[Math] Applications of algebra to analysis

Question

EDIT: I would like to make a list of modern applications of algebra in analysis. By "modern" I will mean developments since the beginning of the 20th century. It is well known that classical linear algebra of the 19th century (or earlier), such as vector spaces, determinants, diagonalization and Jordan form of matrices, have many applications in other areas of mathematics and beyond that, in particular in analysis. Below I will give an example of a modern application.

The choice of areas is motivated by my personal taste.

Now let me describe one of my favorite examples. Let $P$ be real a polynomial in $n$ variables. For any smooth compactly supported test function $\phi$ consider the integral $\int_{\mathbb{R}^n}|P(x)|^\lambda\phi(x)dx$. It converges absolutely for $Re(\lambda)>0$ and is holomorphic in $\lambda$ there. The question posed by I.M. Gelfand was whether it has meromoprhic continuation in $\lambda$ to the whole complex plane.

This problem has positive answer. In full generality it was solved by J. Bernstein and S. Gelfand, and independently by M. Atiyah, in 1969, see http://www.math.tau.ac.il/~bernstei/Publication_list/publication_texts/Bern-Gel-P-lam-FAN.pdf They used the Hironaka's theorem on resolution of singularities. That is a deep important result from algebraic geometry for which Hironaka was awarded Fields medal.

A little later, in 1972, J. Bernstein invented a different proof of the meromorphic continuation theorem which did not use Hironaka's result. The main step was the following statement. There exists a differential operator $D_\lambda$ on $\mathbb{R}^n$ whose coefficients depend polynomially on $x$ and are rational functions of $\lambda$ such that
$$|P|^\lambda=D_\lambda (|P|^{\lambda +1}) \mbox{ for } Re(\lambda) >0.$$
Using this functional equation one can recursively extend the above integral to the whole complex plane.

In order to prove the above functional equation J. Bernstein developed a purely algebraic method which later on became fundamental in the theory of algebraic D-modules.

Answer 1

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 $x_0$.