Convexity Properties of Hamiltonian Group Actions (Crm Monograph Series,)
ISBN-13:
9780821839188
ISBN-10:
0821839187
Author:
Victor Guillemin
Publication date:
2005
Publisher:
American Mathematical Society
Format:
Hardcover
82 pages
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Book details
ISBN-13:
9780821839188
ISBN-10:
0821839187
Author:
Victor Guillemin
Publication date:
2005
Publisher:
American Mathematical Society
Format:
Hardcover
82 pages
Summary
Convexity Properties of Hamiltonian Group Actions (Crm Monograph Series,) (ISBN-13: 9780821839188 and ISBN-10: 0821839187), written by authors
Victor Guillemin, was published by American Mathematical Society in 2005.
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Description
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 Morse-theoretic argument deriving the Klyachko inequalities and some of their generalizations. It reviews various infinite-dimensional 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 Kahler manifold, based on potential-theoretic 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.
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