arXiv research

Study mapping class group action on de Rham quasimorphisms, finding no fixed points.

Study of de Rham cohomology on non-Hausdorff manifolds.

Study de Rham homomorphism for Lipschitz cohomologies on metric simplicial complexes.

We exhibit a cocycle in the simplicial de Rham complex which represents the Euler class. As an application, we construct a Lie algebra cocycle on $L\mathfrak{so}(4)$.

On the basis of A. L. Carey, D. Crowley, M. K. Murray's work, we exhibit a cocycle in the simplicial de Rham complex which represents the Dixmier-Douady class.

Proves cup product homomorphism for bounded cohomology on negatively curved manifolds.

We define quantum exterior product wedge_h and quantum exterior differential d_h on Poisson manifolds (of which symplectic manifolds are an important class of examples). Quantum de Rham cohomology, which is a deformation quantization of de Rham cohomology, is defined as the cohomology of d_h. We also define quantum Dol…

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…

Two de Rham complexes in diffeology are compared using a factor map.

We show a de Rham theory for cubical manifolds, and study rational homotopy type of the classifying spaces of smooth quandles. We also show that secondary characteristic classes in \cite{Dup2,DK} produce cocycles of quandles.

We define quantum exterior product wedge_h and quantum exterior differential d_h on Poisson manifolds, of which symplectic manifolds are an important class of examples. Quantum de Rham cohomology is defined as the cohomology of d_h. We also define quantum Dolbeault cohomology. Quantum hard Lefschetz theorem is proved. …

The paper explores de Rham theory for singular spaces and stacks.

New Lipschitz de Rham theorem for $L_p$-cohomology.

The paper proves formulas and theorems for statistical de Rham Hodge operators on manifolds with boundary.

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…

The paper studies perturbations of de Rham Hodge operators on manifolds with or without boundaries.

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…

The Hodge-de Rham Theorem is introduced and discussed. This result has implications for the general study of several partial differential equations. Some propositions which have applications to the proof of this theorem are used to study some related results concerning a class of partial differential equation in a nove…

We propose a produre of reduction a locally conformal symplectic structure. This procedure of reduction can be applied to wide class of submanifolds. There are no local obstructions for this procedure. But there are global obstructions. We find a necessary and sufficient condition when this reduction holds in terms of …

Study describes global sections of chiral de Rham complex on compact Ricci-flat Kähler manifolds.

De Rham theorem extended to Orlicz cohomology.

Given a compact stratified pseudomanifold with a Thom-Mather stratification and a class of riemannian metrics over its regular part, we study the relationships between the $L^{2}$ de Rham and Hodge cohomology and the intersection cohomology of $X$ associated to some perversities. More precisely, to a kind of metric whi…

Authors prove de Rham cohomology of Poisson and Jacobi manifolds is trivial.

New cohomology theory for diffeological spaces developed.

Develops fractional de Rham theory for Maxwell equations.

The Burde--de Rham theorem is extended to finitely presented pro-$p$ groups with specific conditions.

The paper generalizes current constructions to cohesive modules and characteristic forms.

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.

Promotes spectral functionals to noncommutative fields and proves a theorem.

The primitive cohomology of Calabi-Yau intersections is described using a twisted de Rham complex.

Paper defines spectral triple and computes functional for nonminimal de Rham-Hodge operator.

Study calculates global sections on complex curves.

The paper explores similarities in even and odd-dimensional geometry.

Researchers compute de Rham cohomology of geodesic flow foliations on hyperbolic surfaces.

When a Lie group $G$ has a central $U(1)$-extension, there is a cocycle in the simplicial de Rham complex $Ω^3(NG)$ which represents the Dixmier-Douady class. Mickelsson and Brylinski, McLaughlin constructed a central $U(1)$-extension $\widehat{LSU(2)} \rightarrow LSU(2)$ whose Dixmier-Douady class in $Ω^3(NLSU(2))$ is…

Study Hodge-de Rham numbers for almost complex 4-manifolds, extending properties from complex surfaces.

The original de Rham cohomology due to Souriau and the singular cohomology in diffeology are not isomorphic to each other in general. This manuscript introduces a singular de Rham complex endowed with an integration map into the singular cochain complex which gives the de Rham theorem for every diffeological space. It …

A De Rham model for string topology based on the theory of iterated integrals is presented.

We prove a vanishing theorem for the twisted de Rham cohomology of a compact manifold.

It is shown that the characteristic classes of foliations that were defined by Losik and that take values in the de~Rham cohomology of the space of infinite order frames over the leaf space may be mapped to the characteristic classes with values in the Čech-de~Rham cohomology of the leaf space studied in details by Cra…

Investigates Künneth formula for foliated de Rham cohomology, overcoming non-Hausdorff issues.

We derive a blow-up formula for the de Rham cohomology of a local system of complex vector spaces on a compact complex manifold. As an application, we obtain the blow-up invariance of $E_{1}$-degeneracy of the Hodge-de Rham spectral sequence associated to a local system of complex vector spaces.

In this paper, we study the uniqueness in the de Rham-Wu decomposition for pseudo-Riemannian manifolds.

Study cohomology of odd symplectic manifolds, linking to Lagrangian submanifolds and BV Laplacians.

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 $D_H$, which coincid…

We exhibit the Chern-Simons forms of some characteristic classes in the simplicial de Rham complex.

The study connects curves and cohomology on manifolds.

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…