algebraic analysis

English

Etymology

The area of study in algebraic geometry began as a research programme, instigated in 1959 by Japanese mathematician Mikio Sato.

Noun

algebraic analysis (uncountable)

1. Used other than figuratively or idiomatically: see algebraic,‎ analysis.; the use of techniques from algebra (especially elementary algebra) to analyse and solve problems.
2. (algebraic geometry, algebraic topology) The study of systems of linear partial differential equations, using techniques from sheaf theory and complex analysis to examine functions and certain generalisations of functions (such as hyperfunctions and microfunctions).
   • 1986, Goro Kato (translator), Masaki Kashiwara, Takahiro Kawai, Tatsuo Kimura, Foundations of Algebraic Analysis, Princeton Universty Press, [1].
   • 1999, Goro Kato, Daniele C. Struppa, Fundamentals of Algebraic Microlocal Analysis, Marcel Dekker, page v:

     In their classical work [123], Kashiwara, Kawai, and Kimura pointedly observe that the Japanese algebraic analysis is really the algebraic analysis in the tradition of Euler; we may add that what we mean by algebraic microlocal analysis is the successful attempt to adapt the methods of abstract algebraic geometry to the non-commutative setting in which the base ring is now the ring of variable coefficients^[sic] partial differential operators.

   • 2008, Yoshitsugu Takei, The work of T. Kawai on exact WKB analysis, T. Aoki, H. Majima, Y. Takei, N. Tose (editors), Algebraic Analysis of Differential Equations, Springer, page 19,

     Thus Kawai has been engaged mainly in the research of exact WKB analysis or algebraic analysis of singular perturbation theory since 1989.

Translations

study of systems of linear PDEs

• French: analyse algébrique f
• Japanese: 代数解析
• Spanish: análisis algebraico m

Further reading

• D-module on Wikipedia.Wikipedia
• Hyperfunction on Wikipedia.Wikipedia
• Microlocal analysis on Wikipedia.Wikipedia