The paper defines and analyzes n-fold vector bundles and their decompositions.
problem Understanding and decomposing n-fold vector bundles. method Introducing n-fold vector bundles as functors and studying their cores and decompositions. result Any n-fold vector bundle admits a non-canonical isomorphism to a decomposed n-fold vector bundle. The paper geometrizes N-manifolds using symmetric vector bundles.
problem Geometrizing N-manifolds with a specific symmetry.
method Equivalence between [n]-manifolds and symmetric n-fold vector bundles. result Identical cocycles between symmetric vector bundles and [n]-manifolds. This paper studies graded manifolds of type Δ and their equivalence with n-fold vector bundles.
problem Understanding the relationship between graded manifolds and vector bundles.
method Geometrization process for Zr-graded manifolds of type Δ. result Established an equivalence between a subcategory of n-fold vector bundles and graded manifolds of type Δ.
The abstract generalizes a construction for splitting supermanifolds and studies Lie supergroup cases.
problem Splitting supermanifolds and understanding their structure.
method Using n-fold vector bundles and graded manifolds, the abstract generalizes a construction for splitting supermanifolds. result The images of these embeddings into the category of graded manifolds satisfy universal properties of graded coverings or semicoverings for Lie supergroups and Lie superalgebras.
Study multiplicity-free covering of graded manifolds, proving equivalence of categories.
problem Equivalence of categories of graded manifolds and symmetric vector bundles.
method Defined and computed multiplicity-free covering, showed deck transformation group isomorphic to Sn. result Equivalence of categories of graded manifolds and symmetric n-fold vector bundles. Double vector bundles may be dualized in two distinct ways and these duals are themselves dual. These two dualizations generate a group, denoted DF2, which is the symmetric group S3 on three symbols. In the case of triple vector bundles the authors proved in a previous paper that the correspon…
We calculate the group of dualization operations for triple vector bundles, showing that it has order 96 and not 72 as given in Mackenzie's original treatment. The group is a nonsplit extension of S4 by the Klein group. Dualization operations are interpreted as functors on appropriate categories and are said to be equa…
A natural explicit condition is given ensuring that an action of the multiplicative monoid of non-negative reals on a manifold F comes from homotheties of a vector bundle structure on F, or, equivalently, from an Euler vector field. This is used in showing that double (or higher) vector bundles present in the literatur…
Develops Z2n-supermanifolds theory in math and physics.
problem Challenges in Z2n-supergeometry. method Definition and examples of Z2n-supermanifolds, tangent and cotangent functors, superization of vector bundles. result Fundamental theorem for supermorphisms extended to Z2n-context. In this paper we describe the cohomogeneity one special Lagrangian 3-folds in the cotangent bundle of the 3-sphere, also known in the physics literature as a deformed conifold. Our main result gives a global foliation of the deformed conifold by T^2-invariant special Lagrangian 3-folds, where the generic leaf is topolo…
Let X be a complex toric Fano n-fold and N(T) the normalizer of a maximal torus T in the group of biholomorphic authomorphisms Aut(X). We call X {\em symmetric} if the trivial character is a single N(T)-invariant algebraic character of T. Using an invariant αG(X) introduced by Tian, we …
Cyclic covers of knots uniquely determine the original knot.
problem Determining knots from their branched covers.
method Examining n-fold cyclic branched covers of alternating prime knots.
result The n-fold cyclic cover of an alternating prime knot uniquely determines the original knot.
Decomposes complex manifolds with trivial canonical bundle into homogeneous structures.
problem Decomposing complex manifolds with trivial canonical bundle into homogeneous structures.
method Using MMP and foliation theory, we prove a decomposition theorem and deduce properties of holomorphic geometric structures.
result Holomorphic geometric structures on X are locally homogeneous away from an analytic subset of complex codimension at least two. Lie Calculus connects differential and Lie theory using groupoids.
problem Understanding the relationship between differential and Lie theories.
method Using groupoids to link differential and Lie theories.
result Higher order theory involves higher algebra (n-fold groupoids).
The paper constructs special Lagrangian n-folds in arbitrary dimensions.
problem Developing a construction for special Lagrangian n-folds in arbitrary dimensions.
method Reduction of special Lagrangian condition to a quasilinear elliptic system of 2D non-linear Cauchy-Riemann equations.
result The structure and multiplicity of singularities are governed by an associated polynomial.
