Theorems of existence of local and global solutions of PDE's in the category of noncommutative quaternionic manifolds

S.Marchiafava, P.Piccinni and M.Pontecorvo (eds.), ElibM. European Math. Soc. (http://www.emis.de/proceedings/index.html). In this paper we apply our recent geometric theory of noncommuttive (quantum) manifolds and noncommutative (quantum) PDEs to the category of quantum quaternionic manifolds. These are manifolds modelled on spaces built starting from quaternionic algebras. For PDEs considered in such category we determine theorems of existence of local and global quaternionic solutions. We show also that such a category of quantum quaternionic manifolds properly contains that of manifolds with (almost) quaternionic structure. So our theorems of existence of quantum quaternionic manifolds for PDEs produce a cascade of new solutions with nontrivial topology.