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 the Japanese mathematician Mikio Sato in 1959. This can be seen as an algebraic geometrization of analysis.

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.

From these facts one can obtain a calculus and solutions to linear equations (under appropriate assumptions on solving equations). If the field $ \mathbf F $ is algebraically closed (cf. Algebraically closed field), then solutions of linear equations with scalar coefficients can be calculated by a decomposition of a rational function into vulgar fractions (similarly as in operational calculus). If $ X $ is a commutative algebra, $ \mathbf F = \mathbf C $ and D satisfies the Leibniz condition D ( xy ) = xDy + yDx for x,y \in { \mathop{\rm dom} } D , then the trigonometric identity holds. Some results can be proved also for left-invertible operators, even for operators having either finite nullity or finite deficiency. There is a rich theory of shifts and periodic problems. Recently, logarithms and anti-logarithms have been introduced and studied (even in non-commutative algebras). This means that algebraic analysis is no longer purely linear.

The main advantages of algebraic analysis are:

simplifications of proofs due to an algebraic description of the problems under consideration;

algorithms for solving "similar" problems, although these similarities could be rather far each from another and very formal;

several new results even for the classical operator {d / {dt } } ( which is, indeed, unexpected).

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

Sources

• Kashiwara, Masaki; Kawai, Takahiro (2011). "Professor Mikio Sato and Microlocal Analysis". Publications of the Research Institute for Mathematical Sciences. 47 (1): 11–17. doi:10.2977/PRIMS/29 – via EMS-PH.
• 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