Obviously, I'm forgetting some, but none of these are primarily algebraic. Some algebraic methods are used (orthogonal polynomials in random matrix theory, and determinants and a whole host of other things to study integrable models in the KPZ universality class, etc.), but clearly groups/rings/fields are not playing a major role.
For a more systematic approach you could look at recent issues of Annals of Probability, Annals of Applied Probability, and similar journals. There's not going to be a lot of "modern algebra" (of the flavor you see in algebraic geometry) there.
The same comments apply to PDE and analytic number theory. Both are obviously mature fields (worked on for a long time by many people, with a lot of great discoveries), but again algebra does not play a central role in either. In particular I am not aware of any PDE specialists whose research agenda consists of "trying to turn it into algebra."
There's this rare unclosed survey-like mathoverflow question that highlights some of these points: https://mathoverflow.net/questions/66046/which-nonlinear-pde...
Also, you seem to be overstating the maturity of PDE (people are pretty much still trying to turn it into algebra, but it may not be possible) and number theory (that's an active research area).
I have never seen that saying either, but it seems quite correct, with the exception of the areas where an algebra is impossible, so the field matures without becoming one.