Survey on recent developments in semitoric systems

Abstract

Semitoric systems are a special class of four-dimensional completely integrable systems where one of the first integrals generates an $\mathbb{S}^1$-action. They were classified by Pelayo & Vu Ngoc in terms of five symplectic invariants about a decade ago. We give a survey over the recent progress which has been mostly focused on the explicit computation of the symplectic invariants for families of semitoric systems depending on several parameters and the generation of new examples with certain properties, such as a specific number of singularities of lowest rank.