Foreword

Summary

A Riemann surface is a thing of beauty, possessing geometric shape as well as analytic or algebraic structure. From its introduction in 1851 in Riemann's inaugural dissertation, his first great work establishing the foundations of geometric complex analysis, the concept has exerted an unusual influence as a powerful clarifying mental tool.

Today, the pervasive role of complex analysis in the mathematical and physical sciences has brought these ideas into a significance wider than even their founder could have predicted. In the present book, the reader will find a selection of results which can only indicate the part currently played by surfaces and their spaces of deformations: just as a single convergent power series is enough to generate by continuation an entire Riemann surface structure, so the foundational ideas of our discipline extend and evolve beyond our present view of them.

Central to the contemporary study of Riemann surfaces is the interplay between different aspects, geometric ideas and algebraic or analytical calculations, leading to insights into the deeper properties these objects possess. The basic notion provides a topological base for deploying the most powerful ideas of algebra, geometry and analysis: indeed it establishes a central role for topology in bringing about a unique mathematical synthesis. A single accessible theory serves to interconnect complex analysis and the various algebraic invariants, the fundamental group, field of functions, homology and period lattices. In the reassuring familiarity of a two dimensional framework, we have a global base for complex analytic and covering space methods, interacting with Galois-theoretic properties of the function field.