arXiv research

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

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.

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

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

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…

The paper extends Hodge-de Rham theory to higher-dimensional Sierpinski gaskets.

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.

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…

This is a survey on the equivariant cohomology of Lie group actions on manifolds, from the point of view of de Rham theory. Emphasis is put on the notion of equivariant formality, as well as on applications to ordinary cohomology and to fixed points.

Study on Čech-de Rham obstruction in diffeological spaces.

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…

These lecture notes in the De Rham-Hodge theory are designed for a 1-semester undergraduate course (in mathematics, physics, engineering, chemistry or biology). This landmark theory of the 20th Century mathematics gives a rigorous foundation to modern field and gauge theories in physics, engineering and physiology. The…

Local index density of perturbed de Rham complex is invariant under certain conditions.

We study the de Rham complex on a smooth manifold with a periodic end modeled on an infinite cyclic cover X' \to X. The completion of this complex in exponentially weighted L^2-norms is Fredholm for all but finitely many exceptional weights determined by the eigenvalues of the covering translation map H_*(X') \to H_*(X…

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

Novel Hodge theory developed for Carrollian bundles, addressing black hole event horizons.

Explains Hodge theory and Kodaira embedding theorem for complex manifolds.

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…

Research resolves sign conventions in Floer theory for Morse-Bott case.

Goresky, Kottwitz and MacPherson have recently shown that the computation of the equivariant cohomology ring of a G-manifold can be reduced to a computation in graph theory. This opens up the possibility that many of the fundamental theorems in equivariant de Rham theory may, on closer inspection, turn out simply to be…

New cohomology theory for diffeological spaces developed.

Develops fractional de Rham theory for Maxwell equations.

Study of de Rham cohomology on non-Hausdorff manifolds.

Let G be a general (not necessarily finite dimensional compact) Lie group, let g be its Lie algebra, let Cg be the cone on g in the category of differential graded Lie algebras, and consider the functor which assigns to a chain complex V the V-valued total de Rham complex of G. We describe the G-equivariant de Rham coh…

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…

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

A measured solenoid is a compact laminated space endowed with a transversal measure. The De Rham $L^2$-cohomology of the solenoid is defined by using differential forms which are smooth in the leafwise directions and $L^2$ in the transversal direction. We develop the theory of harmonic forms for Riemannian measured sol…

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

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…

Recent years have witnessed a trend that advanced mathematical tools, such as algebraic topology, differential geometry, graph theory, and partial differential equations, have been developed for describing biological macromolecules. These tools have considerably strengthened our ability to understand the molecular mech…

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

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. …

Graph theory connects automorphisms to cohomology.

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

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

We develop a generalization to non-Witt spaces of the intersection homology theory of Goresky-MacPherson. The second author has described the self-dual sheaves compatible with intersection homology, and the other authors have described a generalization of Cheeger's L2 de Rham cohomology. In this paper we extend both of…

De Rham theorem extended to Orlicz cohomology.

We prove that Chern-Weil forms are the only natural differential forms associated to a connection on a principal G-bundle. We use the homotopy theory of simplicial sheaves on smooth manifolds to formulate the theorem and set up the proof. Other arguments come from classical invariant theory. We identify the Weil algebr…

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

Introduces a framework for rational homotopy theory in diffeological spaces.

We use Hodge theory and a construction of Merkulov to construct $A_{\infty}$ structures on de Rham cohomology and Dolbeault cohomology.

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

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.

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