In 1960s, Arnold conjectured that the number of fixed points of a Hamiltonian diffeomorphism of a symplectic manifold should be (much) greater than its counterpart for a general diffeomorphism. This conjecture has been a constant driven force for major developments in modern symplectic topology, notably various forms of Floer theory. In this lecture, after reviewing previous landmark results, I will present our state-of-art understanding of the Arnold conjecture and the corresponding new ideas in Floer theory. This is joint work with Guangbo Xu.
