Not to be confused with the common phrase "algebraic analysis of [a subject]", meaning "the algebraic study of [that subject]"
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.
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.
algebraic, analysis, confused, with, common, phrase, algebraic, analysis, subject, meaning, algebraic, study, that, subject, area, mathematics, that, deals, with, systems, linear, partial, differential, equations, using, sheaf, theory, complex, analysis, study. Not to be confused with the common phrase algebraic analysis of a subject meaning the algebraic study of that subject 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 Contents 1 Microfunction 2 See also 3 Citations 4 Sources 5 Further readingMicrofunction editThis section needs expansion You can help by adding to it September 2019 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 H n m M O X o r M X displaystyle mathcal H n mu M mathcal O X otimes mathcal or M X nbsp where m M displaystyle mu M nbsp denotes the microlocalization functor o r M X displaystyle mathcal or M X nbsp 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 See also editHyperfunction D module Microlocal analysis Generalized function Edge of the wedge theorem FBI transform Localization of a ring Vanishing cycle Gauss Manin connection Differential algebra Perverse sheaf Mikio Sato Masaki Kashiwara Lars HormanderCitations edit Kashiwara amp Kawai 2011 pp 11 17 Kashiwara amp Schapira 1990 Definition 11 5 1 Sources editKashiwara 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 Further reading editMasaki Kashiwara and Algebraic Analysis Archived 2012 02 25 at the Wayback Machine Foundations of algebraic analysis book review nbsp This mathematical analysis related article is a stub You can help Wikipedia by expanding it vte Retrieved from https en wikipedia org w index php title Algebraic analysis amp oldid 1170642275, wikipedia, wiki, book, books, library,