Simplicial neural networks extend graph neural networks to handle higher-order interactions.
problem Handling higher-order interactions in complex data structures.
method Define a convolution operation for simplicial complexes and use it to construct convolutional neural networks.
result SNNs effectively impute missing data in coauthorship complexes.
The paper proves structures on vector bundles with free actions.
problem Understanding isometry structures on vector bundles.
method Analyzes vector bundles with free canonical isometric actions.
result Total space of vector bundles carries principal bundle structures.
Study of generalized vector bundles and their geometric tools.
problem Extension of differential geometric tools to infinite dimensional vector bundles.
method Analysis of automorphisms, frame bundle, connection 1-forms, and covariant derivatives in diffeological vector pseudo-bundles.
result Non-isomorphism between connection 1-forms and covariant derivatives in infinite dimensional cases.
Proves every equivariant vector bundle over toric manifolds is a Klyachko bundle.
problem Characterizing equivariant vector bundles over toric manifolds.
method Analyzes topological and smooth equivariant vector bundles over toric manifolds.
result Every equivariant vector bundle is a Klyachko bundle.
The paper proves K-stability of special Gushel-Mukai manifolds.
problem Proving K-stability of special Gushel-Mukai manifolds.
method Analyzing the structure of Gushel-Mukai manifolds and their K-stability.
result General special Gushel-Mukai n-folds are K-stable for 3 ≤ n ≤ 6.
Given a compact complex n-fold X satisfying the ∂∂ˉ-lemma and supposed to have a trivial canonical bundle KX and to admit a balanced (=semi-Kähler) Hermitian metric ω, we introduce the concept of deformations of X that are {\bf co-polarised} by the balanced class $[ω^{n-1}]\in H^{n-1,\,n-1…
Study on involution in double jet bundles.
problem Exploring involution in double jet bundles.
method Generalized double tangent bundles to double jet bundles, presented secondary vector bundle structure, proved involution.
result Existence of a natural involution on double jet bundles.
The paper explores grids and warps in triple vector bundles, proving a zero-sum property and applying it to manifold and vector bundle contexts.
problem Understanding the structure and commutativity of triple vector bundles.
method Intrinsic proof of the sum of warps being zero, applied to specific cases like tangent bundles and vector bundles.
result The sum of warps in a triple vector bundle is zero, with applications to manifold and vector bundle structures.
Proves GAGA-style result for toric vector bundles.
problem None explicitly stated in the abstract.
method Algebraic construction of Frölicher approximating vector bundle.
result Proves GAGA-style result for toric vector bundles.
Constructs a functor for graded bundles to vector bundles, characterizing symmetric structures.
problem Characterizing and fully characterizing the image of a functor from graded bundles to vector bundles.
method Constructs a full linearisation functor that takes a graded bundle of degree k to a k-fold vector bundle, fully characterizing the image and discussing related cases.
result Obtains a subcategory of k-fold vector bundles consisting of symmetric k-fold vector bundles equipped with a family of morphisms indexed by the symmetric group Sk. The paper characterizes and contrasts knots with high 4D clasp numbers.
problem Characterizing knots with high 4-dimensional clasp numbers.
method Topological category analysis and construction of counterexamples.
result Characterization and contrast of knots with high 4D clasp numbers.
Classifies equivariant vector bundles over toric manifolds.
problem Classifying vector bundles over toric manifolds.
method Klyachko-type classification over invariant affine charts.
result Generalizes Klyachko's classification of toric vector bundles.
This paper shows vector bundles and differential bundles are equivalent in smooth manifolds.
problem Characterizing vector bundles in smooth manifolds.
method Introducing differential bundles in a tangent category and proving equivalence with vector bundles in smooth manifolds.
result Differential bundles in smooth manifolds are equivalent to vector bundles.
New pushforward operation on vector pseudo-bundles creates new examples.
problem Creating new objects from vector bundle theory in diffeology.
method Introducing pushforward operation on diffeological vector pseudo-bundles.
result Pushforward operation produces new projective diffeological vector spaces.
Introduces new super vector bundles and discusses their properties.
problem Homotopy classification of vector bundles in supergeometry.
method Introduction of ν-grassmannians and Γ, construction of Gauss supermap.
result Generalization of a theorem in homotopy classification for supergeometry.
The study of quotient structures in multi-graded bundles, including double vector bundles.
problem Understanding quotients of multi-graded bundles, especially double vector bundles.
method Analyzing quotients as towers of affine bundles and constructing normal bundles.
result Any quotient of multi-graded bundles fits into a tower of affine bundles.
