(Co)bordism groups in quantum PDE's

A theory of noncommutative manifolds (\textit{quantum manifolds}) is formulated, and for such manifolds a geometric theory of quantum PDEs (QPDEs) is formulated. In particular, a criterion of formal integrability is given that extends to QPDEs previous one given by H. Goldschmidt for PDEs and by A. Prastaro for super PDEs. A general theory of integral (co)bordism for QPDEs is developed, that extends previous one for PDEs formulated by A. Prastaro. Then, non-commutative Hopf algebras, (\textit{full quantum $ p$-Hopf algebras, $ 0\le p\le m-1$}), are canonically associated to any QPDE whose elements represent all the possible invariants that can be recognized for such a structure. Many examples of QPDEs are considered where we apply our theory. In particular, applications to quantum supergravity are considered. Existence of (quantum) tunnel effects for quantum superstrings in supergravity is proved.