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. Theory of smooth relative connections on quiver bundles developed.
problem Existence of smooth relative connections over quiver bundles.
method Developed a theory over RQ on smooth twisted quiver bundles, provided obstructions and necessary/sufficient conditions. result Established a necessary and sufficient condition for the existence of smooth relative connections on tree-type quiver bundles.
Kontsevich's classes distinguish smooth structures on fiber bundles.
problem Distinguishing smooth structures on fiber bundles.
method Using Kontsevich's characteristic classes and real blow-up construction.
result Kontsevich's classes are determined by the topology of the 2-point configuration space bundle.
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.
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 curved metrics on manifolds using smoothing theory.
problem Understanding rational homotopy groups of curved metrics on high-dimensional manifolds.
method Classical results in smoothing theory applied to high-dimensional manifolds.
result Smooth M-bundles with fiberwise negatively curved metrics represent elements of finite order in homotopy groups.
Study smooth bundles over BSU(2) not induced by actions.
problem Characterize smooth bundles over BSU(2) not arising from actions.
method Use technical results on tautological classes and maps induced by actions.
result Exhibit smooth manifold bundles over BSU(2) not induced by actions.
Study smooth loops and loop bundles, relating to G2-structures.
problem Properties of smooth loops and their applications.
method Analyze smooth loops, introduce loop bundles, define torsion and curvature.
result Showed how loop bundles relate to G2-structures. 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.
Study the structure of linear bundle morphisms between vector bundles.
problem Understanding the structure of linear bundle morphisms between vector bundles.
method Analyzing the set of smooth linear bundle morphisms between fibre bundles.
result Detailed structure of linear bundle morphisms between vector bundles.
Classifies Real line bundles with Real connections on manifolds with involution.
problem Classifying Real line bundles with Real connections on manifolds with involution.
method Defines Real smooth Deligne cohomology to interpolate between equivariant sheaf cohomology and smooth imaginary-valued forms.
result Classifies Real line bundles with Real connections on manifolds with involution.
New smooth 2-group extensions from bundle gerbes on manifolds.
problem Classifying equivariant structures on bundle gerbes.
method Global approach to parallel transport, homotopy-coherent associated bundle construction.
result New models for the string group of compact simply-connected Lie groups.
Study Higgs bundles on smooth projective varieties and their restrictions to curves.
problem Interplay between Higgs bundles on smooth projective varieties and their restrictions to curves.
method Investigate the restriction map of Higgs bundles and study branes in moduli spaces.
result Interconnectedness of Higgs bundles and branes on smooth projective varieties and their restrictions.
New examples show not all homology fiber bundles are topological.
problem Disprove conjecture about homology fiber bundles.
method Construct flat, projective morphisms that are Z-homology fiber bundles. result Disprove conjecture about homology fiber bundles.
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…
Article studies symmetry in smooth vector bundles using advanced operations.
problem Symmetry phenomena in smooth vector bundles after two iterations of the normal functor.
method Developed theory of pullback and quotient for double vector bundles and morphisms, focusing on naturality of the normal functor.
result Expected symmetry is obtained through universal behavior and compatibility of operations.
In this paper we aim for a generalisation of the Steenrod Approximation Theorem from, concerning a smoothing procedure for sections in smooth locally trivial bundles. The generalisation is that we consider locally trivial smooth bundles with a possibly infinite-dimensional typical fibre. The main result states that a c…
Theory of 2-vector bundles for smooth manifolds developed.
problem Developing a comprehensive theory for 2-vector bundles over smooth manifolds.
method Based on bicategory of algebras, bimodules, and intertwiners; symmetric monoidal structures; classification via Cech cohomology.
result Unified framework for bundle gerbes and algebra bundles.
Smooth actions of the multiplicative monoid (R,⋅) of real numbers on manifolds lead to an alternative, and for some reasons simpler, definition of a vector bundle, a double vector bundle and related structures like a graded bundle [Grabowski and Rotkiewicz, J. Geom. Phys. 2011]. For these reasons it is n…
Smooth bundles with rough data maintain Hodge kernel isomorphism.
problem Maintaining Hodge kernel isomorphism for smooth bundles with non-smooth geometric data.
method Analyzing nilpotent differential operators and Hodge-Dirac-type operators under perturbations of geometric data.
result Kernels of Hodge-Dirac operators remain isomorphic under uniform perturbations of geometric data.
A subbundle of variable dimension inside the tangent bundle of a smooth manifold is called a smooth distribution if it is the pointwise span of a family of smooth vector fields. We prove that all such distributions are finitely generated, meaning that the family may be taken to be a finite collection. Further, we show …
New configuration space integrals show nontrivial formal smooth structures on 4-manifold bundles.
problem Disproving the 4-dimensional Smale conjecture by constructing nontrivial bundles.
method Defining new configuration space integrals relying on formal smooth structures.
result Discovering a generalized Miller-Morita-Mumford class obstructing formal smooth structures.
A nonassociative generalization of the principal fiber bundles with a smooth loop mapping on the fiber is presented. Our approach allows us to construct a new kind of gauge theories that involve higher ''nonassociative'' symmetries.
Analyzes structure of log smooth pairs when equality holds in Bogomolov-Gieseker inequality.
problem Analyzing log smooth pairs under equality in Bogomolov-Gieseker inequality.
method Examines structure when equality holds in the Bogomolov-Gieseker inequality for semistable logarithmic tangent bundle and canonical extension sheaf.
result Provides insights into the structure of log smooth pairs under specific conditions.
