Related papers: Bounded cohomology via partial differential equati…

We show that the exterior derivative operator on a symplectic manifold has a natural decomposition into two linear differential operators, analogous to the Dolbeault operators in complex geometry. These operators map primitive forms into…

We relate cohomology defined by a system of local Lagrangian with the cohomology class of the system of local variational Lie derivative, which is in turn a local variational problem; we show that the latter cohomology class is zero, since…

We give a definition of differentiable cohomology of a Lie group G (possibly infinite-dimensional) with coefficients in any abelian Lie group. This differentiable cohomology maps both to the cohomology of the group made discrete and to Lie…

We prove the existence of a sequence of commutative diagrams generalizing existing results on the cohomology of the Borel-Serre boundary and well-rounded retract to the context of the well-tempered complex. Our main theorem provides a…

We propose a theorem that extends the classical Lie approach to the case of fractional partial differential equations (fPDEs) of the Riemann--Liouville type in (1+1) dimensions.

We show that bounded cohomology stabilizes along sequences of classical Lie groups, and along sequences of lattices in them. Our method is based on a criterion from (arXiv:2307.12808) which adapts Quillen's stability method to the setting…

We derive a relative version of the local monodromy theorem for ordinary differential equations on an annulus over a mixed-characteristic nonarchimedean field, and give several applications in $p$-adic cohomology and $p$-adic Hodge theory.…

We provide base change theorems, projection formulae and Verdier duality for both cohomology and homology in the context of finite topological spaces

We identify the stochastic processes associated with one-sided fractional partial differential equations on a bounded domain with various boundary conditions. This is essential for modelling using spatial fractional derivatives. We show…

We demonstrate how a simple linear-algebraic technique used earlier to compute low-degree cohomology of current Lie algebras, can be utilized to compute other kinds of structures on such Lie algebras, and discuss further generalizations,…

Complexes of discrete distributional differential forms are introduced into finite element exterior calculus. Thus we generalize a notion of Braess and Sch\"oberl, originally studied for a posteriori error estimation. We construct…

This paper establishes a second vanishing theorem for formal local cohomology modules over Noetherian local rings. We introduce the \textit{formal dimension} invariant and characterize the vanishing of higher formal local cohomology in…

In this thesis, we introduce a new cohomology theory associated to a Lie 2-algebras and a new cohomology theory associated to a Lie 2-group. These cohomology theories are shown to extend the classical cohomology theories of Lie algebras and…

After a self-contained introduction to Lie algebra cohomology, we present some recent applications in mathematics and in physics. Contents: 1. Preliminaries: L_X, i_X, d 2. Elementary differential geometry on Lie groups 3. Lie algebra…

In this paper, we investigate analytical and geometric properties of certain non-compact boundary-manifolds, namely manifolds of bounded geometry. One result are strong Bochner type vanishing results for the L^2-cohomology of these…

If X is a complex projective variety with klt singularities, then the mixed Hodge structures on the first two singular cohomology groups are pure. We describe the pieces of the Hodge decomposition in terms of reflexive differential forms.…

Derived differential manifolds are constructed using the usual homotopy theory of simplicial rings of smooth functions. They are proved to be equivalent to derived differential manifolds of finite type, constructed using homotopy sheaves of…

We provide a general theoretical framework allowing us to extend the classical Lie theory for partial differential equations to the case of equations of fractional order. We propose a general prolongation formula for the study of Lie…

It is shown that the BRST resolution of the spaces of physical states of the systems with anomalies can be consistently defined. The appropriate anomalous complexes are obtained by canonical restrictions of the ghost extended spaces to the…

We show that finite-dimensional Lie algebras over a field of characteristic zero such that their high-degree cohomology in any finite-dimensional non-trivial irreducible module vanishes, are, essentially, direct sums of semisimple and…