arXiv research

Study quantifies geometric complexity of connections on product surfaces.

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

Study reveals a link between Ruelle-Pollicott resonances and cohomology eigenvalues for Anosov diffeomorphisms.

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.

The purpose of this work is to propose a mixed Hodge structure over a CR manifold. As you know, for a CR manifold, Kohn-Rossi cohomology is naturally introduced. However, the relation between Kohn-Rossi cohomology and De Rham cohomology is not so well understood, even in Tanaka's work. We discuss this point.

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…

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.

There are two de Rham complexes in diffeology. The original one is due to Souriau and the other one is the singular de Rham complex defined by a simplicial differential graded algebra. We compare the first de Rham cohomology groups of the two complexes within the Čech--de Rham spectral sequence by making use of the {\i…

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.

Develops a SageMath framework for computing characteristic classes.

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.

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

Develops fractional de Rham theory for Maxwell equations.

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 paper generalizes current constructions to cohesive modules and characteristic forms.

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.

This paper generalizes L2 cohomology theory for complex manifolds.

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.