The paper explores de Rham theory for singular spaces and stacks.
problem Identifying de Rham theory for singular differentiable spaces.
method Identifying two potential answers and studying them, including the exterior algebra of the cotangent complex and de Rham stacks.
result There exists a version of the de Rham theorem for singular differentiable spaces with almost no restrictions.
New cochain algebra for diffeological spaces connects de Rham and singular cohomologies.
problem Incompatibility of de Rham and singular cohomologies in diffeology.
method Introduces a new singular de Rham complex and proves it quasi-isomorphic to the original de Rham complex for manifolds and spaces with singularities.
result The new cochain complex resolves the incompatibility issue in diffeology.
New definition of de Rham spaces changes quotient structure.
problem Traditional de Rham spaces defined using nilpotents.
method Uses infinity nilpotent elements for de Rham spaces.
result De Rham spaces are quotients by germs of diagonals.
Study de Rham theory for cubical manifolds and quandles.
problem Rational homotopy type of classifying spaces of smooth quandles.
method Show de Rham theory for cubical manifolds and study secondary characteristic classes.
result Secondary characteristic classes produce cocycles of quandles.
Study mapping class group action on de Rham quasimorphisms, finding no fixed points.
problem Action of mapping class group on de Rham quasimorphisms.
method Examined the action of mapping class group on de Rham classes in bounded cohomology of a hyperbolic surface.
result No fixed points in the action of mapping class group on de Rham quasimorphisms.
Study calculates global sections on complex curves.
problem Global sections of chiral de Rham complexes on complex curves.
method Calculation on closed complex curves with genus g ≥ 2.
result Space of global sections determined.
We extend the study of the de Rham operator with ideal boundary conditions from the case of isolated conic singularities, as analyzed by Cheeger, to the case of arbitrary stratified pseudomanifolds. We introduce a class of ideal boundary operators and the notion of mezzoperversity, which intermediates between the stand…
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.
Two de Rham complexes in diffeology are compared using a factor map.
problem Comparing two de Rham complexes in diffeology.
method Using a factor map to connect the two de Rham complexes and Čech--de Rham spectral sequence.
result Singular de Rham cohomology of irrational torus is isomorphic to tensor product of original de Rham cohomology and exterior algebra.
Study calculates global sections on special geometric spaces.
problem Calculating global sections on compact Ricci-flat Kähler manifolds.
method Expressed as an invariant subspace of a βγ-bc system under Lie algebra action.
result Space of global sections is an invariant subspace.
Formula derived for cohomology of local systems on complex manifolds.
problem Cohomology of local systems on compact complex manifolds.
method Derive a blow-up formula for de Rham cohomology of local systems.
result Blow-up invariance of E1-degeneracy of Hodge-de Rham spectral sequence. We show that the de Rham theorem, interpreted as the isomorphism between distributional de Rham cohomology and simplicial homology in the dual dimension for a simplicial decomposition of a compact oriented manifold, is a straightforward consequence of elementary properties of currents. The explicit construction of this…
Study shows gap between de Rham and symplectic-Bott-Chern harmonic forms for specific almost-Kähler manifolds.
problem Understanding the gap between de Rham and symplectic-Bott-Chern harmonic forms on specific almost-Kähler manifolds.
method Analyzing the space of de Rham harmonic forms and symplectic-Bott-Chern harmonic forms on closed almost-Kähler manifolds.
result The second non-HLC degree measures the gap between de Rham and symplectic-Bott-Chern harmonic forms.
A classical result in differential geometry states that for a free and proper Lie group action, the quotient map to the orbit space induces an isomorphism between the de Rham complex of differential forms on the orbit space and the basic differential forms on the original manifold. In this paper, this result is general…
Differentiable groupoids and their inertia spaces are studied with a de Rham theorem.
problem Understanding the de Rham cohomology of inertia spaces.
method Introducing differentiable stratified groupoids and proving a de Rham theorem.
result Inertia groupoids of proper Lie groupoids are locally contractible differentiable stratified groupoids.
Defines De Rham operator on diffeological spaces, showing it's unique.
problem No straightforward counterpart of De Rham operator on diffeological spaces.
method Definition based on Levi-Civita connection and Clifford action.
result Only way to define De Rham operator on diffeological spaces.
Study on Čech-de Rham obstruction in diffeological spaces.
problem Obstruction to Čech-de Rham map being an isomorphism in diffeological spaces.
method Higher topos theory, homotopy pullback diagrams, Čech-de Rham bicomplex, ∞-stack cohomology. result New exact sequences in all higher degrees and conceptual proof of cohomology agreement.
