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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.

Exponential sums. --- Differential equations. --- Adjoint representation. --- Algebraic geometry. --- Algebraic integer. --- Algebraically closed field. --- Automorphism. --- Base change. --- Bernard Dwork. --- Big O notation. --- Bijection. --- Calculation. --- Characteristic polynomial. --- Codimension. --- Coefficient. --- Cohomology. --- Comparison theorem. --- Complex manifold. --- Conjugacy class. --- Connected component (graph theory). --- Convolution. --- Determinant. --- Diagram (category theory). --- Differential Galois theory. --- Differential equation. --- Dimension (vector space). --- Dimension. --- Direct sum. --- Divisor. --- Eigenvalues and eigenvectors. --- Endomorphism. --- Equation. --- Euler characteristic. --- Existential quantification. --- Exponential sum. --- Fiber bundle. --- Field of fractions. --- Finite field. --- Formal power series. --- Fourier transform. --- Fundamental group. --- Fundamental representation. --- Galois extension. --- Galois group. --- Gauss sum. --- Generic point. --- Group theory. --- Homomorphism. --- Hypergeometric function. --- Identity component. --- Identity element. --- Integer. --- Irreducibility (mathematics). --- Irreducible representation. --- Isogeny. --- Isomorphism class. --- L-function. --- Laurent polynomial. --- Lie algebra. --- Logarithm. --- Mathematical induction. --- Matrix coefficient. --- Maximal compact subgroup. --- Maximal torus. --- Mellin transform. --- Monic polynomial. --- Monodromy theorem. --- Monodromy. --- Monomial. --- Natural number. --- Normal subgroup. --- P-adic number. --- Permutation. --- Polynomial. --- Prime number. --- Pullback. --- Quotient group. --- Reductive group. --- Regular singular point. --- Representation theory. --- Ring homomorphism. --- Root of unity. --- Scientific notation. --- Set (mathematics). --- Sheaf (mathematics). --- Special case. --- Subcategory. --- Subgroup. --- Subring. --- Subset. --- Summation. --- Surjective function. --- Symmetric group. --- Tensor product. --- Theorem. --- Theory. --- Three-dimensional space (mathematics). --- Torsor (algebraic geometry). --- Trichotomy (mathematics). --- Unitarian trick. --- Unitary group. --- Variable (mathematics).