Agostino Prastaro – författare

This book presents, for the first time, a systematic formulation of the geometric theory of noncommutative PDE's which is suitable enough to be used for a mathematical description of quantum dynamics and quantum field theory. A geometric theory of supersymmetric quantum PDE's is also considered, in order to describe quantum supergravity. Covariant and canonical quantizations of (super) PDE's are shown to be founded on the geometric theory of PDE's and to produce quantum (super) PDE's by means of functors from the category of commutative (super) PDE's to the category of quantum (super) PDE's. Global properties of solutions to (super) (commutative) PDE's are obtained by means of their integral bordism groups.

This book emphasizes the interdisciplinary interaction in problems involving geometry and partial differential equations. It provides an attempt to follow certain threads that interconnect various approaches in the geometric applications and influence of partial differential equations. A few such approaches include: Morse-Palais-Smale theory in global variational calculus, general methods to obtain conservation laws for PDEs, structural investigation for the understanding of the meaning of quantum geometry in PDEs, extensions to super PDEs (formulated in the category of supermanifolds) of the geometrical methods just introduced for PDEs and the harmonic theory which proved to be very important especially after the appearance of the Atiyah-Singer index theorem, which provides a link between geometry and topology.