  We develop some of the ingredients needed for string theory on noncommutative
spacetimes, proposing an axiomatic formulation of T-duality as well as
establishing a very general formula for D-brane charges. This formula is
closely related to a noncommutative Grothendieck-Riemann-Roch theorem that is
proved here. Our approach relies on a very general form of Poincare duality,
which is studied here in detail. Among the technical tools employed are
calculations with iterated products in bivariant K-theory and cyclic theory,
which are simplified using a novel diagram calculus reminiscent of Feynman
diagrams.

High Energy Physics - Theory;K-Theory and Homology