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.
