本书专门论述线性微分方程的伽罗瓦理论，涉及诸多方面：代数理论(尤其是微分伽罗瓦理论)、形式理论、分类、有限项可解性判定算法、单值性、希尔伯特21问题、渐近性和可求和性、反问题以及具正特征值的线性微分方程。附录是本书所用到的代数几何、线性代数群、层及Tannakian范畴中的一些概念。

本书将成为谈领域所有数学家和研究生的标准参考书。

Algebraic Theory

1 Picard-Vessiot Theory

2 Differential Operators and Differential Modules

3 Formal Local Theory

4 Algorithmic Considerations

Analytic Theory

5 Monodromy,the Riemann-Hilbert Problem,and the Differential Galois Group

6 Differential Equations on the Complex Sphere and the Riemann-Hilbert Problem

7 Exact Asymptotics

8 Stoks Phenomenon and Differential Galois Groups

9 Stokes Martrices and Meromorphic Classification

10 Universal Picard-Vessiot Rings and Galois Groups

11 Inverse Problems

12 Moduli for Singular Differential Equations

Appendices

A Algebraic Geometry

B Tannakian Categories

C Sheaves and Cohomology

D Partial Differential Equations

Bibliography

List of Notation

Index