Torus Actions and Integrable Systems

TL;DR

This survey explores how local torus actions relate to integrable dynamical systems, covering topics like normal forms, monodromy, convexity, and integrable PDEs, highlighting their interconnected roles in the theory.

Contribution

It provides a comprehensive overview of the role of torus actions in integrable systems and connects various geometric and analytical aspects.

Findings

Torus actions are central to understanding integrable systems.

Connections between local symmetries and global invariants are elucidated.

Applications to convexity and localization formulas are discussed.

Abstract

This is a survey on natural local torus actions which arise in integrable dynamical systems, and their relations with other subjects, including: reduced integrability, local normal forms, affine structures, monodromy, global invariants, integrable surgery, convexity properties of momentum maps, localization formulas, integrable PDEs.