If a Riemannian 2-dimensional manifold M is not simply connected,
producing a nontrivial closed geodesic is relatively simple: it suffices
to minimize the length among all noncontractible loops. Obviously,
this method does not apply when M is diffeomorphic to the sphere.
Nonetheless, in a pioneering work which appeared in 1917, Birkhoff
showed the existence of at least a nontrivial closed geodesic even in
the latter case. A subsequence celebrated refinement of Ljusternik and
Shnirelman showed indeed that there are always at least three distinct
closed geodesics.


   The proof of Birkhoff is probably the first ""min-max"" argument in
the calculus of variations. In this talk we will address higher dimensional versions of his method, showing how to produce minimal hyper-surfaces, i.e. critical points of the area functional, in a quite general setting.


[Invited by Prof. Ruth Kellerhals]