Joint IAS/Princeton/Columbia Symplectic Geometry Seminar

A point $q$ in a contact manifold $(M,\xi)$ is said to be a translated point of a contactomorphism $\phi$, with respect to a contact form $\alpha$ for $\xi$, if it is a "fixed point modulo the Reeb flow", i.e. if $q$ and $\phi(q)$ are in the same Reeb orbit and $\phi$ preserves $\alpha$ at $q$. Translated points are key objects to look at when studying contact rigidity phenomena such as contact non-squeezing, orderability of contact manifolds and existence of bi-invariant metrics and quasimorphisms on the contactomorphism group. Based on the notion of translated points, in 2011 I proposed a contact analogue of the Arnold conjecture on fixed points of Hamiltonian symplectomorphisms. In my talk I will present a proof of this conjecture under the assumption that there are no closed contractible Reeb orbits, by means of a Floer homology theory for translated points that I am developing ad hoc to study this problem.