Presenting a selection of recent developments in geometrical problems inspired by the N-body
problem these lecture notes offer a variety of approaches to study them ranging from
variational to dynamical while developing new insights making geometrical and topological
detours and providing historical references. A. Guillot's notes aim to describe differential
equations in the complex domain motivated by the evolution of N particles moving on the plane
subject to the influence of a magnetic field. Guillot studies such differential equations using
different geometric structures on complex curves (in the sense of W. Thurston) in order to find
isochronicity conditions. R. Montgomery's notes deal with a version of the planar Newtonian
three-body equation. Namely he investigates the problem of whether every free homotopy class
is realized by a periodic geodesic. The solution involves geometry dynamical systems and the
McGehee blow-up.A novelty of the approach is the use of energy-balance in order to motivate the
McGehee transformation. A. Pedroza's notes provide a brief introduction to Lagrangian Floer
homology and its relation to the solution of the Arnol'd conjecture on the minimal number of
non-degenerate fixed points of a Hamiltonian diffeomorphism.
