Algebraic analysis

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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 the Japanese mathematician Mikio Sato in 1959.[1] This can be seen as an algebraic geometicazation of analysis.

Microfunction[edit]

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[2]

where

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.

See also[edit]

Citations[edit]

  1. ^ Kashiwara & Kawai 2011, pp. 11–17.
  2. ^ Kashiwara & Schapira 1990, Definition 11.5.1.

Sources[edit]

  • 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[edit]