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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...

Peter Mortensen
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david mercurio
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1 Answers1

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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.

Deane Yang
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  • Interesting. Could you be more specific about what this suitable sense of generic characteristic variety is? – Michael Bächtold Oct 04 '18 at 18:04
  • 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". – Deane Yang Oct 04 '18 at 18:12
  • so, for every x, there is a real projective variety... – david mercurio Oct 07 '18 at 07:50
  • (continued) so, for every x, there is a real projective variety...say, V_(p,x). How are these related for different x's? Then we union all these over x's, to form , say, V_p. Is this what you refer to as "the" characteristic variety... of the operator P..? – david mercurio Oct 07 '18 at 08:04
  • (continued) i found many mentions of "characteristic variety of D-modules" but no occurance of characteristic variety of, say, a pDO or ΨDO,... do they refer to it by some other name? – david mercurio Oct 07 '18 at 08:15
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    (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...? – david mercurio Oct 07 '18 at 08:45
  • Hormander's condition is an open one in the space of polynomials of $n$ variables. So if it holds at one point, it holds on a neighborhood of that point , which suffices for local regularity and existence results. If you want to study the operator on a given domain, then you have to assume that it holds at each point in the domain. – Deane Yang Oct 07 '18 at 18:59
  • I never understood D-modules, so maybe someone else can explain. – Deane Yang Oct 07 '18 at 19:01
  • You can find "characterstic variety" or "characteristic set" mentioned in notes or books by Michael Taylor, Francois Treves, Richard Melrose, Chazarain-Piriou on pseudodifferential operators. – Deane Yang Oct 07 '18 at 19:14
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    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. –  Oct 14 '18 at 17:56