One famous example for me is the Kyoto school and the "algebraic analysis" area founded by Mikio Sato. Thanks to sheaf theory and the concepts of derived category, Grothendieck's six operations and homological algebra one can do microlocal analysis (notably the notion of micro-support which generalizes the notion of propagation in PDE), study analytic D-modules (the Riemann-Hilbert correspondance is a well known result proved by algebraic methods), have a good cohomological definition of hyperfunctions, etc ...

A good overview of this theory can be found here. However, these methods are very effective to study linear PDE but seems currently unable to deal with non-linear cases.

Let me perhaps quote a beautiful theorem which highlight the link with PDE. This is theorem 11.3.3 in P. Schapira and M. Kashiwara, Sheaves on manifold.

Let $X$ be a complex manifold, M a coherent $D_X$-module and $Sol_X(M)$ be the solution complex of $M$. Then : $$SS(Sol_X(M)) = \text{char}(M).$$

Here, $SS$ is the micro support and $\text{char}$ the characteristic variety. Actually this theorem is a generalization and sheaf abstraction of the following proposition

Let $P$ be a holomorphic differential operator defined on $X$ and $\phi$ be a real $C^1$-function on $X$ such that $\sigma(P)(x;\partial\phi(x)) \neq 0.$ on a $X$. (Here $\sigma(P)$ denotes the principal symbol) Let $$\Omega = \{x \in X : \phi(x)<0\}$$ and let $f\in \mathcal{O}_X(\Omega)$ be such that $Pf$ extends holomorphically on a neighborhood of $x_0\in \partial \Omega.$ Then $f$ extends holomorphically in a neighborhood of $x_0$.

One famous example for me is the Kyoto school and the "algebraic analysis" area founded by Mikio Sato. Thanks to sheaf theory and the concepts of derived category, Grothendieck's six operations and homological algebra one can do microlocal analysis (notably the notion of micro-support which generalizes the notion of propagation in PDE), study analytic D-modules (the Riemann-Hilbert correspondance is a well known result proved by algebraic methods), have a good cohomological definition of hyperfunctions, etc ...

A good overview of this theory can be found here. However, these methods are very effective to study linear PDE but seems currently unable to deal with non-linear cases.

Let me perhaps quote a beautiful theorem which highlight the link with PDE. This is theorem 11.3.3 in P. Schapira and M. Kashiwara, Sheaves on manifold.

Let $X$ be a complex manifold, M a coherent $D_X$-module and $Sol_X(M)$ be the solution complex of $M$. Then : $$SS(Sol_X(M)) = \text{char}(M).$$

Here, $SS$ is the micro support and $\text{char}$ the characteristic variety. Actually this theorem is a generalization and sheaf abstraction of the following proposition

Let $P$ be a holomorphic differential operator defined on $X$ and $\phi$ be a real $C^1$-function on $X$ such that $\sigma(P)(d\phi_x) \neq 0.$ on a $X$. (Here $\sigma(P)$ denotes the principal symbol) Let $$\Omega = \{x \in X : \phi(x)<0\}$$ and let $f\in \mathcal{O}_X(\Omega)$ be such that $Pf$ extends holomorphically on a neighborhood of $x_0\in \partial \Omega.$ Then $f$ extends holomorphically in a neighborhood of $x_0$.