Algebraic analysis

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. As a research programme, it was started by Mikio Sato in 1959.

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

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

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.

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

References

• Kashiwara, Masaki; Schapira, Pierre (1990). Sheaves on Manifolds. Berlin: Springer-Verlag. ISBN 3-540-51861-4.

Further reading

• Masaki Kashiwara and Algebraic Analysis
• Foundations of algebraic analysis book review