This book is concerned with two areas of mathematics, at first sight
disjoint, and with some of the analogies and interactions between them.
These areas are the theory of linear differential equations in one
complex variable with polynomial coefficients, and the theory of one
parameter families of exponential sums over finite fields. After
reviewing some results from representation theory, the book discusses
results about differential equations and their differential galois
groups (G) and one-parameter families of exponential sums and their
geometric monodromy groups (G). The final part of the book is devoted to
comparison theorems relating G and G of suitably "corresponding"
situations, which provide a systematic explanation of the remarkable
"coincidences" found "by hand" in the hypergeometric case.