"Derivations, cohomology groups and second duals of Banach algebras"

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We learn in a high school (or University) first calculus course that if f and g are two functions whose derivatives D(f) and D(g) exist, then the product function fg also has a derivative D(fg), which satisfies D(fg) = D(f)g + fD(g). In higher Mathematics there are structures comprised of elements-often called vectors-which can be added or multiplied, but, unlike the multiplication of functions, the multiplication of vectors of these structures, in general, may fail to satisfy fg=gf. Let's think of differentiation as being a mapping that assigns to every differentiable function another function (its derivative D(f)) subject to differentiation rules. There are similar mappings D on abstract structures that assign to each vector f in that structure a vector D(f) in the same or another related structure, subject to preservation of addition and multiplication by numbers, and act on the product of two vectors in the same fashion that differentiation does, i.e. D(fg) = D(f)g + fD(g). These mappings are called derivations. One of the simplest derivations is given by the formula D(f) = fg - gf. Here g is a fixed vector and f varies among the vectors of the given structure that D acts upon. These derivations are called inner derivations. We propose to find formulas for derivations on certain structures and to study those abstract structures that have the property that all the derivations into certain related structures are inner or can be approximated by inner derivations. These structures occupy a significant place in research in an area of Mathematics called Banach Algebras. The study of derivations originates from Physics and my research (like every other pure mathematician's research) is theoretical and primarily geared at advancement of knowledge; it may later find its use in applied sciences or industry.