# Integrable systems, symmetries and quantization

## TL;DR

This paper discusses classical integrable systems, their symmetries, and quantization, focusing on non-degenerate singularities, semi-toric systems, and the inverse spectral problem in quantum mechanics.

## Contribution

It presents recent research on semi-toric systems and explores the inverse spectral problem connecting quantum spectra to classical integrable systems.

## Key findings

- Analysis of non-degenerate singularities in integrable systems
- Recent developments in semi-toric systems
- Insights into recovering classical systems from quantum spectra

## Abstract

These notes correspond to a mini-course given at the Poisson 2016 conference in Geneva. Starting from classical integrable systems in the sense of Liouville, we explore the notion of non-degenerate singularity and expose recent research in connection with semi-toric systems. The quantum and semiclassical counterpart will also be presented, in the viewpoint of the inverse question: from the quantum mechanical spectrum, can you recover the classical system?