TITLE: Homological algebra in PDE
QUESTION [13 upvotes]: I have been fascinated by the power and wide applicability of homological methods in algebra and topology. Because I am also interested in PDE, there arises a natural question for me.

What is known about applications of methods from homological algebra to the analysis of solutions of PDE on domains in $\mathbb{R}^n$?

REPLY [5 votes]: Homological techniques are very often seen in the literatures of PDE treated from a differential geometrical point of view. An extensive overview can be found in:
Homological methods in equations of  mathematical physics. Also I recommend one of my favorite book here: The Geometry of Physics: An Introduction by Theodore Frankel. Some simple google-fu gives me a recent book also: Cohomological Analysis of Partial Differential Equations and Secondary Calculus.
I am working on computational physics, and the methodologies arised from de Rham cohomology have been used extensively in construction of the finite element spaces for equations in electromagnetism: Finite element exterior calculus, homological techniques, and applications. Mostly people in my field are interested in solving the Hodge Laplacian acting on a $k$-form(magnetic flux or electric field).