$D$-modules and Microlocal Calculus
About this Title
Masaki Kashiwara, RIMS, Kyoto University, Kyoto, Japan. Translated by Matsumi Saito
Publication: Translations of Mathematical Monographs
Publication Year: 2003; Volume 217
ISBNs: 978-0-8218-2766-6 (print); 978-1-4704-4642-0 (online)
MathSciNet review: MR1943036
MSC: Primary 32C38
Masaki Kashiwara is undoubtedly one of the masters of the theory of $D$-modules, and he has created a good, accessible entry point to the subject. The theory of $D$-modules is a very powerful point of view, bringing ideas from algebra and algebraic geometry to the analysis of systems of differential equations. It is often used in conjunction with microlocal analysis, as some of the important theorems are best stated or proved using these techniques. The theory has been used very successfully in applications to representation theory.
Here, there is an emphasis on $b$-functions. These show up in various contexts: number theory, analysis, representation theory, and the geometry and invariants of prehomogeneous vector spaces. Some of the most important results on $b$-functions were obtained by Kashiwara.
A hot topic from the mid ‘70s to mid ‘80s, it has now moved a bit more into the mainstream. Graduate students and research mathematicians will find that working on the subject in the two-decade interval has given Kashiwara a very good perspective for presenting the topic to the general mathematical public.
Graduate students and research mathematicians.
Table of Contents
- Basic properties of $D$-modules
- Characteristic varieties
- Construction of $D$-modules
- Functorial properties of $D$-modules
- Regular holonomic systems
- Ring of formal microdifferential operators
- Microlocal analysis of holonomic systems
- Microlocal calculus of $b$-functions