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Books > Science & Mathematics > Mathematics > Calculus & mathematical analysis > Complex analysis
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Riemann Surfaces (Hardcover)
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Riemann Surfaces (Hardcover)
Series: Oxford Graduate Texts in Mathematics, 22
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The theory of Riemann surfaces occupies a very special place in
mathematics. It is a culmination of much of traditional calculus,
making surprising connections with geometry and arithmetic. It is
an extremely useful part of mathematics, knowledge of which is
needed by specialists in many other fields. It provides a model for
a large number of more recent developments in areas including
manifold topology, global analysis, algebraic geometry, Riemannian
geometry, and diverse topics in mathematical physics.
This graduate text on Riemann surface theory proves the fundamental
analytical results on the existence of meromorphic functions and
the Uniformisation Theorem. The approach taken emphasises PDE
methods, applicable more generally in global analysis. The
connection with geometric topology, and in particular the role of
the mapping class group, is also explained. To this end, some more
sophisticated topics have been included, compared with traditional
texts at this level. While the treatment is novel, the roots of the
subject in traditional calculus and complex analysis are kept well
in mind.
Part I sets up the interplay between complex analysis and topology,
with the latter treated informally. Part II works as a rapid first
course in Riemann surface theory, including elliptic curves. The
core of the book is contained in Part III, where the fundamental
analytical results are proved. Following this section, the
remainder of the text illustrates various facets of the more
advanced theory.
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