Consider a symplectic manifold $(M,\omega)$, a closed coisotropic submanifold $N$ of $M$, and a Hamiltonian diffeomorphism $\phi$ on $M$. A leafwise fixed point for $\phi$ is a point $x\in N$ that under $\phi$ is mapped to its isotropic leaf. These points generalize fixed points and Lagrangian intersection points. In classical mechanics leafwise fixed points correspond to trajectories that are changed only by a time-shift, when an autonomous mechanical system is perturbed in a time-dependent way.

J. Moser posed the following problem: Find conditions under which leafwise fixed points exist. A special case of this problem is V.I. Arnold's conjecture about fixed points of Hamiltonian diffeomorphisms.

The aim of this minicourse is to provide an overview over solutions to Moser's problem by other people and by myself. I will also discuss applications of the existence of leafwise fixed points, including the following:

* Non-existence of certain presymplectic embeddings, including a spherical nonsqueezing result that improves Gromov's nonsqueezing.

* Existence of a discontinuous capacity. This answers a question by Hofer et al.

I will also explain ways in which Floer theory can be used to address Moser's problem.