Quantum geometry of PDE's.

In this paper we present the formal quantization of PDE's, as introduced by A. Prastaro, is recast in categorial language. Formal quantization results as a canonical functor defined on the category of differential equations. Furthermore, a Dirac-quantization can be interpreted as a covering in the category of differential equations. A quantum (pre-)spectral measure is a functor that can be factorized by means of formal quantization and a (pre-)spectral measure. A relation between canonical Dirac-quantization and singular solutions of PDE's is given. It is also proved that the knowledge of B\"aklund correspondences, as well as the conservation laws, can aid the procedure of canonical quantization of PDE's. Physically interesting examples are considered. In particular, we give the canonical quantization of an anharmonic oscillator. A general theory of quantum tunneling effects in PDE's is given. In particular, quantum cobordism has been related with Leray-Serre spectral sequences of PDE's.