Ellis Kolchin developed differential Galois theory in the 1950s. It seems to be a powerful tool that can decide the solvability and the form of the solutions to a given differential equation.

Why isn't differential Galois theory widely used in differential geometry? It is plausible that we can solve some problems of differential/integral geometry using this theory.

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

  • $\begingroup$ I haven't had much exposure to Galois theory yet, but I have heard that it is not widely taught, so I conjecture that the teachers of these courses aren't sufficiently comfortable with it. $\endgroup$ – user142198 Mar 27 '15 at 0:32
  • $\begingroup$ @Incurrence The question is about differential Galois theory, not ordinary (algebraic) Galois theory, and it's about research in differential geometry, not courses. When you write "I have heard that it is not widely taught" and "the teachers of these courses aren't sufficiently comfortable with it", if you're talking about algebraic Galois theory, you're wrong on both counts. $\endgroup$ – Alex Kruckman Apr 25 '16 at 2:04
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    $\begingroup$ The question is now posted on MathOverflow: Why is differential Galois theory not widely used? $\endgroup$ – Martin Sleziak Nov 19 '18 at 14:01
  • $\begingroup$ @MartinSleziak thanks for the reminder, I am doing a few edits on some of my old questions, please leave it to me before further moderation, thank you! $\endgroup$ – Henry.L Nov 19 '18 at 14:07
  • $\begingroup$ @HenryL The only thing I wanted was to have link to MO question at least in the comments (so that the users who stumble upon this question find also the question on MO). As you probably now, it is recommended to link both copies to each other when posting on multiple sites. (So I was planning no further actions other than posting the above comment.) $\endgroup$ – Martin Sleziak Nov 19 '18 at 14:09

Among others, there is a nice concrete application differential Galois theory to the Non-Integrability of Hamiltonian Systems :


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