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 $\mathbb{R}^n$?
On
You might be interested in reading Chang's "Infinite Dimensional Morse Theory" or Jost's "Riemannian Geometry and Geometric Analysis". Moreover, I guess it would be useful to give you an example: there is a result due to Bahri and Coron that states as follows: if $\Omega \subset \mathbb{R}^N$, $N\ge 3$ is a smooth bounded domain, you can find a solution of
$$\begin{cases}-\Delta u = u^{\frac{N+2}{N-2}}&\text{in }\Omega\\ u > 0 & \text{in }\Omega\\ u = 0&\text{on }\partial \Omega,\end{cases}$$
provided that the domain $\Omega$ is "topologically nontrivial", that is if the homology group $H_q(\Omega, \mathbb{Z}_2) \ne 0$ for some $q > 0$. Of course, there are a lot of more recent works in the study of pdes which involves techniques from algebraic topology.
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).