Cohomological Analysis of Partial Differential Equations and Secondary Calculus
American Mathematical Soc., Oct 16, 2001
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".
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acyclic analog c-density C-field C-spectral sequence C-theory called canonical Cartan distribution CDiff Chapter characteristic classes classical cochain mapping cohomology classes coincides compute conservation laws Consider constructions contact structure coordinates Corollary corresponding defined definition denoted Diff diffeomorphism differential calculus differential forms differential operators diffiety domain elements equivalent Euler operator fact fibers filtration first-order functions functor Green formula Hence homomorphism homotopy horizontal identified infinitesimal symmetries integral manifolds isomorphism Jk(n Lagrangian Lemma Lie algebra Lie derivative Lie field Lie transformation linear morphism n-dimensional natural 5-module nonlinear notation Note obtained obviously partial differential equations particular problem projective projective module PROOF Proposition proved quotient representing object resp respect restriction rioc(7r scalar Secondary Calculus secondary differential operators secondary module secondary vector shows smbl smooth solution spectral Spencer submanifold submodule Subsection tangent term theory trivial vector fields