Short Course: An introduction to the geometry of integrable Hamiltonian systems
Speaker: Daniele Sepe
Dates: to be confirmed.
Syllabus: An important question in Hamiltonian mechanics is to describe qualitative properties of the dynamics under consideration. In general, this is a hard problem but it can be tackled for those systems that are known to be integrable, i.e. that admit the "largest number" of constants of motion. In spite of their relatively simple dynamical behaviour, integrable Hamiltonian systems play a prominent role in Hamiltonian mechanics and beyond, ranging from symplectic geometry and Lie theory to quantum mechanics.
The aim of this course is to provide an introduction to the geometry of such systems from a symplectic perspective. After introducing the necessary tools from symplectic geometry, we will study the structure of integrable Hamiltonian systems near regular points and regular (connected components of) fibres, proving the Darboux-Carathéodory and the Liouville-Arnol'd theorems. The theory will be illustrated by some (simple) examples.
Possible extra topics (if time permits and/or the course is extended to 18 hours): Lagrangian fibrations and their global invariants, an introduction to singular orbits and fibres of integrable systems, the local and global geometry of non-commutative integrable Hamiltonian systems.
This is a joint course with Università di Verona, and it will be taught remotely and in English.