Geometric Invariant Theory (GIT) is developed in this text within
the context of algebraic geometry over the real and complex
numbers. This sophisticated topic is elegantly presented with
enough background theory included to make the text accessible to
advanced graduate students in mathematics and physics with diverse
backgrounds in algebraic and differential geometry. Throughout the
book, examples are emphasized. Exercises add to the reader's
understanding of the material; most are enhanced with hints. The
exposition is divided into two parts. The first part, 'Background
Theory', is organized as a reference for the rest of the book. It
contains two chapters developing material in complex and real
algebraic geometry and algebraic groups that are difficult to find
in the literature. Chapter 1 emphasizes the relationship between
the Zariski topology and the canonical Hausdorff topology of an
algebraic variety over the complex numbers. Chapter 2 develops the
interaction between Lie groups and algebraic groups. Part 2,
'Geometric Invariant Theory' consists of three chapters (3-5).
Chapter 3 centers on the Hilbert-Mumford theorem and contains a
complete development of the Kempf-Ness theorem and Vindberg's
theory. Chapter 4 studies the orbit structure of a reductive
algebraic group on a projective variety emphasizing Kostant's
theory. The final chapter studies the extension of classical
invariant theory to products of classical groups emphasizing recent
applications of the theory to physics.
General
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