We consider one possible definition of a diffeological connection on a diffeological vector pseudo-bundle. It is different from the one proposed in [7] and is in fact simpler, since it is obtained by a straightforward adaption of the standard definition of a connection as an operator on the space of all smooth sections…
We prove that all smooth sphere bundles that admit fiberwise 1/4-pinched metrics are induced bundles of vector bundles, so their structure groups reduce from the diffeomorphism group of the sphere to the orthogonal group. This result implies the existence of many smooth n-sphere bundles over a k-sphere that do not supp…
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.
Study bundles over surfaces with specific fibers, determining characteristic numbers and obstructions.
problem Characterizing bundles over surfaces with highly connected fibers.
method Analyzing smooth and topological bundles, providing necessary and sufficient conditions, and computing characteristic numbers.
result Determine characteristic numbers and divisibility constraints on signatures and genera for bundles of this type.
Parallel transport of a connection in a smooth fibre bundle yields a functor from the path groupoid of the base manifold into a category that describes the fibres of the bundle. We characterize functors obtained like this by two notions we introduce: local trivializations and smooth descent data. This provides a way to…
New solutions found for Hull-Strominger system on torus bundles.
problem Finding solutions to the Hull-Strominger system with torus symmetry.
method Constructing solutions on torus bundles over K3 orbifolds.
result Smooth manifolds with complex structures have solutions to the Hull-Strominger system.
New geometric objects generalize Lie groupoids, with nontrivial tangent bundle properties.
problem Generalizing Lie groupoids to nonassociative structures.
method Introducing quasiloopoids and loopoids, proving properties of their tangent bundles, and reformulating discrete mechanics.
result Tangent bundles of loopoids are canonically loopoids, but cotangent bundles are not.
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…
Proves smooth torsion invariants match across methods.
problem Matching higher torsion invariants in smooth bundles.
method Explicit comparison of constructions by Igusa/Klein vs. homotopy theory by Badzioch et al.
result Higher torsion invariants agree across methods.
Classifies smooth manifolds homotopy equivalent to sphere products
problem Classifying smooth manifolds homotopy equivalent to sphere products
method Using normal-invariant map and explicit families of manifolds
result Classifies smooth manifolds up to almost diffeomorphism
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.
We prove vanishing results for the generalized Miller-Morita-Mumford classes of some smooth bundles whose fiber is a closed manifold that supports a nonpositively curved Riemannian metric. We also find, under some extra conditions, that the vertical tangent bundle is topologically rigid.
Let K be a a Lie group, modeled on a locally convex space, and M a finite-dimensional paracompact manifold with corners. We show that each continuous principal K-bundle over M is continuously equivalent to a smooth one and that two smooth principal K-bundles over M which are continuously equivalent are also smoothly eq…
The study classifies Kähler threefolds with special fiber bundles.
problem Classifying Kähler threefolds with specific fiber bundles.
method Generalized from projective case, using fiber bundles over the circle and étale covers.
result Proves Kotschick's conjecture in dimension 3.
New exotic 4-manifolds found with fiber bundles.
problem Finding non-diffeomorphic exotic 4-manifolds.
method Smooth fiber bundles over the circle.
result Exotic 4-manifolds with non-simply-connected fiber.
In this paper, we construct round fold maps or stable fold maps with concentric singular value sets introduced by the author on smooth bundles over spheres or bundles over more general manifolds. The class of round fold maps includes special generic maps on spheres and such maps have been constructed on smooth bundles …
Study Lie algebroid connections on principal bundles over complex projective varieties.
problem Existence and properties of Lie algebroid connections on principal bundles.
method Definition and study of Lie algebroid valued connections on holomorphic principal G-bundles, investigation of existence criteria.
result Investigation of criteria for existence of Lie algebroid connections on principal G-bundles over smooth complex projective curves.
Confirming operator characterization on smooth manifolds.
problem Characterizing an operator on smooth sections of tangent bundles.
method Using algebraic axioms and H^1(M, R) = {0} assumption.
result Operator can be characterized universally for any smooth manifold.
We recall and partially improve four versions of smooth, non-abelian gerbes: Cech cocycles, classifying maps, bundle gerbes, and principal 2-bundles. We prove that all these four versions are equivalent, and so establish new relations between interesting recent developments. Prominent partial results we prove are a bij…
This thesis predicts the distribution of smoothed zeros of random sections on line bundles.
problem Predicting the distribution of smoothed zeros of random sections on line bundles.
method Developing smoothing operators on discrete surfaces and computing the expected sum of indices on each face.
result Predictions on the distribution of smoothed section's signed zeros with multiplicity.
Solves Demailly's system for direct sums of ample line bundles on Riemann surfaces.
problem Proving the existence of smooth solutions for Demailly's system.
method Used Demailly's system and Leray-Schauder degree theory to reduce the problem.
result Proved existence of smooth solutions for direct sums of ample line bundles.
In this note we introduce the notion of a smooth structure on a conical pseudomanifold M in terms of C∞-rings of smooth functions on M. For a finitely generated smooth structure C∞(M) we introduce the notion of the Nash tangent bundle, the Zariski tangent bundle, the tangent bundle of M, and the …
Extends Gelfand duality to various geometric and analytical categories.
problem Generalizing Gelfand duality to different types of manifolds and bundles.
method Unified cohomological argument for manifolds and suitable classes of functions for bundles.
result Gelfand duality extended to real analytic and Stein manifolds, and to vector, affine, and jet bundles.
The paper explores smooth equivariant rigidity and finds infinitely many exotic smooth structures.
problem Smooth equivariant rigidity for certain group actions on manifolds.
method Analysis of G-vector bundles and Atiyah-Singer index theorem. result Infinitely many exotic smooth structures for certain group actions.