The Moment-Weight Inequality and the Hilbert-Mumford Criterion: GIT from the Differential Geometric Viewpoint

by Valentina Georgoulas, Joel W. Robbin, Dietmar Arno Salamon

Overview

This book provides an introduction to geometric invariant theory from a differential geometric viewpoint.  It is inspired by certain infinite-dimensional analogues of geometric invariant theory that arise naturally in several different areas of geometry. The central ingredients are the moment-weight inequality relating the Mumford numerical invariants to the norm of the moment map, the negative gradient flow of the moment map squared, and the Kempf--Ness function. The exposition is essentially self-contained, except for an appeal to the Lojasiewicz gradient inequality. A broad variety of examples illustrate the theory, and five appendices cover essential topics that go beyond the basic concepts of differential geometry. The comprehensive bibliography will be a valuable resource for researchers.

The book is addressed to graduate students and researchers interested in geometric invariant theory and related subjects.  It will be easily accessible to readers with a basic understanding of differential geometry and does not require any knowledge of algebraic geometry. 

Product Details

ISBN-13:	9783030893002
Publisher:	Springer International Publishing
Publication date:	01/01/2022
Series:	Lecture Notes in Mathematics , #2297
Sold by:	Barnes & Noble
Format:	eBook
File size:	13 MB
Note:	This product may take a few minutes to download.

About the Author

Valentina Georgoulas was born in Zürich in 1978. She studied theoretical physics at ETH Zürich, where she received her master degree in 2008. She then went on to doctoral studies in mathematics and in 2016 completed her PhD in mathematics at ETH Zürich under the direction of Dietmar Salamon. After further training, she now works as a high school teacher for mathematics at MNG Rämibühl in Zürich. She is the mother of two children.
Joel W. Robbin was born in Chicago in 1941 and completed his PhD at Princeton University in 1965 under the direction of Alonzo Church. After a postdoctoral position in Princeton he took up an Assistant Professorship at the University of Wisconsin-Madson in 1967, where he became full Professor in 1973, and emeritus in 2010. Joel Robbin began his research in mathematical logic (and wrote a text book on this subject) and later moved on to dynamical systems and symplectic topology. In 1970 he proved a conjecture by Stephen Smale which asserts that Axiom A implies structural stability. His publications include a book on "Matrix Algebra" and a joint book with Ralph Abraham on "Transversal Mappings and Flows". He is a Fellow of the American Mathematical Society.
Dietmar A. Salamon was born in Bremen in 1953 and completed his PhD at the University of Bremen in 1982 under the direction of Diederich Hinrichsen. After postdoctoral positions in Madison and Zrich, he took up a position at the University Warwick in 1986, and moved to ETH Zurich in 1998. His field of research is symplectic topology and related subjects. He was an invited speaker at the ECM 1992 in Paris, at the ICM 1994 in Zurich, and at the ECM 2000 in Barcelona. He delivered the Andrejewski Lectures in Goettingen (1998) and at the Humboldt Unversity Berlin (2005), and the Xth Lisbon Summer lectures in Geometry (2009). He is the author of several text books and research momgraphs including two joint books with Dusa McDuff entitled "Introduction to Symplectic Topology" and "J-holomorphic Curves and Symplectic Topology" for which they received the Leroy P. Steele Prize for Mathematical Exposition in 2017.

Table of Contents

- Introduction. - The Moment Map. - The Moment Map Squared. - The Kempf–Ness Function. - μ-Weights. - The Moment-Weight Inequality. - Stability in Symplectic Geometry. - Stability in Algebraic Geometry. - Rationality. - The Dominant μ-Weight. - Torus Actions. - The Hilbert–Mumford Criterion. - Critical Orbits.

Editorial Reviews

The book is carefully written with detailed proofs and should be accessible to anyone with a basic background knowledge of (real and complex) differential geometry. … This book is an excellent addition to the literature on GIT and symplectic reduction and could serve as an ideal reference for a graduate lecture course or seminar, but also as a valuable source for researchers interested in GIT or in the problems in geometry where the ideas appear in an infinite-dimensional context.” (Markus Röser, Mathematical Reviews, May, 2023)

From the Publisher