Group theory and differential equations [Lecture notes]

Group Theory and Differential Equations
TABLE OF CONTENTS

I Lie Theory of Transformation Groups
Historical background
Examples of different types of differential equations and transformation groups
Examples of abstract groups
One parameter transformation groups on R2
Invariants of one parameter transformation groups on R2
Canonical forms for certain second order differential equations
Topological groups
Lie groups
Lie algebras
Infinitesimal transformation groups
Differential invariants of transformation groups
The complete transformation group of a second order differential system
Solvable infinitesimal transformation groups and the solution of differential equations by quadrature
Problems
Bibliography


II The Monodromy Group and Fuchsian Differential Equations
Introduction An example of the monodromy group
Survey of properties of analytic functions of several variables
Existence uniqueness continuation, and singularities of solutions
Complex manifolds
First order nonlinear differential equations
Linear differential equations and the monodromy group
Fuchsian differential equations on Riemann surfaces
Local theory of Fuchsian differential equations
Fuchsian differential equations on the Riemann sphere
Remarks on the second order Fuchsian equation on St
The hypergeometric differential equation on the Riemann sphere
Hypergeometric equations with a finite monodromy group
Existence and uniqueness theorems for Fuchsian differential equations with a prescribed monodromy group
Problems
Bibliography

III The Galois or Rationality Group of a Linear Homogeneous Differential Equation
Integration of differential equations in finite terms containing elementary functions
Differential field extensions and Liouville elementary functions
Transcendental and Hypertranscendental field extensions Holder's theorem on the Gamma function
Functions of finite order and Liouville's principle
Liouville‘s theory of the Bessel and Riccati differential equations
Liouville, generalized Liouville, and Picard-Vessiot extensions of differential fields and solvability of differential equations
Galois group and Galois correspondence
Examples of the Galois groun of a Picard-Vessiot extension
Ideals and algebraic varieties Zariski topology
Algebraic matrix groups
Solvable algebraic matrix groups
The Galois group of a Picard-Vessiot extension is an algebraic matrix group
A Picard-Vessiot extension is normal
Completion of the Galois correspondence
Solution of differential equations in elementary functions
Problems
Bibliography