Ever since the concepts of Galois groups in algebra and fundamental
groups in topology emerged during the nineteenth century,
mathematicians have known of the strong analogies between the two
concepts. This book presents the connection starting at an
elementary level, showing how the judicious use of algebraic
geometry gives access to the powerful interplay between algebra and
topology that underpins much modern research in geometry and number
theory. Assuming as little technical background as possible, the
book starts with basic algebraic and topological concepts, but
already presented from the modern viewpoint advocated by
Grothendieck. This enables a systematic yet accessible development
of the theories of fundamental groups of algebraic curves,
fundamental groups of schemes, and Tannakian fundamental groups.
The connection between fundamental groups and linear differential
equations is also developed at increasing levels of generality. Key
applications and recent results, for example on the inverse Galois
problem, are given throughout.
General
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