Introduction

Summary

This volume is the outcome of the MSRI special semester on Galois Groups and Fundamental Groups, held in the fall of 1999. Respecting the famous Greek requirements of unity of place, time, and action, the semester was an unforgettable, four-month-long occasion for all mathematicians interested in and responsible for the developments of the connections between Galois theory and the theory of fundamental groups of curves, varieties, schemes and stacks to interact, via a multitude of conferences, lectures and conversations.

Classical Galois theory has developed a number of extensions and ramifications into more specific theories, which combine it with other areas of mathematics or restate its main problems in different situations. Three of the most important of these extensions are geometric Galois theory, differential Galois theory, and Lie Galois theory, all of which have undergone very rapid development in recent years. Each of these theories can be developed in characteristic zero, over the field ℂ of complex numbers, over number fields or p-adic fields, or in characteristic p > 0; various versions of the classical and inverse Galois problems can be posed in each situation. The purpose of this introduction is to give a brief overview of these three themes, which form the framework for all the articles contained in this book.

The main focus of study of geometric Galois theory is the theory of curves and the many objects associated to them: curves with marked points, their fields of moduli and their fundamental groups, covers of curves with their ramification information and their fields of moduli, and the finite quotients of the fundamental group which are the Galois groups of the covers, as well as the moduli spaces and Hurwitz spaces which parametrize all these objects.