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, algebraic analysis is the application of algebraic operations on analytic quantities. As a research programme, it was started by the Japanese mathematician Mikio Sato in 1959. 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
H n ( μ M ( O X ) ⊗ o r M / X ) {\displaystyle {\mathcal {H}}^{n}(\mu _{M}({\mathcal {O}}_{X})\otimes {\mathcal {or}}_{M/X})}
where
μ M {\displaystyle \mu _{M}} denotes the microlocalization functor,
o r M / X {\displaystyle {\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.
See also Hyperfunction 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 Hörmander Microdifferential operator
Citations
Sources
Further reading Masaki Kashiwara and Algebraic Analysis Archived 25 February 2012 at the Wayback Machine Foundations of algebraic analysis book review