Integrable systems on compact Hermitian symmetric spaces

In this talk I shall first briefly discuss the so called Poisson-Nijenhuis structure (a Poisson structure plus a (1,1) tensor with some compatibility conditions), that underlies all integrable systems. In particular, it allows us to find the Hamiltonian in involution as a spectral problem of the Nijenhuis tensor. Then I will focus on a class of manifolds known as the Hermitian symmetric spaces that exhibit such a structure and we shall explore the relation of the integrable system so obtained with the other celebrated integrable system, called the Gelfand-Ceitlin system. I will make the talk accessible to all, even those without any previous experience in Poisson and symplectic geometry (myself included).