Classical and quantum W-Algebras and applications to Hamiltonian equations

We start by giving an overview of the four fundamental physical theories, namely classical mechanics, quantum mechanics, classical field theory and quantum field theory, and the corresponding algebraic structures, namely Poisson algebras, associative algebras, Poisson vertex algebras and vertex algebras. We then focus on classical and quantum algebras, with a particular emphasis on their application to integrable Hamiltonian PDE.