Geometric Invariant Theory: Over the Real and Complex Numbers (Universitext)

December 8, 2020
Geometric Invariant Theory: Over the Real and Complex Numbers (Universitext)

Designed for non-mathematicians, physics students as well forexample, who want to learn about this important area ofmathematicsWell organized and touches upon the main subjects, which offera deeper understanding of the orbit structure of an algebraicgroupPainless presentation places the subject within reasonablereach for mathematics and physics student at the graduatelevel——————————Geometric Invariant Theory (GIT) is developed in this textwithin the context of algebraic geometry over the real and complexnumbers. This sophisticated topic is elegantly presented withenough background theory included to make the text accessible toadvanced graduate students in mathematics and physics with diversebackgrounds in algebraic and differential geometry. Throughout thebook, examples are emphasized. Exercises add to the reader’sunderstanding 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 ofthe book. It contains two chapters developing material in complexand real algebraic geometry and algebraic groups that are difficultto find in the literature. Chapter 1 emphasizes the relationshipbetween the Zariski topology and the canonical Hausdorff topologyof an algebraic variety over the complex numbers. Chapter 2develops 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 andcontains a complete development of the Kempf–Ness theorem andVindberg’s theory. Chapter 4 studies the orbit structure of areductive algebraic group on a projective variety emphasizingKostant’s theory. The final chapter studies the extension ofclassical invariant theory to products of classical groupsemphasizing recent applications of the theory to physics.