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Lectures on Symplectic Manifolds
Alan Weinstein, University of California, Berkeley, CA
A co-publication of the AMS and CBMS.
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CBMS Regional Conference Series in Mathematics
1977; 48 pp; softcover
Number: 29
Reprint/Revision History:
reprinted with corrections 1979; fifth printing 2000
ISBN-10: 0-8218-1679-9
ISBN-13: 978-0-8218-1679-0
List Price: US$19 Member Price: US$15.20
All Individuals: US\$15.20
Order Code: CBMS/29

The first six sections of these notes contain a description of some of the basic constructions and results on symplectic manifolds and lagrangian submanifolds. Section 7, on intersections of largrangian submanifolds, is still mostly internal to symplectic geometry, but it contains some applications to machanics and dynamical systems. Sections 8, 9, and 10 are devoted to various aspects of the quantization problem. In Section 10 there is a feedback of ideas from quantization theory into symplectic geometry itslef.

Reviews

"This volume of lecture notes is devoted to some problems in differential topology arising from the study of Hamiltonian systems and geometrical quantization ... After the necessary definitions are given, some topological facts and problems concerning symplectic and Lagrangian manifolds are presented and their origin in abstract Hamiltonian mechanics is outlined. The author treats the classification problem for symplectic manifolds and discusses the relevant theorems of Darboux and Moser, and comments on the embedding problem and intersection theory for Lagrangian submanifolds. The last three chapters are devoted to geometric quantization in connection with representation theory of Lie groups, the quasi-classical approximation in quantum mechanics, and the theory of Fourier integral operators. The author states several open problems and supplies a detailed bibliography."

-- Mathematical Reviews

• Introduction
• Symplectic manifolds and lagrangian submanifolds, examples
• Lagrangian splittings, real and complex polarizations, Kähler manifolds
• Reduction, the calculus of canonical relations, intermediate polarizations
• Hamiltonian systems and group actions on symplectic manifolds
• Normal forms
• Lagrangian submanifolds and families of functions
• Intersection theory of lagrangian submanifolds
• Quantization on cotangent bundles
• Quantization and polarizations
• Quantizing lagrangian submanifolds and subspaces, construction of the Maslov bundle
• References