D. Bennequin and P. Baudot introduced a cohomological construction adapted to information theory, called ”information cohomology” (see ”The homological nature of Entropy”, 2015). Our text serves as a detailed introduction to information cohomology, containing the necessary background in probability theory and homological algebra. It makes explicit the link with topos theory, as introduced by Gr...

The definition and properties of the Euler-Lagrange cohomology groups H EL , 1 6 k 6 n, on a symplectic manifold (M2n, ω) are given and studied. For k = 1 and k = n, they are isomorphic to the corresponding de Rham cohomology groups H1 dR(M 2n) and H dR (M 2n), respectively. The other Euler-Lagrange cohomology groups are different from either the de Rham cohomology groups or the harmonic cohomo...

Using local cohomology and algebraic D-Modules, we generalize a comparison theorem between relative de Rham cohomology and Dwork cohomology due to N. Katz, A. Adolphson and S. Sperber.

In this note we describe aspects of the cohomology of coherent sheaves on a complete toric variety X over a field k and, more generally, the local cohomology, with supports in a monomial ideal, of a finitely generated module over a polynomial ring S. This leads to an efficient way of computing such cohomology, for which we give explicit algorithms. The problem is finiteness. The ith local cohom...

We relate the cohomology of the Orlik-Solomon algebra of a discriminantal arrangement to the local system cohomology of the complement. The Orlik-Solomon algebra of such an arrangement (viewed as a complex) is shown to be a linear approximation of a complex arising from the fundamental group of the complement, the cohomology of which is isomorphic to that of the complement with coefficients in ...

Let f be a real-valued real analytic function in several variables. Then we associate with each algebraic local cohomology class u with support in f = 0 a distribution (generalized function) ρ(u) in terms of the residue of fλ + with respect to λ at a negative integer. Then ρ constitutes a homomorphism of modules over the sheaf of analtyic functions but not over the sheaf of differential operato...

In this paper we consider the cohomology of a closed arithmetic hyperbolic 3-manifold with coefficients in the local system defined by the even symmetric powers of the standard representation of SL.2;C/. The cohomology is defined over the integers and is a finite abelian group. We show that the order of the 2nd cohomology grows exponentially as the local system grows. We also consider the twist...

Hecke operators play an important role in the theory of automorphic forms, and automorphic forms are closely linked to various cohomology groups. This paper is mostly a survey of Hecke operators acting on certain types of cohomology groups. The class of cohomology on which Hecke operators are introduced includes the group cohomology of discrete subgroups of a semisimple Lie group, the de Rham c...