Is there a way to view "the space of all possible linear PDE's" as an algebraic variety with singularities?

This is in connection with a quote from someone on the web that I saw a long time ago. At that time I had contacted the author, but they chose not to answer.

The quote:

In some sense, the space of all possible linear PDE's can be viewed as a singular algebraic variety, where Hormander's theory applies only to generic (smooth) points and the most interesting and heavily studied PDE's all lie in a lower-dimensional subvariety and mostly in the singular set of the variety.

Any pointers/refs on any of the points made in the quote would be gratefully received...

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    $\begingroup$ It is not finite dimensional, but I suppose that won't stop some daring people. $\endgroup$ – Ben McKay Oct 4 '18 at 14:48
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    $\begingroup$ The original quote is here. $\endgroup$ – j.c. Oct 4 '18 at 15:41
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    $\begingroup$ Geez. It was me who said it? I'm not sure why I said most of them lie in the singular set of the variety. $\endgroup$ – Deane Yang Oct 4 '18 at 21:06
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    $\begingroup$ I just checked and did not find any email from anyone named Mercurio. Nevertheless, if you did send an email to me and I did not respond, I apologize. $\endgroup$ – Deane Yang Oct 4 '18 at 21:07
  • $\begingroup$ @deane yang...no need to apoligize, ma'am. maybe the message got lost in transit. Btw thanks for the answer below [have questions following your answer] $\endgroup$ – david mercurio Oct 7 '18 at 7:45

Hormander showed that there is a generic set of scalar linear PDE's that can be studied using general techniques, known as microlocal analysis. This can be linked to algebraic geometry as follows: Any scalar linear partial differential operator of order $k$ on an open set in $\mathbb{R}n$ can be written as

$$ Pu = \sum_{|\alpha|\le k} a^\alpha\partial_\alpha u, $$

where each coefficient $a^\alpha$ is a smooth function, $\alpha = (\alpha_1, \dots, \alpha_n)$ and $\partial_\alpha = (\partial_1)^{\alpha_1}\cdots(\partial_n)^{\alpha_n}$. If this is studied using the Fourier transform, then a natural object to study turns out to the principal symbol $$ \sigma(x,\xi) = \sum_{|\alpha| = k} a^\alpha(x)\xi_\alpha, $$ where $\xi = (\xi_1, \dots, \xi_n) \in \mathbb{R}^n$ and $\xi_\alpha = (\xi_1)^{\alpha_1}\cdots(\xi_n)^{\alpha_n}$. For each $x$, this is a homogeneous polynomial of degree $k$ and therefore its zero set is a real algebraic variety on $\mathbb{R}P^{n-1}$. This is known as the characteristic variety. Hormander proved, if the characteristic variety is generic in a suitable sense, regularity estimates, local existence of solutions, and many other things about solutions to equations defined using such operators. However, PDEs most studied have symbols lying in a subvariety of very high codimension, and the techniques used by Hormander are used outside the field of microlocal analysis in only a few specialized areas (e.g., scattering theory, inverse problems). The PDEs with most impact are elliptic, hyperbolic, and parabolic PDEs. Elliptic and most hyperbolic PDEs are generic in Hormander's sense, but parabolic PDEs are not.

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  • $\begingroup$ Interesting. Could you be more specific about what this suitable sense of generic characteristic variety is? $\endgroup$ – Michael Bächtold Oct 4 '18 at 18:04
  • $\begingroup$ There's a more general condition, but the simplest version is that the real projective variety be nonsingular. You can look up "real principal type". $\endgroup$ – Deane Yang Oct 4 '18 at 18:12
  • $\begingroup$ so, for every x, there is a real projective variety... $\endgroup$ – david mercurio Oct 7 '18 at 7:50
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    $\begingroup$ (continued) Also i am confused between two versions of microlocal analysis---the algebraic analysis (analytic) sato-kashiwara version and the smooth version--in particular, i have a quote, "The breakthrough of microlocal analysis quickly spread from the analytic framework to the C^∞-framework, under the impulse of Lars Hörmander who replaced the use of holomorphic functions by that of the Fourier transform." ....where would i begin to read if i want to understand all this...? $\endgroup$ – david mercurio Oct 7 '18 at 8:45
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    $\begingroup$ As to the two versions of microlocal analysis I suggest to read the talks of Sato and of Hörmander at the ICM 1970 in Nice. Hörmander translates Sato's sheaf C into the concept of wavefront set. Pseudodifferential operators are the main technique. $\endgroup$ – Sönke Hansen Oct 14 '18 at 17:56

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