The Arnold conjecture and Floer homology
We shall present the context of the conjecture by V.I. Arnol'd in the 1960's about the least number of fixed points a Hamiltonian symplectomorphism on a compact manifold must have, and will outline the main ideas of Floer's theory in a context as little technical as possible.
This theory builds a homology for compact symplectic manifolds, whose underlying complex is spanned by fixed points of a generic Hamiltonian symplectomorphism. It answers Arnol'd's question with brio but also raises questions as to what its natural generalizations actually compute.