Symplectic theory of completely integrable Hamiltonian systems

TL;DR

This paper reviews recent advances in the symplectic theory of completely integrable Hamiltonian systems on 4-manifolds, emphasizing singular affine structures and their historical development, and discusses a spectral conjecture for quantum systems.

Contribution

It provides a comprehensive survey of recent symplectic developments and connects classical results with modern quantum spectral conjectures.

Findings

Understanding of singular affine structures in integrable systems

Survey of key historical results by Arnold, Duistermaat, and Eliasson

Discussion of a spectral conjecture for quantum integrable systems

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, 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.