**Translations of Mathematical Monographs**

2001;
247 pp;
Hardcover

MSC: Primary 35; 37;
Secondary 58

**Print ISBN: 978-0-8218-2922-6
Product Code: MMONO/204**

List Price: $122.00

AMS Member Price: $97.60

MAA Member Price: $109.80

**Electronic ISBN: 978-1-4704-4629-1
Product Code: MMONO/204.E**

List Price: $115.00

AMS Member Price: $92.00

MAA Member Price: $103.50

# Cohomological Analysis of Partial Differential Equations and Secondary Calculus

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*A. M. Vinogradov*

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".

#### Readership

Graduate students and research mathematicians interested in all areas of mathematics where nonlinear PDE's are used and studied, including algebraic and differential geometry and topology, variational calculus and control theory, mechanics of continua, mathematical and theoretical physics.

#### Table of Contents

# Table of Contents

## Cohomological Analysis of Partial Differential Equations and Secondary Calculus

- Cover Cover11
- Title page iii4
- Contents v6
- Foreword vii8
- Introduction ix10
- From symmetries of partial differential equations to Secondary Calculus 118
- Elements of differential calculus in commutative algebras 2542
- Geometry of finite-order contact structures and the classical theory of symmetries of partial differential equations 5774
- Geometry of infinitely prolonged differential equations and higher symmetries 95112
- 𝒞-spectral sequence and some applications 127144
- Introduction to Secondary Calculus 189206
- Bibliography 237254
- Index 243260
- Back Cover Back Cover1265