(CO)Bordisms in PDEs and quantum PDEs

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 technique 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 Ek ⊂ Jnk(W) is the boundary of a (p + 1)-dimensional integral compact manifold contained also in Ek (integral bordism) or eventually in the jet-space Jnk(W) containing Ek (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 also “quantum PDE”) we mean a new structure that extends globally usual concepts of quantum spaces, and that is very useful for physical applications.