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 did not find anything on math.SE about diffieties. I found this short introduction to the idea of a diffiety on the ncat-lab but it does not say much about the applications of the idea and does not contrast it with differential algebraic geometry.
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?