Homological Methods in Equations of Mathematical Physics

Abstract

These lecture notes are a systematic and self-contained exposition of the cohomological theories naturally related to partial differential equations: the Vinogradov C-spectral sequence and the C-cohomology, including the formulation in terms of the horizontal (characteristic) cohomology. Applications to computing invariants of differential equations are discussed. The lectures contain necessary introductory material on the geometric theory of differential equations and homological algebra.

150 pages, AmS-LaTeX 1.2 (based on LaTeX 2e), needs to run through LaTeX four times, no figures, Lectures given in August 1998 at the International Summer School in Levoca, Slovakia. Revised version 2 (Mon, 21 Dec 1998): misprints corrected, Definition 4.3 on page 70 changed

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• A geometric derivation of KdV-type hierarchies from root systems
• Remarks on two approaches to the horizontal cohomology: compatibility complex and the Koszul-Tate resolution
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• Functors of Differential Calculus in Diolic Algebras
• On weakly non-local, nilpotent, and super-recursion operators for N=1 super-equations
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