Algebraic analysis is an area of
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
that deals with systems of
linear partial differential equations by using
sheaf theory
In mathematics, a sheaf (: sheaves) is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the d ...
and
complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic ...
to study properties and generalizations of
functions such as
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 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. 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 In mathematics, an analytic manifold, also known as a C^\omega manifold, is a differentiable manifold with analytic transition maps. The term usually refers to real analytic manifolds, although complex manifolds are also analytic. In algebraic geo ...
of
dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coo ...
''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
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 ...
. 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 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 de ...
s on ''X'' to ''M''.
See also
*
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 ...
*
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. S ...
*
Microlocal analysis
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 ...
*
Generalized function
In mathematics, generalized functions are objects extending the notion of functions on real or complex numbers. There is more than one recognized theory, for example the theory of distributions. Generalized functions are especially useful for tr ...
*
Edge-of-the-wedge theorem
*
FBI transform
*
Localization of a ring
Localization or localisation may refer to:
Biology
* Localization of function, locating psychological functions in the brain or nervous system; see Linguistic intelligence
* Localization of sensation, ability to tell what part of the body is aff ...
*
Vanishing cycle In mathematics, vanishing cycles are studied in singularity theory and other parts of algebraic geometry. They are those homology (mathematics), homology cycles of a smooth fiber in a family which vanish in the singular fiber.
For example, in a map ...
*
Gauss–Manin connection
In mathematics, the Gauss–Manin connection is a connection on a certain vector bundle over a base space ''S'' of a family of algebraic varieties V_s. The fibers of the vector bundle are the de Rham cohomology groups H^k_(V_s) of the fibers V_s ...
*
Differential algebra
*
Perverse sheaf
*
Mikio Sato
*
Masaki Kashiwara
is a Japanese mathematician and professor at the Kyoto University Institute for Advanced Study (KUIAS). He is known for his contributions to algebraic analysis, microlocal analysis, ''D''-module theory, Hodge theory, sheaf theory and represent ...
*
Lars Hörmander
Citations
Sources
*
*
Further reading
Masaki Kashiwara and Algebraic Analysis
Foundations of algebraic analysis book review
Complex analysis
Fourier analysis
Generalized functions
Partial differential equations
Sheaf theory
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