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.
Graded bundles are a class of graded manifolds which represent a natural generalisation of vector bundles and include the higher order tangent bundles as canonical examples. We present and study the concept of the linearisation of graded bundle which allows us to define the notion of the linear dual of a graded bundle.…
Analyzes Kodaira-Iitaka dimension and multiplicity using intersection theory.
problem Understanding Kodaira-Iitaka dimension and multiplicity in analytic terms.
method Expresses dimensions and multiplicity in terms of intersection theory of plurisubharmonic envelopes.
result Introduces non-pluripolar numerical Kodaira-Iitaka dimension and shows it dominates the classical dimension.
Develops non-semisimple ETQFTs for 3-manifolds.
problem Creating ETQFTs for non-semisimple categories.
method Introducing relative modular categories and using 2-categorical universal construction.
result Extends ETQFTs to non-semisimple cases.
We apply the integral formula of volumes to the family of graded linear series constructed from any test configuration. This solves the conjecture raised by Witt--Nyström so that the sequence of spectral measures for the induced C∗-action on the central fiber converges to the canonical Duistermatt--Heckman …
Motivated by topology, we develop a general theory of traces and shadows for an endobicategory, which is a~pair: bicategory C and endobifunctor Σ:C→C. For a graded linear bicategory and a fixed invertible parameter q, we quantize this theory by using the endofunctor Σq such th…
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 double principal bundles with applications in geometry.
problem Understanding structures in double vector bundles.
method Definition and analysis of double principal bundles.
result Double vector bundles can be realized as associated bundles of their frame bundles.
Study of semi-principal bundles using group actions and wreath products.
problem Understanding bundles with fibers as free G-spaces. method Defining semi-principal bundles, bases, and frame bundles; using wreath products and functors.
result Semi-principal bundles can be retracted to principal bundles, preserving parallel transport.
Establishes a framework for stringor bundles, proving their canonical isomorphism to Stolz-Teichner's.
problem Defining and rigorously studying higher differential geometric objects like stringor bundles.
method Developed a framework of 2-Hilbert bundles, including an associated bundle construction.
result Proves the Stolz-Teichner stringor bundle is canonically isomorphic to the 2-Hilbert bundle.
Generalizes graded bundles with more flexible transformation laws.
problem No specific problem stated; focuses on generalization of graded bundles.
method Introduces filtered bundles with more general polynomial transformation laws.
result Linearisation of filtered bundles is well-defined.
In this study, we generalize double tangent bundles to double jet bundles. We present a secondary vector bundle structure on a 1-jet of a vector bundle. We show that 1-jet of a vector bundle carries two vector bundle structures, namely primary and secondary structures. We also show that the manifold charts induced by p…
Proves correspondence between harmonic and Higgs bundles.
problem Connecting harmonic and Higgs bundles for study.
method Kobayashi-Hitchin correspondence for polystable bundles.
result Establishes correspondence between good wild harmonic bundles and polystable good filtered λ-flat 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.
On a complex manifold, a co-Higgs bundle is a holomorphic vector bundle with an endomorphism twisted by the tangent bundle. The notion of generalized holomorphic bundle in Hitchin's generalized geometry coincides with that of co-Higgs bundle when the generalized complex manifold is ordinary complex. Schwarzenberger's r…
Introduces Darboux-Lie derivative for fiber bundles.
problem None explicitly stated; focuses on introducing a new derivative.
method Study of Darboux-Lie derivative for fiber-bundle maps.
result Properties of Darboux-Lie derivative for fiber bundles.
A stratified bundle is a fibered space in which strata are classical bundles and in which attachment of strata is controlled by a structure category of fibers. Well known results on fibre bundles are shown to be true for stratified bundles; namely the pull back theorem, the bundle theorem and the principal bundle theor…
Optimizes edge coloring in graph bundling for better edge differentiation.
problem Difficulty in identifying origins and destinations of individual edges in strongly bundled graphs.
method Optimizes edge coloring based on pairwise edge strength and origin-destination dissimilarity, solving a nonlinear optimization problem.
result Peacock bundles enhance graph layout comprehensibility with edge differentiation.
Categorical bundles provide a natural framework for gauge theories involving multiple gauge groups. Unlike the case of traditional bundles there are distinct notions of triviality, and hence also of local triviality, for categorical bundles. We study categorical principal bundles that are product bundles in the categor…
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.
Study curvature of determinant bundle over Teichmüller space.
problem Curvature of determinant bundle over varying conformal structures.
method Analyzing the curvature of the determinant bundle over the Teichmüller space.
result Curvature properties of determinant bundle as conformal structure varies.
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.
Formula found for minimum Ricci curvature in Fano bundles.
problem Finding the minimum Ricci curvature in Fano homogeneous toric bundles.
method Derived a formula for the greatest lower bound on Ricci curvature.
result Formula for the minimum Ricci curvature in Fano bundles.
Introduces Weyl bundle gerbe and shows it's not D-stably isomorphic.
problem Identifying and distinguishing bundle gerbes.
method Introduces cup product bundle gerbe, defines Weyl bundle gerbe, and uses holonomy to show non-isomorphism.
result Weyl bundle gerbe and basic bundle gerbe are not D-stably isomorphic.
Symplectic fillings of prequantization bundles are shown to be disk bundles under certain conditions.
problem Characterizing symplectic fillings of prequantization bundles with finite capacities.
method Analysis of symplectic capacities and diffeomorphisms.
result Symplectic fillings of prequantization bundles are diffeomorphic to disk bundles under finite capacity conditions.
