Algebraic analysis: Difference between revisions

Content deleted Content added

Inline

Latest revision as of 08:52, 16 August 2023

Algebraic analysis is an area of mathematics that deals with systems of linear partial differential equations by using sheaf theory and complex analysis to study properties and generalizations of functions such as hyperfunctions and microfunctions. Semantically, it is the application of algebraic operations on analytic quantities. As a research programme, it was started by the Japanese mathematician Mikio Sato in 1959.^[1] This can be seen as an algebraic geometrization of analysis. It derives its meaning from the fact that the differential operator is right-invertible in several function spaces.

It helps in the simplification of the proofs due to an algebraic description of the problem considered.

Microfunction

Let M be a real-analytic manifold of dimension n, and let X be its complexification. The sheaf of microlocal functions on M is given as^[2]

{\mathcal {H}}^{n}(\mu _{M}({\mathcal {O}}_{X})\otimes {\mathcal {or}}_{M/X})

where

$\mu _{M}$ denotes the microlocalization functor,
${\mathcal {or}}_{M/X}$ is the relative orientation sheaf.

A microfunction can be used to define a Sato's hyperfunction. By definition, the sheaf of Sato's hyperfunctions on M is the restriction of the sheaf of microfunctions to M, in parallel to the fact the sheaf of real-analytic functions on M is the restriction of the sheaf of holomorphic functions on X to M.

Citations

^ Kashiwara & Kawai 2011, pp. 11–17.
^ Kashiwara & Schapira 1990, Definition 11.5.1.

Sources

Kashiwara, Masaki; Kawai, Takahiro (2011). "Professor Mikio Sato and Microlocal Analysis". Publications of the Research Institute for Mathematical Sciences. 47 (1): 11–17. doi:10.2977/PRIMS/29 – via EMS-PH.
Kashiwara, Masaki; Schapira, Pierre (1990). Sheaves on Manifolds. Berlin: Springer-Verlag. ISBN 3-540-51861-4.

