On what basis do you say that probability has "morphed into an algebra"? Consider all of the popular research topics today: the KPZ equation, Schramm-Loewner evolution, spin glasses, percolation and critical phenomena, rigorous statistical mechanics in general [1], random matrix theory, large deviations, stochastic analysis and (P)SDEs.
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."
I know vaguely from a few friends that there is some cross pollination between algebraic geometry and PDE's but both fields are so huge that this doesn't characterize either field. I think the gist of it is that the moduli space of PDE solutions are frequently interesting as varieties.
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."
[1]: https://www.unige.ch/math/folks/velenik/smbook/