CRM Monograph Series 2005; 82 pp; softcover Volume: 26 ISBN10: 0821842366 ISBN13: 9780821842362 List Price: US$38 Member Price: US$30.40 Order Code: CRMM/26.S
 This is a monograph on convexity properties of moment mappings in symplectic geometry. The fundamental result in this subject is the Kirwan convexity theorem, which describes the image of a moment map in terms of linear inequalities. This theorem bears a close relationship to perplexing old puzzles from linear algebra, such as the Horn problem on sums of Hermitian matrices, on which considerable progress has been made in recent years following a breakthrough by Klyachko. The book presents a simple local model for the moment polytope, valid in the "generic" case, and an elementary Morsetheoretic argument deriving the Klyachko inequalities and some of their generalizations. It reviews various infinitedimensional manifestations of moment convexity, such as the Kostant type theorems for orbits of a loop group (due to Atiyah and Pressley) or a symplectomorphism group (due to Bloch, Flaschka and Ratiu). Finally, it gives an account of a new convexity theorem for moment map images of orbits of a Borel subgroup of a complex reductive group acting on a Kähler manifold, based on potentialtheoretic methods in several complex variables. This volume is recommended for independent study and is suitable for graduate students and researchers interested in symplectic geometry, algebraic geometry, and geometric combinatorics. Titles in this series are copublished with the Centre de Recherches Mathématiques. Readership Graduate students and researchers interested in symplectic geometry, algebraic geometry, and geometric combinatorics. Reviews "This book is written by the hands of two masters in the field and will be useful to all those interested in these topics. ...[T]he graphical conditions are excellent."  Zentralblatt MATH "This monograph gives a beautiful account of a fundamental theme in symplectic geometry: convexity of moment map images ... Guillemin and Sjamaar give a succinct survey of convexity results in symplectic geometry. It is an excellent starting point for someone new to the field, as well as a useful reference for the experts."  Mathematical Reviews Table of Contents  Introduction
 The convexity theorem for Hamiltonian \(G\)spaces
 A constructive proof of the nonabelian convexity theorem
 Some elementary examples of the convexity theorem
 Kähler potentials and convexity
 Applications of the convexity theorem
 Bibliography
