On the exotic De-Rham cohomology. Perturbation theory as a spectral sequence

图上的德拉姆上同调与自同构群

Graph de Rham Cohomology and the Automorphsim Group

该研究将Aut(Γ)的诱导作用与有限图的离散德拉姆上同调关联，证明微分几何相关结果的图论类似结论，拓展了图论与微分几何的交叉研究。

Multivalued functionals, one-forms and deformed de Rham complex

We discuss some applications of the Morse-Novikov theory to some problems in modern physics, where appears a non-exact closed 1-form $ω$ (a multi-valued functional). We focus mainly our attention to the cohomology of the de Rham complex of a compact manifold $M^n$ with a deformed differential $d_ω=d +λω$. Using Witten's approach to the Morse theory one can estimate the number of critical points of $ω$ in terms of the cohomology of deformed de Rham complex with sufficiently large values of $λ$ (torsion-free Novikov's inequalities). We show that for an interesting class of solvmanifolds this coh…

扭曲 de Rham 上同调上的高阶留数与典范配对

Higher residues and canonical pairing on the twisted de Rham cohomology

本文通过给出扭曲 de Rham 上同调上典范配对的显式公式，并建立其与 Saito 高阶留数配对的关系，从而重新证明了 Shklyarov 猜想。

Cech and de Rham Cohomology of Integral Forms

We present a study on the integral forms and their Cech/de Rham cohomology. We analyze the problem from a general perspective of sheaf theory and we explore examples in superprojective manifolds. Integral forms are fundamental in the theory of integration in supermanifolds. One can define the integral forms introducing a new sheaf containing, among other objects, the new basic forms delta(dtheta) where the symbol delta has the usual formal properties of Dirac's delta distribution and acts on functions and forms as a Dirac measure. They satisfy in addition some new relations on the sheaf. It tu…

Optimal space of linear classical observables for Maxwell $k$-forms via spacelike and timelike compact de Rham cohomologies

Being motivated by open questions in gauge field theories, we consider non-standard de Rham cohomology groups for timelike compact and spacelike compact support systems. These cohomology groups are shown to be isomorphic respectively to the usual de Rham cohomology of a spacelike Cauchy surface and its counterpart with compact support. Furthermore, an analog of the usual Poincaré duality for de Rham cohomology is shown to hold for the case with non-standard supports as well. We apply these results to find optimal spaces of linear observables for analogs of arbitrary degree $k$ of both the vect…

De Rham cohomology of configuration spaces with Poisson measure

The space $Γ_X$ of all locally finite configurations in a Riemannian manifold $X$ of infinite volume is considered. The deRham complex of square-integrable differential forms over $Γ_X$, equipped with the Poisson measure, and the corresponding deRham cohomology are studied. The latter is shown to be unitarily isomorphic to a certain Hilbert tensor algebra generated by the $L^2$-cohomology of the underlying manifold $X$.