Survey on recent developments in semitoric systems

TL;DR

This survey reviews recent progress in semitoric systems, focusing on computing symplectic invariants, generating new examples, and understanding their properties within four-dimensional integrable systems.

Contribution

It summarizes recent advances in explicit computation of symplectic invariants and the construction of new semitoric system examples.

Findings

Explicit computation of symplectic invariants for parameter-dependent families

Development of new semitoric system examples with specific singularity properties

Enhanced understanding of the structure and classification of 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.