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I ask primarily because despite not having taken that many math classes (up to two semesters of a PDE class in college), it would be very interesting if maybe we could gain intuition regarding nonlinear PDE's by looking at properties of their algebraic counterparts (i.e, from properties of parabolas for example, we could gain a better understanding of parabolic PDE's). This would extend to looking at other algebraic curves to study many other differential equations, at least, that is what I am thinking.

Also, if this actually is a field of research currently, what would be a good way to get into it (books and papers to read, and possible people who are doing this kind of thing to look into)?

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    $\begingroup$ In the field "integrable system", there is a correspondence between solution to the differetial equation (the Lax equation) and something called "spectral curve". Solutions to the equation can be given by some line bundles on that algebraic curve. You may find the book "Integrable Systems: Twistors, Loop Groups, and Riemann Surfaces" interesting. $\endgroup$ Commented Jun 11, 2015 at 22:58
  • $\begingroup$ A good book to accompany a standard PDE course to explore concepts from a geometric perspective is "Lectures on PDEs by V.I.Arnol'd" amazon.com/Lectures-Partial-Differential-Equations-Universitext/… $\endgroup$ Commented Jun 11, 2015 at 23:03
  • $\begingroup$ Thanks. I'll be sure to get both books. $\endgroup$ Commented Jun 12, 2015 at 17:02

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