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
In mathematics, a sheaf 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 data could ...
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
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
is a Japanese mathematician known for founding the fields of algebraic analysis, hyperfunctions, and holonomic quantum fields. He is a professor at the Research Institute for Mathematical Sciences in Kyoto.
Education
Sato studied at the Univ ...
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
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010) ...
-
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 ge ...
of
dimension
In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
''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 derivativ ...
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
*
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. There is more than one recognized theory, for example the theory of distributions. Generalized functions are especially useful in making discontinuous functions ...
*
Edge-of-the-wedge theorem
In mathematics, Bogoliubov's edge-of-the-wedge theorem implies that holomorphic functions on two "wedges" with an "edge" in common are analytic continuations of each other provided they both give the same continuous function on the edge. It is u ...
*
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
In commutative algebra and algebraic geometry, localization is a formal way to introduce the "denominators" to a given ring or module. That is, it introduces a new ring/module out of an existing ring/module ''R'', so that it consists of fraction ...
*
Vanishing cycle In mathematics, vanishing cycles are studied in singularity theory and other parts of algebraic geometry. They are those homology cycles of a smooth fiber in a family which vanish in the singular fiber.
For example, in a map from a connected comp ...
*
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 o ...
*
Differential algebra
In mathematics, differential rings, differential fields, and differential algebras are rings, fields, and algebras equipped with finitely many derivations, which are unary functions that are linear and satisfy the Leibniz product rule. A n ...
*
Perverse sheaf The mathematical term perverse sheaves refers to a certain abelian category associated to a topological space ''X'', which may be a real or complex manifold, or a more general topologically stratified space, usually singular. This concept was intro ...
*
Mikio Sato
is a Japanese mathematician known for founding the fields of algebraic analysis, hyperfunctions, and holonomic quantum fields. He is a professor at the Research Institute for Mathematical Sciences in Kyoto.
Education
Sato studied at the Univ ...
*
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
Lars Valter Hörmander (24 January 1931 – 25 November 2012) was a Swedish mathematician who has been called "the foremost contributor to the modern theory of linear partial differential equations". Hörmander was awarded the Fields Medal ...
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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