The theory of $D$-modules is powerful, bringing ideas from algebra and algebraic geometry to the analysis of systems of differential equations. This text emphasises $b$-functions, some of the most important results of which were obtained by the author.

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.

Basic properties of $D$-modules
Characteristic varieties
Construction of $D$-modules
Functorial properties of $D$-modules
Regular holonomic systems
$b$-functions
Ring of formal microdifferential operators
Microlocal analysis of holonomic systems
Microlocal calculus of $b$-functions
Appendix
Bibliography
Index
Index of notations