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The Idea-section at microlocal analysis could be optimized to transport an actual idea more efficiently. Somebody should try a sentence that starts with “Microlocal analysis is…”.
Maybe an issue is to put a bound on the subject. What I know of microlocal analysis is all about studying wavefront sets of distributions, and I would be tempted to say “Microlocal analysis is the study of distributions, notably of fundamental solutions to differential equations, with explicit attention to the singularity structure of these distributions, as encoded in their wavefront sets. “
That certainly captures the idea of the subject as presented in volume I of Hörmander’s book, but I gather the other volumes (which I am not that familiar with) enarge the scope of the subject.
I think the key word is the geometric use of cotangent vectors in the study of operators. You can do some elementary study of distributions without ever mentioning Fourier transform, oscillatory integrals and other gadgets involving the “momentum variables” and that would not be microlocal. There are parts of microlocal analysis which deal with various operators which while related to distributions, do not need a point of view of distribution theory. While local refers to points and their neighborhoods, the microlocal refers to points of cotangent bundle and their (formal or true) neighborhoods, I think. In microlocal geometry one thinks of a cotangent codirection attached to every point.
I added some sort of idea section.
Thanks, Zoran.
This idea
“microlocal = points + cotangent vectors”
goes in the direction I was hoping for. As in:
“wave front set = singular locus + cotangents directions of non-rapid Fourier mode decay”.
Which concept of microlocal analysis besides the wave front set involves cotangent vectors?
Zoran, if the answer to my last question is “none”, then I propose once more to make the Idea-statetment in the entry more to the point. Here is a suggestion:
Microlocal analysis is the study of the functional analysis of generalized functions/distributional densities with attention paid not just to their singular support, i.e. to the points around which they are singular as generalized functions, but also the directions of propagation of the singularities at each singular point, which is the set of covectors known as their wave front set. This extra directional (“microlocal”) information governs the basic operations on distributions, notably the pullback of distributions and the product of distributions.
Since the wave front set is the set of (co-)directions along which, locally, the Fourier transform of distributions is not rapidly decreasing (the set of “UV divergencies” in applications to perturbative quantum field theory), much of microlocal analysis is concerned with constructions related to Fourier transformation, such as the discussion of pseudodifferential operators.
The answer is far from “none” but I am not an expert to give a balanced, or even a reliable, list. The first one which comes to mind (and which is getting more popular) is the more recent theory of microlocal sheaves of Kashiwara-Schapira. In this theory, the sheaves over manifolds are given the their microsupport which is a cone over some set within the sphere cotangent bundle.
As far as your intro I don’t know why word “functional analysis” would be exclusive, there are so many geometric works on microlocal theory which emphasize on operations on various sheaves, supports, characteristic subvarieties, vanishing cycles and so on. When mentioning Fourier transformation, one could say “and generalizations”.
Thanks for mentioning “microlocal sheaves”, that helped me locate the relevant sources.
Maybe we need to distinguish “microlocal geometry” from “microlocal analysis”. At least some authors, like David Nadler (here: pdf) say “microlocal geometry” for the study of microlocal sheaves, and Pierre Shapira in his review (here: pdf) considers in the last section the application of microlocal sheaves to the special case of analysis, where the first thing he finds it wave front sets.
So it looks to me like mentioning of “functional analysis” is safe in an Idea section on “microlocal analysis” (and I find that reassuring) while the more general story about microlocal sheaves should maybe go to an entry titled “microlocal geometry” or, in fact, “microlocal sheaves”.
I am not trying to be a pain, I am just trying for our entry to convey some actual kind of idea. If the topic of “microlocal analysis” cannot be said to be about analysis and if all we can say is that it somehow involves cotangent vectors, that would seem to be overly vague and uninformative terminology.
How about we split off an entry “microlocal geometry” where you discuss the general picture, and I keep something like my suggestion in #5 for the entry on “microlocal analysis” proper?
Urs. It may help to look at Pierre Schapira’s article for the New Spaces book. He mentions both microlocal analysis and microlocal sheaves. I expect you have access to this whilst Zoran may not as I do not know if it is available outside that collection as yet.
Tim, thanks. I maybe need to remember where that Dropbox folder sits. But is it likely he will state it much differently than in his older exposition mentioned above (pdf)?
I found that a chapter “Microlocal analysis” book by Alexander Strohmaier (here) gives just the kind of “Idea” of microlocal analysis that I am suggesting we should state in the article: In his abstract he writes;
Microlocal Analysis deals with the singular behavior of distributions in phase space. $[...]$ It turns out that the singularities of distributions can be localized in phase space. This leads to the notion of wavefront sets, a refinement of the notion of singular support.