Smooth classifying spaces for groups defined using diffeological spaces.
problem Classifying smooth principal bundles for a smooth group G. method Developed the theory of smooth principal bundles using diffeological spaces, defining D-numerable bundles and proving classification results. result Smooth structures on Milnor's spaces EG and BG classify all D-numerable principal bundles over any diffeological space. Identifies images of determinant morphism for specific co-Higgs bundles.
problem Determining images of determinant morphism for co-Higgs bundles.
method Identifying images of the determinant morphism of trace-free co-Higgs bundles modeled on rank 2 Schwarzenberger bundles.
result Identified images of the determinant morphism for specific co-Higgs bundles.
The paper defines and explores vector bundles that can be turned and their properties.
problem Understanding which vector bundles can be turned and their properties.
method Defining and investigating vector bundles that can be turned, and developing their theory and obstructions.
result Rank-2k bundles over the 2k-sphere are turnable, and this implies orientability. Study projective flat vector bundles over Riemann surfaces using Wronskian line bundles.
problem Understanding projective flat holomorphic vector bundles over Riemann surfaces.
method Assigning Wronskian line bundles to vector bundles and interpreting Abel's identity.
result Abel's identity is the first Chern class of the Wronskian line bundle.
The paper generalizes bundle gerbes over groupoids and their correspondence with PB groupoids.
problem Generalizing bundle gerbes over groupoids and their properties.
method Developed a functorial correspondence between PB groupoids and bundle gerbes over groupoids.
result Built a correspondence between PB groupoids and bundle gerbes over groupoids, including partial quotients.
Generalizes Higgs bundles theory using a vector bundle twist.
problem Extending Higgs bundles theory to incorporate vector bundle twists.
method Defined a Hitchin map and spectral correspondence, stated Hitchin-Kobayashi correspondence.
result Established a theory halfway between curve and higher-dimensional variety Higgs bundles.
Study geometric characterization of asymptotic pseudodifferential calculus on spinor bundles.
problem Geometric characterization of asymptotic pseudodifferential calculus on spinor bundles.
method Groupoid approach to pseudodifferential calculus, rescaled bundle.
result Rescaled bundle provides geometric characterization to asymptotic pseudodifferential calculus on spinor bundles.
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.
The paper extends orthogonal decomposition results to hermitian Higgs bundles.
problem Classical propositions on holomorphic vector bundles do not always extend to Higgs bundles.
method The approach involves extending propositions on orthogonal decompositions and the second fundamental form to hermitian Higgs bundles.
result Extended propositions concerning orthogonal decompositions and the second fundamental form have applications in Higgs bundles.
Master's thesis reviews bundle gerbe theory and introduces new isomorphic gerbe.
problem Understanding bundle gerbe theory and its applications.
method Introduced cup product bundle gerbe and showed its isomorphism to the pullback of the basic bundle gerbe by the Weyl map.
result Stable isomorphism between cup product bundle gerbe and pullback of basic bundle gerbe by the Weyl map.
Introduces generalized principal bundles and connections, linking them to standard gauge theories.
problem Generalized principal bundles and connections in field theories.
method Local coordinate transformation laws and horizontal lifts.
result Generalized principal connections are associated to Lie group fiber bundle connections.
Investigates connections in Lie group bundles, focusing on geometric reduction.
problem Geometric reduction of gauge field theories.
method Definition and analysis of equivariant connections in Lie group bundles.
result Provides conditions for the existence and properties of equivariant connections.
Study proves existence of precotangent bundles for Grassmannians.
problem Existence of precotangent bundles for Grassmannians.
method Proof for Grassmannians of reflexive Banach spaces and p-restricted Grassmannians of polarized Hilbert space. result Existence of bundle predual to tangent bundle (precotangent bundle).
Study of Pascal algebra matrices and their jet bundle map for vector bundles.
problem Defining and studying Pascal algebra matrices and their map on jet bundles.
method Identifying Pascal algebra matrices, showing generator well defines Pascal map, using it for intrinsic contact definition.
result Intrinsic definition of point-wise contact between Hermitian vector bundles using unitary equivalence of Pascal maps.
A bundle gerbe is constructed from an oriented smooth vector bundle of even rank with a fiberwise inner product, over a compact connected orientable smooth manifold with Riemannian metric. From a trivialization of the bundle gerbe is constructed an irreducible Clifford module bundle, a spinor bundle over the smooth fre…
Introduces Chern-Dirac bundles on non-Kähler Hermitian manifolds.
problem No specific problem stated; focuses on new mathematical structures.
method Introduces Chern-Dirac bundles and operators, showing isomorphisms with cohomology.
result Spaces of harmonic spinors on V-spinor bundle isomorphic to Dolbeault cohomology.
Stable Higgs bundles on elliptic fibrations are decomposed into simpler components.
problem Understanding stable Higgs bundles on complex manifolds with elliptic fibrations.
method Proved that every stable Higgs bundle on a compact complex manifold with a positive principal elliptic fibration can be decomposed into a simpler form.
result Every stable Higgs bundle on M is of the form (L⊗Π∗B0,Π∗ΦX), where (B0,ΦX) is a stable Higgs bundle on X and L is a holomorphic line bundle on M. 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.
In \cite{BR1}, \cite{BR2}, a parabolic determinant line bundle on a moduli space of stable parabolic bundles was constructed, along with a Hermitian structure on it. The construction of the Hermitian structure was indirect: The parabolic determinant line bundle was identified with the pullback of the determinant line b…
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 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.
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.
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.