  We present some ideas for a possible Noncommutative Topological Quantum Field
Theory (NCTQFT) and Noncommutative Floer Homology (NCFH). Our motivation is
two-fold and it comes both from physics and mathematics: On the one hand we
argue that NCTQFT is the correct mathematical framework for a quantum field
theory of all known interactions in nature (including gravity). On the other
hand we hope that a possible NCFH will apply to practically every 3-manifold
(and not only to homology 3-spheres as ordinary Floer Homology currently does).
The two motivations are closely related since, at least in the commutative
case, Floer Homology Groups constitute the space of quantum observables of
(3+1)-dim Topological Quantum Field Theory. Towards this goal we present some
"Noncommutative" Versions of Hodge Theory for noncommutative differentail forms
and tangential cohomology for foliations.