The purpose of this paper is to present a ``Cech-De Rham'' model for the cohomology of leaf spaces. This model lends itself to the construction of characteristic classes (in the cohomology of classifying spaces) by explicit geometrical constructions which are immediate extensions of the standard constructions for manif…
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.
Using the language and terminology of relative homological algebra, in particular that of derived functors, we introduce equivariant cohomology over a general Lie-Rinehart algebra and equivariant de Rham cohomology over a locally trivial Lie groupoid in terms of suitably defined monads (also known as triples) and the a…
New Lipschitz de Rham theorem for Lp-cohomology.
problem Developing a new de Rham theorem for Lp-cohomology. method Regularization procedure in Lipschitz de Rham calculus applied to metric simplicial complexes.
result Established Lipschitz de Rham theorem for Lp-cohomology. Extended de Rham theorem for manifolds with boundaries.
problem Applying de Rham decomposition to manifolds with boundaries.
method Using development of curves to extend classical theorem.
result Extended de Rham theorem for manifolds with boundaries.
The chiral de Rham complex of Malikov, Schechtman, and Vaintrob, is a sheaf of differential graded vertex algebras that exists on any smooth manifold Z, and contains the ordinary de Rham complex at weight zero. Given a closed 3-form H on Z, we construct the twisted chiral de Rham differential DH, which coincid…
The paper proves formulas and theorems for statistical de Rham Hodge operators on manifolds with boundary.
problem Formulating Lichnerowicz type formulas and Kastler-Kalau-Walze theorems for statistical de Rham Hodge operators.
method Developed Lichnerowicz type formulas and proved Kastler-Kalau-Walze type theorems for statistical de Rham Hodge operators on compact manifolds with boundary.
result Proved Lichnerowicz type formulas and Kastler-Kalau-Walze type theorems for statistical de Rham Hodge operators on compact manifolds with boundary.
The paper studies perturbations of de Rham Hodge operators on manifolds with or without boundaries.
problem Analyzing perturbations of de Rham Hodge operators on manifolds with boundaries.
method Lichnerowicz type formulas and Kastler-Kalau-Walze type theorems for 4D and 6D compact manifolds with or without boundaries.
result Proves Kastler-Kalau-Walze type theorems for perturbations of de Rham Hodge operators on 4D and 6D manifolds with or without boundaries.
New cohomology theory for diffeological spaces developed.
problem Cohomology of diffeological spaces.
method Diffeological Čech cohomology theory.
result Established connections between diffeological Čech cohomology and de Rham cohomology.
We prove an analogue of the de Rham theorem for the extended L^2-cohomology introduced by M. Farber. This is done by establishing that the de Rham complex over a compact closed manifold with coefficients in a flat Hilbert bundle E of A-modules over a finite von Neumann algebra A is chain-homotopy equivalent (with bound…
Proves Mayer-Vietoris sequence for diffeological spaces using generating families.
problem No specific problem stated; focuses on extending a sequence.
method Uses generating families instead of coverings in diffeological spaces.
result Proves a Mayer-Vietoris sequence for diffeological spaces.
New A∞ structures derived from equivariant de Rham complex for S1-action.
problem Deriving new A∞ structures from equivariant de Rham complex. method Applying Witten's deformation and homological perturbation to get new A∞ structures; extending and proving Fukaya's conjecture. result Proved Fukaya's conjecture relating Witten's deformed equivariant de Rham complexes to new Morse theoretical A∞ complexes. Study de Rham homomorphism for Lipschitz cohomologies on metric simplicial complexes.
problem Triviality of de Rham homomorphism kernel and non-increasing monotonicity of parameters.
method Regularization in Lipschitz de Rham calculus on metric simplicial complexes with bounded geometry.
result Explicit specification of non-trivial cohomology classes for a sequence of parameters.
New classification for certain 4-manifolds using quasiregular mappings.
problem Classifying closed simply connected quasiregularly elliptic 4-manifolds.
method Using quasiregular mappings and de Rham cohomology.
result Closed simply connected quasiregularly elliptic 4-manifolds are homeomorphic.
New projection operators for multipatch spaces with stable properties.
problem Problems with non-matching interfaces in multipatch spaces.
method Construction of commuting projection operators on de Rham sequences of multipatch spaces with local tensor-product parametrization.
result Local and stable projection operators in any Lp norm for shape-regular spline patches with different mappings and local refinements. Study the structure of Kähler foliations with negative Ricci curvature.
problem Characterize the structure of Kähler foliations with negative Ricci curvature.
method Prove a de Rham type theorem decomposition on the leaf space.
result Characterize each factor in the decomposition of the leaf space.
