Hardcover ISBN:  9780821829226 
Product Code:  MMONO/204 
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Electronic ISBN:  9781470446291 
Product Code:  MMONO/204.E 
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Book DetailsTranslations of Mathematical MonographsVolume: 204; 2001; 247 ppMSC: Primary 35; 37; Secondary 58;
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".ReadershipGraduate 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

Chapters

From symmetries of partial differential equations to Secondary Calculus

Elements of differential calculus in commutative algebras

Geometry of finiteorder contact structures and the classical theory of symmetries of partial differential equations

Geometry of infinitely prolonged differential equations and higher symmetries

$\mathcal {C}$spectral sequence and some applications

Introduction to Secondary Calculus


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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".
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.

Chapters

From symmetries of partial differential equations to Secondary Calculus

Elements of differential calculus in commutative algebras

Geometry of finiteorder contact structures and the classical theory of symmetries of partial differential equations

Geometry of infinitely prolonged differential equations and higher symmetries

$\mathcal {C}$spectral sequence and some applications

Introduction to Secondary Calculus