Symplectic theory of completely integrable Hamiltonian systems

Álvaro Pelayo, San Vũ Ngoc

Published 2013-06-01Version 1

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, of which we also give a concise survey. As a motivation, we present a collection of remarkable results proven in the early and mid 1980s in the theory of Hamiltonian Lie group actions by Atiyah, Guillemin-Sternberg and Delzant among others, and which inspired many people, including the authors, to work on more general Hamiltonian systems. The paper concludes discussing a spectral conjecture for quantum integrable systems.