Study describes global sections of chiral de Rham complex on compact Ricci-flat Kähler manifolds.
problem None explicitly stated; focuses on description of global sections.
method Complete description of vertex algebra of global sections.
result Complete description of vertex algebra of global sections of chiral de Rham complex.
De Rham theorem extended to Orlicz cohomology.
problem Extending de Rham's theorem to a broader class of cohomology.
method Proving isomorphism between de Rham Lφ-cohomology and simplicial ℓφ-cohomology. result Isomorphism between de Rham Lφ-cohomology and simplicial ℓφ-cohomology. The paper extends Hodge-de Rham theory to higher-dimensional Sierpinski gaskets.
problem Analyzing differential forms and Laplacians on higher-dimensional fractal structures.
method Constructing sequences of graphs approximating Sierpinski gaskets, defining k-forms, de Rham derivatives, and their duals, proving harmonic properties, and exploring 2-forms.
result Obtained a basis for the space of harmonic 1-forms on level-3 Sierpinski gasket.
Authors prove de Rham cohomology of Poisson and Jacobi manifolds is trivial.
problem Understanding algebraic structures on de Rham cohomology of Poisson and Jacobi manifolds.
method Using DG operads and quasi-isomorphisms, they show the de Rham cohomology structure is trivial.
result The de Rham cohomology of Poisson and Jacobi manifolds has no higher structure beyond commutativity.
Unveils fermions' geometric nature in the Standard Model as noncommutative forms.
problem Understanding the geometric structure of fermions in the Standard Model.
method Uses noncommutative geometry to represent fermion multiplet as de Rham forms.
result Fermions in the Standard Model are represented as noncommutative de Rham forms.
Develops fractional de Rham theory for Maxwell equations.
problem Formulating fractional calculus for Maxwell equations.
method Fractional tangent functionals, Riemann-Liouville integral, polynomial algebra, exterior algebra.
result Fractional de Rham complex for Maxwell equations.
Study proves formulas linking curvature to de Rham cohomology of foliations.
problem Understanding de Rham cohomology of totally geodesic foliations.
method Proved transverse Weitzenböck identities for horizontal Laplacians.
result Nullity theorems for de Rham cohomology under curvature positivity.
New proof of Wolpert's Magic Formula using de Rham cohomology.
problem Proving Wolpert's Magic Formula for Teichmüller space.
method Using de Rham cohomology and Goldman's symplectic form.
result Derive a new proof of Wolpert's Magic Formula.
The Burde--de Rham theorem is extended to finitely presented pro-p groups with specific conditions.
problem Extending the Burde--de Rham theorem to pro-p groups with certain constraints. method Assumption of total degrees of relators being 0, concrete examples, and cohomological interpretations.
result The theorem is extended to finitely presented pro-p groups under specified conditions. Promotes spectral functionals to noncommutative fields and proves a theorem.
problem Spectral functionals on noncommutative fields and manifolds with boundary.
method Carries out promotion to spectral functionals, associates with noncommutative residue, and proves theorem.
result Proves Dabrowski-Sitarz-Zalecki type theorem for statistical de Rham Hodge operators on manifolds with boundary.
The primitive cohomology of Calabi-Yau intersections is described using a twisted de Rham complex.
problem Describing the primitive cohomology of Calabi-Yau intersections.
method Using a twisted de Rham complex and formal flat F-manifold structures.
result Constructs formal flat F-manifold structures on the primitive cohomology of Calabi-Yau intersections.
This paper extends de Rham theory of smooth manifolds to exploded manifolds. Included are versions of Stokes' theorem, De Rham cohomology, Poincare duality, and integration along the fiber. The resulting cohomology theory is used to define Gromov Witten invariants of exploded manifolds in a separate paper.
Paper defines spectral triple and computes functional for nonminimal de Rham-Hodge operator.
problem Computing spectral functions for nonminimal de Rham-Hodge operators.
method Definitions and computations of spectral triple and functional.
result Computed spectral Einstein functional for even-dimensional compact manifolds.
Researchers compute de Rham cohomology of geodesic flow foliations on hyperbolic surfaces.
problem Answering a problem posed by Haefliger and Li about geodesic flow foliations.
method Unitary representation theory of PSL(2, R) and Hodge decompositions of de Rham complexes.
result Computed de Rham cohomology of weak stable foliations for various coefficients.
The paper finds Chern-Simons forms for specific classes in simplicial de Rham complex.
problem None explicitly stated; focuses on finding forms.
method Exhibiting Chern-Simons forms of characteristic classes in simplicial de Rham complex.
result Chern-Simons forms for specific characteristic classes identified.