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Symplectic theory of completely integrable Hamiltonian systems


Authors: Álvaro Pelayo and San Vũ Ngọc
Journal: Bull. Amer. Math. Soc. 48 (2011), 409-455
MSC (2010): Primary 37J35; Secondary 37J05, 37J15, 53D35, 37K10, 53D20, 14H70
DOI: https://doi.org/10.1090/S0273-0979-2011-01338-6
Published electronically: April 25, 2011
MathSciNet review: 2801777
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Abstract: This paper explains the recent developments on the symplectic theory of Hamiltonian completely integrable systems on symplectic $ 4$-manifolds, compact or not. One fundamental ingredient of these developments has been the understanding of singular affine structures. These developments make use of results obtained by many authors in the second half of the twentieth century, notably Arnold, Duistermaat, and Eliasson; we also give a concise survey of this work. As a motivation, we present a collection of remarkable results proved in the early and mid-1980s in the theory of Hamiltonian Lie group actions by Atiyah, Guillemin and Sternberg, and Delzant among others, and which inspired many people, including the authors, to work on more general Hamiltonian systems. The paper concludes with a discussion of a spectral conjecture for quantum integrable systems.


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Additional Information

Álvaro Pelayo
Affiliation: Department of Mathematics, Washington University, One Brookings Drive, Campus Box 1146, St. Louis, Missouri 63130-4899; and School of Mathematics, Institute for Advanced Study, Einstein Drive, Princeton, New Jersey 08540
Email: apelayo@math.wustl.edu; apelayo@math.ias.edu

San Vũ Ngọc
Affiliation: Institut de Recherches Mathématiques de Rennes, Université de Rennes 1, Campus de Beaulieu, 35042 Rennes cedex, France
Email: san.vu-ngoc@univ-rennes1.fr

DOI: https://doi.org/10.1090/S0273-0979-2011-01338-6
Received by editor(s): July 16, 2010
Received by editor(s) in revised form: November 29, 2010, and March 21, 2011
Published electronically: April 25, 2011
Dedicated: In memory of Professor Johannes (Hans) J. Duistermaat (1942–2010)
Article copyright: © Copyright 2011 American Mathematical Society

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