Some applications of parameterized Picard–Vessiot theory

This is an expository article describing some applications of parameterized Picard–Vessiot theory. This Galois theory for parameterized linear differential equations was Cassidy and Singer's contribution to an earlier volume dedicated to the memory of Andrey Bolibrukh. The main results we present here were obtained for families of ordinary differential equations with parameterized regular singularities in joint work with Singer. They include parametric versions of Schlesinger's theorem and of the weak Riemann–Hilbert problem as well as an algebraic characterization of a special type of monodromy evolving deformations illustrated by the classical Darboux–Halphen equation. Some of these results have recently been applied by different authors to solve the inverse problem of parameterized Picard–Vessiot theory, and were also generalized to irregular singularities. We sketch some of these results by other authors. The paper includes a brief history of the Darboux–Halphen equation as well as an appendix on differentially closed fields.