In the 1960s, the KdV equation was discovered to have infinitely many conserved quantities, explained by a "Lax pair" formalism. Due to this, the KdV equation is often described as completely integrable. Similar features were soon found in other nonlinear equations, spurring the field of integrable PDEs in which the KdV equation continues to be one of the flagship models.
These ideas were originally implemented for sufficiently fast decaying initial data and, in the 1970s, for periodic initial data. In this talk, we will describe recent progress for almost periodic initial data, centered around a conjecture of Percy Deift that the solution is almost periodic in time. The case of almost periodic initial data is strongly motivated by the periodic case but carries significant challenges so, beyond a class of algebro-geometric solutions, rigorous results have remained scarce. We will discuss the proof of existence, uniqueness, and almost periodicity in time, in the regime of absolutely continuous and sufficiently "thick" spectrum (in a sense made precise in the talk), and in particular, the proof of Deift's conjecture for small analytic quasiperiodic initial data.