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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):

  1. Differential algebraic geometry
  2. Diffiety Theory

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:

  1. Does anyone know both approaches well enough to compare their methods and applications? If so, please share this knowledge.
  2. If I want to learn one of these approaches, which one should I choose?
  3. 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.

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Disclaimer: I don't know much about this, and I hope more knowledgeable people weigh in.

You might want to take a look at Ayoub's differential Galois theory for schemes and the foliated topology (see preprint).

If we are interested in solutions of a single polynomial equation in one variable (over a field and its algebraic extensions), the relevant part of algebra is Galois theory. There is a well-known version of Galois theory for differential equations over (differential) fields, also called Picard-Vessiot theory.

From the point of view of algebraic geometry, Galois theory is the study of the etale site of the spectrum of a field. Thus a natural version of your question (though maybe not exactly what you wanted) asks for a version of differential Galois theory for varieties.

Since the extension of the classical Galois theory to varieties (or schemes) relies on the etale topology, a natural lead is to look for a topology with an analogous relation to Picard-Vessiot theory and its natural extension to varieties. As far as I understand, this is what Ayoub's foliated topology does (among other things).

A somewhat relevant quote from Ayoub (p. 4)

En un sens, la topologie étale est une approximation profinie de la topologie transcendante. Un slogan que j’aimerai proposer est le suivant: la topologie feuilletée est une approximation pro-algébrique de la topologie transcendante!

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  • $\begingroup$ Wow, thank you, this is very interesting. Unfortunately, I do not speak french though. Is there an English version? I can't mark it as answer because the question also demands a comparison of the two approaches I gave above but that you mention yet another way to approach PDEs more abstractly in connection with Galois groups is very useful and valuable! $\endgroup$ – exchange Jun 20 '18 at 19:22
  • $\begingroup$ Yes, I addressed only the "if someone knows more, please let me know as well" part... $\endgroup$ – Piotr Achinger Jun 20 '18 at 19:26
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    $\begingroup$ Roughly translated: "in a sense, the étale topology is a profinite approximation to the transcendental topology. A slogan I ... propose is the following: the foliated topology is a proalgebraic approximation to the transcendental topology!" $\endgroup$ – David Roberts Jun 20 '18 at 21:57

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