Vinogradov's Cohomological Geometry of Partial Differential Equations

TL;DR

This paper reviews Secondary Calculus, a homotopy-based formalism developed by Vinogradov for analyzing the geometric and algebraic structures of PDE solution spaces, emphasizing coordinate-free methods.

Contribution

It provides a comprehensive overview of Secondary Calculus, highlighting its foundational principles, key concepts like horizontal cohomologies, and the role of homotopy in PDE geometry.

Findings

Secondary Calculus offers a coordinate-free framework for PDE analysis.

Horizontal cohomologies encode symmetries and conservation laws.

Homotopy plays a central role in the calculus on diffieties.

Abstract

Secondary Calculus is a formal replacement for differential calculus on the space of solutions of a system of possibly non-linear partial differential equations and it is essentially due to Alexandre M. Vinogradov and his collaborators. Many coordinate free properties of PDEs find their natural place in Secondary Calculus including: symmetries and conservation laws, variational principles and the coordinate free aspects of the calculus of variations, recursion operators and Hamiltonian structures, etc. The building blocks of this language are horizontal cohomologies of diffieties, i.e. infinite prolongations of PDEs, and their versions with local coefficients. The main paradigm of Secondary Calculus is the principle, due to A. M. Vinogradov, roughly stating that: differential calculus on the space of solutions of a PDE is calculus up to homotopy on the horizontal De Rham algebra of the associated diffiety. We will review the fundamentals of Secondary Calculus including its main motivations. In the last part of the paper, we will try to explain the role of homotopy in the theory.