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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

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Let M be a real-analytic manifold and X its complexification. (The definition of microfunctions here).

A microfunction can be used to define a hyper function. 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

• Masaki Kashiwara and Pierre Schapira: Sheaves on Manifolds. Springer-Verlag. Berlin Heidelberg New York.1990: ISBN 3-540-51861-4.

Further reading

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

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• Algebraic analysis
• Generalized functions
• Sheaf theory
• Complex analysis
• Fourier analysis
• Partial differential equations
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