Title: Surviving without the shape functions

Abstract: The use of $0$-Cochains in the approximation of boundary value problems
allows the use of elements (in 2 and in 3 dimensions) of very general shape, leaving
the patch-test intact and hence, somehow, ensuring first order convergence. In the
construction of the corresponding stiffness matrix the procedure allows a certain
amaount of freedom. The best use of this freedom is still under question.
The lecture will present the result of a recent paper (joint work with A. Buffa
and K. Lipnikov) and discuss the use of $0$-Cochains in the general frameworks of
cochain approximation and of Mimetic Finite Differences. A usuful survey of
"what is known and what is not known, what has been proved, and what has been
conjectured" will also be presented.