Algebraic analysis

{{Short description|Technique of studying linear partial differential equations}}

{{Hatnote|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.{{sfn|Kashiwara|Kawai|2011|pp=11–17}} This can be seen as an algebraic geometrization of analysis. According to Schapira, parts of Sato's work can be regarded as a manifestation of Grothendieck's style of mathematics within the realm of classical 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{{sfn|Kashiwara|Schapira|1990|loc=Definition 11.5.1}}

: $\mathcal{H}^n(\mu_M(\mathcal{O}_X) \otimes \mathcal{or}_{M/X})$

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

Sources

{{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

}}

Algebraic analysis

Microfunction

See also

Citations

Sources

Further reading