@@ Line 1: / Line 1: @@
+{{Short description|Technique of studying linear partial differential equations}}
-{{dablink|Not to be confused with the common phrase "algebraic analysis of <nowiki>[a subject]</nowiki>", meaning "the algebraic study of <nowiki>[that subject]</nowiki>"}}
+{{Hatnote|Not to be confused with the common phrase "algebraic analysis of <nowiki>[a subject]</nowiki>", meaning "the algebraic study of <nowiki>[that subject]</nowiki>"}}
-'''Algebraic analysis''' is an area of [[mathematics]] that deals with systems of linear [[partial differential equation]]s by using [[sheaf theory]] and [[complex analysis]] to study properties and generalizations of functions such as [[hyperfunction]]s and microfunctions.  As a research programme, it was started by [[Mikio Sato]] in 1959.<ref>{{cite article|title=Professor Mikio Sato and Microlocal Analysis|first1=Masaki |last1=Kashiwara|author-link1=Masaki Kashiwara|first2=Takahiro |last2=Kawai|author-link2=Takahiro Kawai|journal=PRIMS|volume=47|issue=1|year=2011|url=http://www.ems-ph.org/journals/show_pdf.php?issn=0034-5318&vol=47&iss=1&rank=2|doi=10.2977/PRIMS/29|via=EMS-PH}}</ref>
+'''Algebraic analysis''' is an area of [[mathematics]] that deals with systems of [[Partial differential equation|linear partial differential equations]] by using [[sheaf theory]] and [[complex analysis]] to study properties and generalizations of [[Function (mathematics)|functions]] such as [[hyperfunction]]s and microfunctions. Semantically, it is the application of algebraic operations on analytic quantities. As a research programme, it was started by the Japanese mathematician [[Mikio Sato]] in 1959.{{sfn|Kashiwara|Kawai|2011|pp=11–17}} This can be seen as an algebraic geometrization of analysis. It derives its meaning from the fact that the differential operator is right-invertible in several function spaces.
+It helps in the simplification of the proofs due to an algebraic description of the problem considered.
 == Microfunction ==
 {{expand section|date=September 2019}}
-Let ''M'' be a real-analytic manifold of dimension ''n'' and ''X'' its complexification. The sheaf of '''microlocal functions''' on ''M'' is given as<ref>{{harvnb|Kashiwara–Schapira|loc=Definitin 11.5.1.}}</ref>
+Let ''M'' be a [[Real number|real]]-[[analytic manifold]] of [[Manifold#Mathematical definition|dimension]] ''n'', and let ''X'' be its complexification. The sheaf of '''microlocal functions''' on ''M'' is given as{{sfn|Kashiwara|Schapira|1990|loc=Definition 11.5.1}}
-:<math>\mathcal{H}^n(\mu_M(\mathcal{O}_X) \otimes \mathcal{or}_{M/X})</math>.
+:<math>\mathcal{H}^n(\mu_M(\mathcal{O}_X) \otimes \mathcal{or}_{M/X})</math>
 where
-*<math>\mu_M</math> denotes the [[microlocalization functor]],
+* <math>\mu_M</math> denotes the [[microlocalization functor]],
-*<math>\mathcal{or}_{M/X}</math> is the [[relative orientation sheaf]].<!-- need to give a more gentle definition -->
+* <math>\mathcal{or}_{M/X}</math> is the [[Orientation sheaf|relative orientation sheaf]].<!-- need to give a more gentle definition -->
-A microfunction can be used to define a Sato's hyperfunction. By definition, the sheaf of [[Sato's hyperfunction]]s on ''M'' is the restriction of the sheaf of microfunctions to ''M'', in parallel to the fact the sheaf of real-analytic functions on ''M'' is the restriction of the sheaf of holomorphic functions on ''X'' to ''M''.
+A microfunction can be used to define a Sato's [[hyperfunction]]. By definition, the sheaf of [[Sato's hyperfunction]]s on ''M'' is the restriction of the sheaf of microfunctions to ''M'', in parallel to the fact the sheaf of [[analytic function|real-analytic functions]] on ''M'' is the restriction of the sheaf of [[holomorphic function]]s on ''X'' to ''M''.
 ==See also==
-*[[Hyperfunction]]
+* [[Hyperfunction]]
-*[[D-module]]
+* [[D-module]]
-*[[Microlocal analysis]]
+* [[Microlocal analysis]]
-*[[Generalized function]]
+* [[Generalized function]]
-*[[Edge-of-the-wedge theorem]]
+* [[Edge-of-the-wedge theorem]]
-*[[FBI transform]]
+* [[FBI transform]]
-*[[Localization of a ring]]
+* [[Localization of a ring]]
-*[[Vanishing cycle]]
+* [[Vanishing cycle]]
-*[[Gauss–Manin connection]]
+* [[Gauss–Manin connection]]
-*[[Differential algebra]]
+* [[Differential algebra]]
-*[[Perverse sheaf]]
+* [[Perverse sheaf]]
-*[[Mikio Sato]]
+* [[Mikio Sato]]
-*[[Masaki Kashiwara]]
+* [[Masaki Kashiwara]]
-*[[Lars Hörmander]]
+* [[Lars Hörmander]]
-== References ==
+== Citations ==
-{{reflist}}
+{{Reflist}}
-*{{cite book |first1=Masaki |last1=Kashiwara |first2=Pierre |last2=Schapira |author-link2=Pierre Schapira |title=Sheaves on Manifolds |publisher=Springer-Verlag |location=Berlin |year=1990 |isbn=3-540-51861-4}}
+==Sources==
+{{refbegin}}
+*{{cite journal | title = Professor Mikio Sato and Microlocal Analysis
+ | last1 = Kashiwara | first1 = Masaki
+ | last2 = Kawai | first2 = Takahiro
+ | author1-link = Masaki Kashiwara
+ | author2-link = Takahiro Kawai
+ | journal = Publications of the Research Institute for Mathematical Sciences | via = EMS-PH
+ | year = 2011 | volume = 47 | issue = 1 | pages = 11–17
+ | url = http://www.ems-ph.org/journals/show_pdf.php?issn=0034-5318&vol=47&iss=1&rank=2
+ | doi = 10.2977/PRIMS/29
+ | doi-access = free
+}}
+*{{cite book| title = Sheaves on Manifolds
+ | last1 = Kashiwara | first1 = Masaki
+ | last2 = Schapira | first2 = Pierre
+ | author2-link = Pierre Schapira (mathematician)
+ | year = 1990
+ | publisher = Springer-Verlag | location = Berlin
+ | isbn = 3-540-51861-4
+}}
+{{refend}}
 ==Further reading==
-*[http://people.math.jussieu.fr/~schapira/mispapers/Masaki.pdf Masaki Kashiwara and Algebraic Analysis]
+* [http://people.math.jussieu.fr/~schapira/mispapers/Masaki.pdf Masaki Kashiwara and Algebraic Analysis] {{Webarchive|url=https://web.archive.org/web/20120225173659/http://people.math.jussieu.fr/~schapira/mispapers/Masaki.pdf |date=2012-02-25 }}
-*[http://projecteuclid.org/euclid.bams/1183554451 Foundations of algebraic analysis book review]
+* [http://projecteuclid.org/euclid.bams/1183554451 Foundations of algebraic analysis book review]
+{{Authority control}}
 [[Category:Algebraic analysis| ]]
-[[Category:Generalized functions]]
-[[Category:Sheaf theory]]
 [[Category:Complex analysis]]
 [[Category:Fourier analysis]]
+[[Category:Generalized functions]]
 [[Category:Partial differential equations]]
+[[Category:Sheaf theory]]
 {{mathanalysis-stub}}