Among the topics are genus change in inseparable extensions of functional fields, the homology of noetherian rings and local rings, the

cohomology groups of tori in infinite Galois extensions of number fields, an algorithm for determining the type of a singular fiber in an elliptic pencil, variation of the canonical height of a point depending on a parameter, the non-existence of certain Galois extensions of Q unramified outside two, and refining Gross' conjecture on the values of abelian L-functions.

Basic forms are preserved by the exterior derivative and are used to define basic de-Rham

cohomology groups [H*.

The endomorphism algebra of a tilting module preserves many significant invariants, for example, the center of an algebra, the number of nonisomorphic simple modules, the Hochschild

cohomology groups, and Cartan determinants.

Knowing that the coherent cohomology of polydisks vanishes also opens the road towards computing global

cohomology groups for projective analytic spaces over ring of integers of number fields.

In [1], we studied Lie-Rinehart cohomology of singularities, and we gave an interpretation of these

cohomology groups in terms of integrable connections on modules of rank one defined on the given singularities.

The orbicycle index polynomial can be use to compute the even dimensions of the orbifold

cohomology groups for global orbifolds of the form [M.

It surveys several algebraic invariants: the fundamental group, singular and Cech homology groups, and a variety of

cohomology groups.

N,S]) consisting of those homomorphisms which induce isomorphisms between the respective Tate

cohomology groups of [X.

1](G) is amenable as a Banach algebra; that is, the first

cohomology groups [H.

Here, a topological module V over a topological ring R is said to have vanishing U-cohomology, where U is a topological group which acts continuously and faithfully on V, if the

cohomology groups [H.

Specific topics include the signed mean curvature measure, discrete holomorphicity and Ising model operator formalism, scaling asymptotics of heat kernels of line bundles, the topology of Dolbeault

cohomology groups, and the point source inverse back-scatter problem.

Their topics include moduli spaces of twisted sheaves on a projective variety, integral Hodge classes on uni-ruled or Calabi-Yau threefolds, birational geometry of symplectic resolutions of nilpotent orbits, moduli stacks of second-rank Giesker bundles with a fixed determinate on a nodal curve, vector bundles on curves and theta functions, Abelian varieties with bounded modular height, the moduli of regular holonomic Dx-modules with natural parabolic stability,

cohomology groups of stable quasi-Abelian degenerations, semi-stable extensions on arithmetic surfaces, cusp form motives, polarized K3 surfaces, rigid geometry and applications, and moduli of stable parabolic conventions with Riemann-Hilbert correspondence and other features.