Linear Differential Equations and Group Theory from Riemann to Poincare by J. Gray
introduction
The story begins with the hypergeometric series, studied by Euler and Gauss. This is a power series in x depending on three parameters. It is deeply rooted in classical analysis and it solves a linear differential equation, the hypergeometric equation. Kummer pushed the classical approach to its end by finding the 24 explicit solutions to this equation. These solutions are intricately related to each other; and one solution is defined here, another there, and so on. In short, the situation is clamouring for a Riemann to explain that all of this makes perfect sense complexly in terms of analytic continuation and monodromy relations. This is the way to go. Fuchs developed a general theory of linear differential equations along these lines. Then it's back to the hypergeometric series for more inspiration. For which parameter values is the hypergeometric series an algebraic function? Schwarz discovered that this condition on the three parameters may be expressed as that they correspond to a triangular tessellation. What is this clamouring for if not group theory? Well, that's easy for us to say. Actually, generalising Schwarz's results became a battle between the old and the new. Fuchs and Gordan went at it with invariant theory, but Klein carried the day with group theory and geometry. And the victorious march of these ideas was only just beginning. Dedekind and Klein used them to transform the theory of elliptic modular functions, which old fossils like Fuchs and Hermite had only been able to approach via elliptic functions. Indeed, the basic idea, that of periodicity with respect to a group, "was to prove to be the way historically towards the 'right' generalization of elliptic functions", namely automorphic functions. This is the culmination of the book, and here the story is told with more zeal, through correspondence highlights and so on. Poincare's interest in differential equations lead him to Fuchs's work. Despite "ignorance, even quite astounding ignorance", of much of the above literature, he still immediately discovered the connection with hyperbolic geometry (while boarding a bus, no less). This naturally caught the eye of Klein, who, being "deliberately well-read", felt that he had to inform Poincare about these works and his own perspective "that the task of modern analysis was to find all functions invariant under linear transformations". The famous competition that followed was really "more of a cooperative effort". Eventually Poincare's papers concluded this whole remarkable development, through which solid problems of classical analysis prompted a beautiful theory of complex functions deeply unified with group theory and geometry.