*note time change*

We will present some recent results on the long-time regularity of solutions to the free
boundary Euler equations.

First, we will discuss a result for the full problem with vorticity and gravity in three space
dimensions. For small and localized initial data, we show regularity up to a time that is (almost)
of the order $1/\epsilon$, where $\epsilon$ is the size of the initial vorticity. This is a
natural time scale for the evolution of the vorticity and, importantly, it is independent of the
size of the irrotational components. This is joint work with Daniel Ginsberg (CUNY).

We will then discuss recent and ongoing joint works with Yu Deng (USC) and Alexandru Ionescu
(Princeton) on irrotational solutions on large tori. We obtain new deterministic results for
gravity waves in one space dimension and a long-time result for random solutions. These works are
motivated by the theory of weak wave turbulence.