Defines connections on parabolic vector bundles for Lie algebroids.
problem Characterizing parabolic vector bundles with Lie algebroid connections.
method Constructs Lie algebroid connections on parabolic vector bundles, uses Atiyah exact sequence.
result Characterizes stable Lie algebroid vector bundles with connections.
The paper extends positivity results from vector bundles to Kobayashi positive ones.
problem Extending positivity results from vector bundles to Kobayashi positive ones.
method Using convexity of Kobayashi positive Finsler metrics and duality for convex Finsler metrics.
result The quotient and tensor product of Kobayashi positive vector bundles are also Kobayashi positive.
Study on singularities of bundle homomorphisms induced by Morin maps.
problem Characterizing singular points of bundle homomorphisms.
method Analyzing conditions for singularities induced by Morin maps, using Hamilton vector fields for contact structures.
result Characterization of singularities in bundle homomorphisms induced by Morin maps.
Linearizes nonlinear connections on vector and affine bundles.
problem Defining linear connections on nonlinear bundles.
method Linearization procedure for vector fields.
result Applications in Classical Mechanics.
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.
Criteria for lifting manifold diffeomorphisms to vector bundle automorphisms.
problem Lifting diffeomorphisms to vector bundles.
method Criteria for lifting diffeomorphisms to linear automorphisms of vector bundles.
result Criteria for lifting diffeomorphisms to linear automorphisms of vector bundles.
Introduces higher algebroids via vector bundle comorphisms.
problem Generalizing Lie algebroids and higher tangent bundles.
method Defines higher algebroids as vector bundle comorphisms of graded-linear bundles with specific axioms.
result Provides natural examples and applications in geometric mechanics.
In this paper, we construct a category of short exact sequences of vector bundles and prove that it is equivalent to the category of double vector bundles. Moreover, operations on double vector bundles can be transferred to operations on the corresponding short exact sequences. In particular, we study the duality theor…
Proof of smooth radial sections in vector bundles.
problem Existence of smooth radial sections in vector bundles.
method Proof of existence using linear connection.
result Existence of smooth radial sections in vector bundles.
Study on topological rigidity of ALE vector bundles with specific conditions.
problem Classifying ALE vector bundles with asymptotically conical total spaces.
method Topological classification and geometric analysis of ALE vector bundles.
result Only 2-sphere, projective plane, and open contractible manifolds admit ALE tangent bundles.
Develops mixed quantization for graph vector bundles.
problem Solving asymptotic spectral problems on graph vector bundles.
method Mixed quantization technique for graph vector bundles.
result Applications to various spectral problems.
The paper examines the limit of harmonic flow on flat vector bundles.
problem Understanding the limiting behavior of harmonic flow on flat complex vector bundles.
method Analyzes the harmonic flow and proves the limit is isomorphic to a graded flat complex vector bundle.
result The limit of the harmonic flow on flat complex vector bundles is isomorphic to a graded flat complex vector bundle.
Proves a formula for push-forward of polynomial Chern forms in universal vector bundles.
problem Positivity of characteristic forms in vector bundles.
method Explicit computation of Chern curvature and use of flag bundles.
result Positivity of polynomials in Chern forms for Griffiths semipositive bundles.
Study on stable vector bundles over Gauduchon manifolds.
problem Existence and stability of vector bundles over Gauduchon manifolds.
method Uhlenbeck--Yau's continuity method for approximate Hermitian--Einstein structures.
result Equivalence of semi-stability and existence of Hermitian--Einstein structures.
Parseval frames can be thought of as redundant or linearly dependent coordinate systems for Hilbert spaces, and have important applications in such areas as signal processing, data compression, and sampling theory. We extend the notion of a Parseval frame for a fixed Hilbert space to that of a moving Parseval frame for…
Defines linear weightings for vector bundles and explores their applications.
problem Understanding and extending the concept of weightings in vector bundles.
method Constructs weighted normal bundles and deformation spaces; explains the relationship between weightings and differential operators.
result Captures the rescaled spinor bundle and related constructions.
New dHYM connections found on complex vector bundles.
problem Existence of dHYM connections on higher rank vector bundles.
method Constructing explicit non-trivial examples and providing algebraic conditions.
result First explicit non-trivial dHYM connections on higher rank holomorphic vector bundles.