Algebraic analysis is an area of
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
that deals with systems of
linear partial differential equations by using
sheaf theory and
complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
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. 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.
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
:
where
*
denotes the
microlocalization functor
In mathematical analysis, microlocal analysis comprises techniques developed from the 1950s onwards based on Fourier transforms related to the study of variable-coefficients-linear and nonlinear partial differential equations. This includes genera ...
,
*
is the
relative orientation sheaf.
A microfunction can be used to define a Sato's
hyperfunction. By definition, the sheaf of
Sato's hyperfunction
In mathematics, hyperfunctions are generalizations of functions, as a 'jump' from one holomorphic function to another at a boundary, and can be thought of informally as distributions of infinite order. Hyperfunctions were introduced by Mikio Sato ...
s 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 function
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivativ ...
s on ''X'' to ''M''.
See also
*
Hyperfunction
*
D-module
In mathematics, a ''D''-module is a module (mathematics), module over a ring (mathematics), ring ''D'' of differential operators. The major interest of such ''D''-modules is as an approach to the theory of linear partial differential equations. Sin ...
*
Microlocal analysis
*
Generalized function
*
Edge-of-the-wedge theorem
*
FBI transform
The Federal Bureau of Investigation (FBI) is the domestic intelligence and security service of the United States and its principal federal law enforcement agency. Operating under the jurisdiction of the United States Department of Justice, t ...
*
Localization of a ring
*
Vanishing cycle
*
Gauss–Manin connection
*
Differential algebra
*
Perverse sheaf
*
Mikio Sato
*
Masaki Kashiwara
is a Japanese mathematician. He was a student of Mikio Sato at the University of Tokyo. Kashiwara made leading contributions towards algebraic analysis, microlocal analysis, D-module, ''D''-module theory, Hodge theory, sheaf theory and represent ...
*
Lars Hörmander
Citations
Sources
*
*
Further reading
Masaki Kashiwara and Algebraic AnalysisFoundations of algebraic analysis book review
Complex analysis
Fourier analysis
Generalized functions
Partial differential equations
Sheaf theory
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