Timeline for Moduli space of linear partial differential equations
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| Oct 14, 2018 at 17:56 | comment | added | user80744 | 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 7, 2018 at 19:14 | comment | added | Deane Yang | 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. | |
| Oct 7, 2018 at 19:01 | comment | added | Deane Yang | I never understood D-modules, so maybe someone else can explain. | |
| Oct 7, 2018 at 18:59 | comment | added | Deane Yang | 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. | |
| Oct 7, 2018 at 8:45 | comment | added | david mercurio | (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...? | |
| Oct 7, 2018 at 8:15 | comment | added | david mercurio | (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? | |
| Oct 7, 2018 at 8:04 | comment | added | david mercurio | (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..? | |
| Oct 7, 2018 at 7:50 | comment | added | david mercurio | so, for every x, there is a real projective variety... | |
| Oct 4, 2018 at 18:12 | comment | added | Deane Yang | 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". | |
| Oct 4, 2018 at 18:04 | comment | added | Michael Bächtold | Interesting. Could you be more specific about what this suitable sense of generic characteristic variety is? | |
| Oct 4, 2018 at 15:11 | history | answered | Deane Yang | CC BY-SA 4.0 |