Homological Methods in Equations of Mathematical Physics
Joseph Krasil'shchik, Alexander Verbovetsky
TL;DR
This paper provides a comprehensive introduction to homological methods, specifically cohomological theories like the Vinogradov C-spectral sequence, applied to analyze and compute invariants of differential equations in mathematical physics.
Contribution
It offers a systematic, self-contained exposition of cohomological theories related to PDEs, including applications to invariants, with necessary background in geometry and homological algebra.
Findings
Development of the Vinogradov C-spectral sequence framework
Methods for computing invariants of differential equations
Integration of homological algebra with geometric PDE analysis
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.
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Taxonomy
TopicsNonlinear Waves and Solitons · Advanced Topics in Algebra · Numerical methods for differential equations