Algebraic geometry is said to be useful to study not only specific solutions of polynomial equations but to understand the intrinsic properties of the totality of solutions of a system of equations.
However, varieties (which are the central object of study in algebraic geometry) are generalisations of solution sets of polynomial equations and thus do not seem suitable for investigating (intrinsic properties of solutions of a system of) differential equations.
But precisely the investigation of differential equations in such an abstract manner would probably enable great applications in physics and computer science.
While looking for approaches that generalise the notions of algebraic geometry to the study of differential equations I basically came across the following two (if someone knows more, please let me know as well):
However, the information provided on the wiki-sides are not very helpful. Furthermore, I only found this post on mathoverflow about diffieties which is not directly addressing my question. I found a short introduction to the idea of a diffiety on the ncat-lab but it does not contrast the idea and its applications.
My questions are:
- Does anyone know both approaches well enough to compare their methods and applications? If so, please share this knowledge.
- If I want to learn one of these approaches, which one should I choose?
- And what would be good books to start?
PS: I found a link to a so-called diffiety institute which provides access to a lot of writings on diffiety theory. And here are some more information on differential algebra.
EDIT: The diffiety institute site is not being updated anymore except for the library but I found a more up-to-date website carrying on the promotion of the idea of a diffiety: The Levi-Civita Institute.