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E.R. Kolchin has developed the differential Galois theory in 1950s. And it seems powerful a tool which can decide the solvability and the form of solutions to a given differential equation.(e.g. Whether it is of closed form or not. see)

My question is why differential Galois theory is not widely used in differential geometry. It is plausible that we can solve some problems of differential/integral geometry using this set of theory.

I have read some answers provided here like Why do we need admissible isomorphisms for differential Galois theory? and other stuffs. I have read Kaplansky's and Buium's books. My question follows:

So what is the major 'pullback' in this theory that prevents its wide application to other situations rather than discrete geometry (e.g. Diophantine geometry)?

My original question on Mathematics Stack Exchange is: Why differential Galois theory is not widely used? which yields no satisfying answers.

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    $\begingroup$ Differential Galois theory is concerned with linear differential equations in one variable. Specialists in differential equations tend to care about more than that (e.g., PDE, nonlinear equations), and the aspect of solvability addressed by differential Galois theory is largely of no interest to them even for linear differential equations. Even ordinary Galois theory is not really of much importance to analysts needing roots of a polynomial in C. Nobody really cares about being able write roots in a very restrictive form related to solvable Galois groups. It is mostly of historic interest. $\endgroup$
    – KConrad
    Commented Apr 3, 2015 at 2:54
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    $\begingroup$ The uses of algebraic Galois theory today (e.g., in number theory) have nothing to do with the classical application to solvability of roots in terms of radicals. Even algebraists don't care about that application very much. So it is no surprise that analysts would also have little use for differential Galois theory to help them solve (linear) differential equations. The analysts have a much wider scope for what it means to solve a differential equation than anything that can be offered from differential Galois theory. $\endgroup$
    – KConrad
    Commented Apr 3, 2015 at 3:00
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    $\begingroup$ @KConrad Beyond that, can we simply obtain any information from the restrictive form of differential equations? Because I know that later researchers digged deeper into some sub-branch called partial differential Galois extension which relates to topics about PDEs. Let's say if I know the solution's form to the Frenet-Serret Equations for some R^3 surface S, can I say anything nontrivial about this surface S in geometry?(This is NOT a good example since DGalois theory always requires algebraically closed constant field...) $\endgroup$
    – Henry.L
    Commented Apr 3, 2015 at 3:04
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    $\begingroup$ Do whatever you want, but I suggest looking at the chapter by Beukers on differential Galois theory in "From Number Theory to Physics" (Waldschmidt et al., eds.). $\endgroup$
    – KConrad
    Commented Apr 3, 2015 at 4:02
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    $\begingroup$ Since the question has been bumped, I will mention that the tag (abstract-algebra) is deprecated on MathOverflow and should not be used - see the tag-info. And the question is also missing a top-level tag. Maybe somebody who knows more about this area would be able to suggest suitable tags for this question. $\endgroup$
    – Martin Sleziak
    Commented Nov 19, 2018 at 14:42

2 Answers 2

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The theory of differential Galois theory is used, but in algebraic, not differential geometry, under the name of D-modules. A D-module is an object that is somewhat more complicated than a representation of the differential Galois group, in the same way that a sheaf is a more complicated than just a Galois representation, but I think it is cut from the same cloth. A D-module describes not just the solutions of a differential equation but also how they behave at singularities.

D-modules are used in many different algebraic geometry situations.

While differential Galois theory may seem analytic it is actually much more algebraic. For instance, in analysis and differential geometry you tend to care how large things are, while in algebra you don't, and differential Galois theory says nothing about size. In algebra you hope for exact solutions, while in analysis approximate solutions are usually good enough, and differential Galois theory good for describing exact solutions. In differential geometry you often have great freedom in gluing together local pieces to get a global structure, where in algebra local pieces are rigid and hard to glue together, and differential Galois theory describes rigid structures where one tiny piece controls everything.

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    $\begingroup$ Is there any reference about this relation between D-modules and differential Galois theory? I am not familiar with algebraic geometry, I have to say. $\endgroup$
    – Henry.L
    Commented Apr 3, 2015 at 3:40
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    $\begingroup$ I'd recommend looking at books by Michael F. Singer: www4.ncsu.edu/~singer/ms_papers.html, $\endgroup$
    – Dima Pasechnik
    Commented Apr 3, 2015 at 18:28
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As indicated by KConrad in his comments, differential Galois theory is used in the part of transcendental number theory that tries to establish algebraic/linear independence of values of special functions at algebraic numbers. Examples are given by the theorems of Siegel-Shidlovski, Nesterenko, etc. Roughly speaking, its rôle is to guarantee that an auxiliary function constructed in the arithmetic part of the proof is non-zero.

Contrary to what he writes, there is a nonlinear differential Galois theory, namely Malgrange's theory of the differential groupoid of a foliation. It is not widely used in transcendental number theory for the moment.

A reason why differential Galois theory does not explicitly appears in differential geometry is that DGT works in the framework of differential fields, and smooth functions on a connected open subset of $\mathbf R^n$ do not form a field (unless $n=0$).

However, as explained in the book of Thomas Hawkins (Emergence of the theory of Lie groups), the works of Lie were explicitly motivated by generalizing Galois theory to differential equations.

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    $\begingroup$ Besides the historical account of Hawkins, a math book covering Lie's ideas for solving differential equations through ideas of symmetry is Olver's "Applications of Lie Groups to Differential Equations." $\endgroup$
    – KConrad
    Commented Apr 3, 2015 at 11:49
  • $\begingroup$ I am curious about 'differential Galois theory is used in the part of transcendental number theory', so I really want some reference to have a read, thanks! $\endgroup$
    – Henry.L
    Commented Apr 4, 2015 at 4:29
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    $\begingroup$ For those who are interested in the connection between Lie theory and DGT, see also: mathoverflow.net/questions/175761/… $\endgroup$
    – Henry.L
    Commented Apr 4, 2015 at 4:44

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