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Bulletin of the American Mathematical Society

The Bulletin publishes expository articles on contemporary mathematical research, written in a way that gives insight to mathematicians who may not be experts in the particular topic. The Bulletin also publishes reviews of selected books in mathematics and short articles in the Mathematical Perspectives section, both by invitation only.

ISSN 1088-9485 (online) ISSN 0273-0979 (print)

The 2020 MCQ for Bulletin of the American Mathematical Society is 0.84.

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Symplectic theory of completely integrable Hamiltonian systems
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by Álvaro Pelayo and San Vũ Ngọc PDF
Bull. Amer. Math. Soc. 48 (2011), 409-455 Request permission


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
  • MR Author ID: 731609
  • Email:;
  • San Vũ Ngọc
  • Affiliation: Institut de Recherches Mathématiques de Rennes, Université de Rennes 1, Campus de Beaulieu, 35042 Rennes cedex, France
  • Email:
  • 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)
  • © Copyright 2011 American Mathematical Society
  • Journal: Bull. Amer. Math. Soc. 48 (2011), 409-455
  • MSC (2010): Primary 37J35; Secondary 37J05, 37J15, 53D35, 37K10, 53D20, 14H70
  • DOI:
  • MathSciNet review: 2801777