This makes me re-iterate my suggestion to split the entry into one on the general idea of microlocal sheaves, and one on the use of microlocal analysis in distribution theory. I want to do this because I want to be able to point my readers to an entry that gives them a quick good idea of microlocal analysis in the sense of Hörmander, Strohmaier, etc., and our entry presently does not give that.
Ah, Scharpia’s contribution to “New Spaces” is of course on the arXiv: arXiv:https://arxiv.org/abs/1701.08955. And the introduction agrees with what I am after:
The microlocal point of view first appeared in Analysis with Mikio Sato soon followed by Hörmander who both introduced among others the notion of wave front set. The singularities of a hyperfunction or a distribution on a manifold M are viewed as the projection on M of singularities living in the cotangent bundle
$[...]$
This microlocal point of view was then extended to sheaf theory by Masaki Kashiwara and the author in the eighties who introduced the notion of microsupport of sheaves giving rise to microlocal sheaf theory.
I suggest we say it just this way in the entry on microlocal analysis, and then make “microlocal sheaf theory” a pointer to an entry dedicated to that more general perspective.
The Encyclopedia of Mathematics also agrees that “microlocal analysis” is about distributions and their wave front sets: here.
Okay, I have taken the liberty of splitting it up:
At microlocal analysis I have:
first put in the Idea from #5
followed that by Schapira’s historical remark on how this was the original idea which was then generalized to microlocal sheaf theory,
gave microlocal sheaf theory a stub entry, and moved all references to it which used to be at “microlocal analysis” but which seem to be about the sheaf perspective instead.
Microlocal analysis and microlocal geometry like in general analysis on manifold like objects and geometry of such differs on wheather the emphasis is on maps from the manifold (say into the field) or maps to the manifold (which in special cases gives submanifolds and so on). Weather you use analytic category or smooth category or some other class of maps it is less essential. The school of Sato and Kashiwara in microlocal analysis was part of what was called by themselves in 1960-1980s “algebraic analysis” and was characterized by prevalent usage of D-modules and nonlinear tools. Some are nowdays trying to capture microlocal content by algebraic and categorical means. Indeed, the Fourier harmonics and alike can be to some extent simulated in some algebraic settings.
It seems you now advocate particular formalism and not the content of microlocal analysis. The more essential point is what is studied and not the particular packing of definitions.
moved all references to it which used to be at “microlocal analysis” but which seem to be about the sheaf perspective instead
As I see this is not a good job. You moved mostly Japanese and Russian schools (and anything else what you are not familiar) rather than what is related to microlocal sheaves. Most of references which you moved have nothing to with microlocal sheaves. Some of them have some sheaves, like D-modules, what is just a point of view on some classes of differential equations, bit those have nothing to do with microlocal sheaves.
I moved the Egorov’s reference back to microlocal analysis. Being Russian does not disqualify this reference in PDE and Hoermander’s language for the VINITI encyclopaedia article not being microlocal analysis and it talks nothing about sheaves except that it has hyperfunctions. Hyperfunctions are about boundary values of analytic functions and it is just a variant of distribution theory. The fact that analytic continuation does prefer the sheaf language in that small part of the article does not make connection to microlocal sheaves. But Egorov’s article as far as I inspected did NOT even mention a single instance of a sheaf anywhere within the text. See its MR entry, which is cited in the citation in $n$Lab: MR1175403. The same about Safarov’s lecture notes pdf which I also moved back to microlocal analysis.
It is also unnatural to separate Tamarkin’s work from main entry on microlocal analysis. You see, Weinstein defined the symplectic “category” but the problem is that the composition is not well defined if some transversality is not met. This is similar to the problems with Fukaya category. Tamarkin and Tsygan go toward the setup where such problems are removed, as far as I understand. It is too some extent also related to problems of quantization.
On the other hand it is problem that many references moved to microlocal sheaf have not even a letter on the topic in the title.
When you move the references back, maybe you could use the occasion to equip them with a brief hint of what they actually discuss. That makes the list of references more useful and avoids the impression that it is a random collection.
Some categories related to microlocal analysis
Given a compact symplectic manifold whose symplectic form has integer periods, one associates to it a dg category based on the microlocal analysis according to Kashiwara-Schapira.
…we try to explain how standard symplectic techniques, for instance, generating function, capacities, symplectic homology, etc., are elegantly packaged in the language of sheaves as well as related intriguing sheaf operators. In addition, many concepts developed in Tamarkin category theory are natural generalizations of persistent homology theory…
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