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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.

Mathematics. --- Algebraic geometry. --- Group theory. --- Algebraic Geometry. --- Group Theory and Generalizations. --- Geometry, algebraic. --- Algebraic geometry --- Geometry --- Groups, Theory of --- Substitutions (Mathematics) --- Algebra --- Invariants.

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Real Productive Groups I

Topological groups. Lie groups --- Lie groups. --- Representations of groups. --- Group representation (Mathematics) --- Groups, Representation theory of --- Group theory --- Groups, Lie --- Lie algebras --- Symmetric spaces --- Topological groups

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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.

Group theory --- Algebraic geometry --- Geometry --- landmeetkunde --- wiskunde --- geometrie

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Real Productive Groups I.

Topological groups. Lie groups

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Mathematical physics --- 514.7 --- Symplectic manifolds --- Fourier analysis --- Lie groups --- Quantum theory --- Quantum dynamics --- Quantum mechanics --- Quantum physics --- Physics --- Mechanics --- Thermodynamics --- Groups, Lie --- Lie algebras --- Symmetric spaces --- Topological groups --- Analysis, Fourier --- Mathematical analysis --- Manifolds, Symplectic --- Geometry, Differential --- Manifolds (Mathematics) --- Differential geometry. Algebraic and analytic methods in geometry --- Fourier analysis. --- Lie groups. --- Quantum theory. --- Symplectic manifolds. --- 514.7 Differential geometry. Algebraic and analytic methods in geometry --- Variétés symplectiques --- Global differential geometry --- Géométrie différentielle globale --- Fourier, Analyse de --- Théorie quantique --- Géométrie différentielle globale. --- Théorie quantique --- Variétés symplectiques