Cohomological Analysis of Partial Differential Equations and Secondary
Calculus

By:
Alexandre Vinogradov,
University of Salerno, Italy.
Translations of Mathematical Monographs,
vol. 204,
American Mathematical Society,
2001, 247 pages, ISBN: 0-8218-2922-X,
ISSN: 0065-9282.
Description

This book is dedicated to fundamentals of a new theory, which is an analog of
affine algebraic geometry for (nonlinear) partial differential equations. This
theory grew up from the classical geometry of PDE's originated by S. Lie and his
followers by incorporating some nonclassical ideas from the theory of integrable
systems, the formal theory of PDE's in its modern cohomological form given by D.
Spencer and H. Goldschmidt and differential calculus over commutative algebras
(Primary Calculus). The main result of this synthesis is Secondary Calculus on
diffieties, new geometrical objects which are analogs of algebraic varieties in
the context of (nonlinear) PDE's.

Secondary Calculus surprisingly reveals a deep cohomological nature of the
general theory of PDE's and indicates new directions of its further progress.
Recent developments in quantum field theory showed Secondary Calculus to be its
natural language, promising a nonperturbative formulation of the theory.

In addition to PDE's themselves, the author describes existing and potential
applications of Secondary Calculus ranging from algebraic geometry to field
theory, classical and quantum, including areas such as characteristic classes,
differential invariants, theory of geometric structures, variational calculus,
control theory, etc. This book, focused mainly on theoretical aspects, forms a
natural dipole with Symmetries and Conservation Laws
for Differential
Equations of Mathematical Physics, Volume 182 in this same series,
Translations of Mathematical Monographs, and shows the theory "in
action".

Contents

From symmetries of partial differential equations to Secondary Calculus

Elements of differential calculus in commutative algebras

Geometry of finite-order contact structures and the classical theory of
symmetries of partial differential equations

Geometry of infinitely prolonged differential equations and higher
symmetries