing the group of orientation-preserving quasi-isometries of the real line.
result The group of quasi-isometries of the real line cannot act effectively on the line.
Study Blaschke's asymptotic lines on surfaces in 3D space.
problem Characterize Blaschke's asymptotic lines on surfaces in 3D.
method Analyze binary differential equations near cusp and umbilic points.
result Describe Blaschke's asymptotic lines near Euclidean parabolic set.
Co-PLNet combines point and line predictions to improve wireframe parsing accuracy and efficiency.
problem Separate line and point predictions lead to inconsistent wireframes.
method Co-PLNet uses a Point-Line Prompt Encoder to convert early point detections into spatial prompts, which guide line refinement.
result Co-PLNet achieves better accuracy and robustness in wireframe parsing compared to existing methods.
Classifies metric lines in Engel-type groups, a step towards solving sub-Riemannian manifold problems.
problem Classifying metric lines in Engel-type groups.
method Sequence method to study metric lines in jet space.
result Classified metric lines of Engel-type groups $\Eng(n)$.
New Calabi-Yau metrics with conical singularities are created near complex lines.
problem Creating Calabi-Yau metrics with conical singularities near complex lines.
method Using branched covering arguments to construct metrics with conical singularities.
result Calabi-Yau metrics with unstable conical singularities are successfully created.
A line arrangement of 3n lines in CP2 satisfies Hirzebruch property if each line intersect others in n+1 points. Hirzebruch asked if all such arrangements are related to finite complex reflection groups. We give a positive answer to this question in the case when the line arrangement in CP2 is…
The paper examines asymptotic lines of plane fields in 3D space.
problem Qualitative properties of asymptotic lines in plane fields.
method Analysis of null directions and Gaussian curvature.
result Asymptotic lines coincide with classical ones in completely integrable fields.
The paper explores reflection principles for lightlike line segments on maximal surfaces.
problem Reflection property does not hold for lightlike line segments on maximal surfaces.
method Analyzes reflection properties for lightlike line segments connecting shrinking singularities.
result Shows a kind of reflection principle for lightlike line segments on maximal surfaces.
Circular nets with spherical parameter lines have geometric properties related to Darboux cyclides and terminating Laplace sequences.
problem Discretizing surfaces with spherical curvature lines.
method Lie-geometric discretisation in terms of principal contact element nets.
result Circular nets with two families of spherical parameter lines are related to Darboux cyclides.
The paper explores graphons of line graphs from sparse finite graphs.
problem Estimating graph limits from sparse finite graphs.
method Mapping finite graphs to their line graphs and analyzing graphs with the square-degree property.
result Graphons of line graphs can distinguish between sparse graphs like star graphs and superlinear preferential attachment graphs.
New method describes entanglement of straight lines in 3D space.
problem Tackles the geometry and topology of configurations of straight lines.
method Introduces direction matrices and a discrete motion principle.
result Shows n-crosses as links of pairwise connected unknots.
Shows CM line bundles are ample on K-stable varieties.
problem Ensuring CM line bundles are ample on K-stable varieties.
method Analyzes CM line bundles on K-stable varieties and their families.
result CM line bundles are ample on K-stable varieties with maximal variation.
Study proves Hodge symmetry on Oeljeklaus-Toma manifolds with line bundles.
problem Hodge symmetry on complex manifolds with line bundles.
method Analyzes Dolbeault cohomology of Oeljeklaus-Toma manifolds with holomorphic line bundles.
result Proves Hodge symmetry and vanishing/non-vanishing of Dolbeault cohomology.
Braided vector fields on spatial subdomains homeomorphic to the cylinder play a crucial role in applications such as solar and plasma physics, relativistic astrophysics, fluid and vortex dynamics, elasticity, and bio-elasticity. Often the vector field's topology -- the entanglement of its field lines -- is non-trivial,…
We define a pseudo-inverse for line graphs using linear integer programming.
problem Not all graphs have a corresponding root graph, making the line graph operation non-invertible.
method Propose a linear integer program to edit the smallest number of edges in the line graph to recover a root graph.
result The pseudo-inverse operation is well-behaved and works in practice as shown by empirical experiments.
The paper finds two types of metric lines in curve spaces.
problem Classifying metric lines in jet spaces of curves.
method Established the existence of two families of metric lines in the 2-jet space of plane curves.
result Found precise criteria for identifying metric lines in sub-Riemannian geodesics.
We explain the bundle structures of the {\it Determinant line bundle} and the {\it Quillen determinant line bundle} considered on the connected component of the space of Fredholm operators including the identity operator in an intrinsic way. Then we show that these two are isomorphic and that they are non-trivial line …
The authors study smooth lines on projective planes over the algebra C of complex numbers, the algebra C^1 of double numbers, and the algebra C^0 of dual numbers. In the space RP^5, to these smooth lines there correspond families of straight lines describing point three-dimensional tangentially degenerate submanifolds …
Geometric quantization extended to big line bundles.
problem Quantization of line bundles with large curvature.
method Proving asymptotic isometry and submultiplicative norms equivalence, showing Mabuchi geodesic rays.
result Bounded submultiplicative filtrations on big line bundles lead to Mabuchi geodesic rays.
There is a natural duality between line congruences in R3 and surfaces in R4 that sends principal lines into asymptotic lines. The same correspondence takes the discriminant curve of a line congruence into the parabolic curve of the dual surface. Moreover, it takes the ridge curves to the flat r…
Line graph transformation aids graph isomorphism tests by excluding challenging graph properties.
problem Limited theoretical understanding of line graph transformation's impact on GNN models.
method Examined CFI and strongly regular graphs, showing line graph transformation helps WL tests distinguish these graphs.
result Line graph transformation aids WL tests in distinguishing challenging graph properties.