(Co)bordisms in PDE's and quantum PDE's.

In this paper we announce some recent results on the quantum and integral (co)bordism in PDEs and quantum PDEs. We shall essentially prove that the tecnique of (co)bordism, introduced by Pontrjagin and Thom in algebraic topology, can be generalized in the framework of partial differential equations in order to obtain sufficient criteria that allow to decide when a $p$-dimensional compact closed integral manifold contained in a PDE $ E_k\subset J^k_n(W)$, is the boundary of a $ (p+1)$-dimensional integral compact manifold contained also in $ E_k$ (integral bordism) or eventually in the jet-space $ J^k_n(W)$ containing $ E_k$ (quantum bordism). Furthermore, we shall prove that such results can be extended to the category of quantum PDEs. Here, by the term "quantum manifold" (and as a consequence of "quantum PDEs") we mean a new structure that extends globally usual concepts of quantum spaces, and that is very useful for physical applications.