nLab algebraic analysis

Algebraic analysis is a program introduced by Mikio Sato from around 1958, based on the idea that the study of differential equations should be done in a coordinate-free manner, and operations should follow general nonsense geometric and algebraic constructions. One of the first steps was the introduction of the concept of D-module, and of a holonomic D-module, hyperfunctions (as a sheaf theoretic approach to distribution theory), then relying on homological algebra in derived categories (it seems that Sato introduced them independently from Grothendieck-Verdier, without publication at the time). Study of nonlinear and nonholonomic system was supposed to reduce on study of holonomic systems on more complicated spaces, e.g. on the cartesian square of the original space and so on. This depends on subtle properties of the study of singularities and other aspects and works only in some generality, with ongoing progress. Singularity theory and the lagrangian geometry are very important aspects of the algebraic analysis. Later Sato introduced microlocalization and his program joined young Masaki Kashiwara around 1968. Numerous connections to mathematical physics (e.g. holonomic quantum fields, integrable systems) and Hodge theory gradually entered into the program.

It seems that the vision of this program fits well with nPOV. See also D-geometry.

• interview with Mikio Sato in Notices AMS

• Masaki Kashiwara, Takahiro Kawai, Tatsuo Kimura, Foundations of algebraic analysis, Transl. from Japanese by Goro Kato. Princeton Mathematical Series 37, 1986. xii+255 pp.

  MR87m:58156; J.-L. Brylinski, Book Review: Foundations of algebraic analysis. Bull. Amer. Math. Soc. (N.S.) 18 (1988), no. 1, 104–108, doi

• Tadao Oda, Introduction to algebraic analysis on complex manifolds, Algebraic varieties and analytic varieties (Tokyo, 1981), 29–48, Adv. Stud. Pure Math., 1, North-Holland, Amsterdam (1983) $[$doi:10.2969/aspm/00110029, MR85a:14010$]$

A short but quite complete overview of algebraic analysis can be found in:

• Pierre Schapira Triangulated categories for the analysts. In “Triangulated categories” London Math. Soc. LNS 375 Cambridge University Press, pp 371-389 (2010) pdf;