The evolution in time of a quantum system is characterized by the Hamiltonian. The talk will focus on the Hamiltonian related to the N-body problem. After introducing some notations and context for the N-body problem, I will present a result on the decomposition of the essential spectrum of a class of operators that contained the Hamiltonian. This result is a generalization of the well-known HVZ-theorem. This decomposition is obtained using the theory of algebra of operators with, in particular, the use of C*-algebras. In a second step, I will give results on the regularity of the eigenfunctions of the Hamiltonian. This regularity result is obtained using the theory of differentials geometry, with a focus on the blow-up of manifolds with corners. I will end with a result connecting these two approaches: the operators algebra's one and the differential's geometry one's. This connecting result show that these two approach are actually built on the same space. These are joint work with B. Amman, N. Prudhon and V. Nistor.

In this talk I will present some recent improvements in the study of homogenisation of compressible viscous fluid in the case of tiny holes and the connection with fluid structure problems. In particular I highlight a situation where the fluid + rigid body problem has more flexibility respect to case of the fluid alone. This flexibility helps us to deduce new ideas to study the case of the fluid alone.