soft: Reference/ Suggested Read: Homological Algebraic techniques in PDEs I was reading this article on wikipiedia and was interested by the apparent link between Homological Algebra and PDEs.  What is an accessible reference which showcases the link between these topics?   
 A: First, a critique. Both homological algebra and PDEs are far from being monolithic subjects. So, a well-posed question about the relation between the two should be quite specific. Though you don't explicitly say it, your question suggests that you are interested in the homological algebra methods that appeared in the work of Spencer and others influenced by him. So, I'll restrict my answer to that.
The early ideas appeared already in the work of Kodaira and Spencer on the deformation of complex structures. The equations describing a complex structure are non-linear over-determined PDEs. One way to study the deformations of a given complex structure is to linearize these equations and then try to complete a linear solution to a 1-parameter family of non-linear (true) deformations. As Kodaira and Spencer discovered, not all linear solutions can be continued to non-linear deformations. A problem can appear as follows. Solving the defining equations of a complex structure perturbatively, at the quadratic order one needs to solve a linear inhomogeneous equation with a source quadratic in the linear order solution. If this source lies outside the range of the linear operator in the quadratic order equation, there is obviously no solution and the linear order solution cannot be continued to a non-linear one (not even to second order). In other segments of the PDE literature, this phenomenon is called linearization instability. So, some obstructions to continuing the linear order deformation of a complex structure to a $1$-parameter family of non-linear deformations lies in the discrepancy between the range of the linear differential operator appearing at the quadratic perturbative order and the non-linear map that is used to construct its source term from the linear order solution. Kodaira and Spencer figured out that this discrepancy is absent if some cohomology of the compatibility complex of the linearized defining equations of a complex structure vanishes. The same cohomology could also be described as the cohomology of the sheaf of complex vector fields on the underlying complex manifold. All of this is described in Kodaira's book:


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*Kodaira, Kunihiko, Complex manifolds and deformation of complex structures (Springer, 1986).


Later Spencer encouraged people to further apply ideas from homological algebra to the theory of over-determined PDEs. The basic problem, given a PDE system, is to get a complete list of its integrability conditions (equations, other than the ones already given, of the same or lower differential order that automatically hold for any solution). Integrability conditions are usually identified by taking derivatives of the given PDE and trying to eliminate as many of the highest derivative terms as possible. Once all integrability conditions have been identified the system is said to have been completed to involution. It has been known since the early 20th century that completion to involution can be established by an algorithm in a definite number of steps. However, the existing methods required either the use of an explicit coordinate system (Janet, Riquier) or the conversion of the PDE system into an equivalent first order Pfaffian system (Cartan, Kuranishi). Making liberal use of some homological complexes constructed from the symbol of a given PDE system (Spencer cohomology) a geometric theory of the completion of over-determined PDEs to involution was then created (Spencer, Bott, Quillen, Goldschmidt). Seminal early papers on this subject include


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*Goldschmidt, H. Existence theorems for analytic linear partial differential equations. The Annals of Mathematics 86, 246-270 (1967).

*Goldschmidt, H. Integrability criteria for systems of nonlinear partial differential equations. Journal of Differential Geometry 1, 269-307 (1967).

*Spencer, D. C. Overdetermined systems of linear partial differential equations. Bulletin of the American Mathematical Society 75, 179-240 (1969).


A modern comprehensive treatment of this theory, invoking even stronger ties to homological algebra can be found in


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*Seiler, W. M. Involution: The Formal Theory of Differential Equations and its Applications in Computer Algebra (Springer, 2010).

