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