Disproving the Deift conjecture: the loss of almost periodicity

In this note we present a result from [CKV24] related to a conjecture of Deift from 2008, who posited that almost periodic initial data leads to almost periodic solutions to the Korteweg-de Vries equation (KdV). We show that this is not always the case. Building on the new observation that the conjecture fails for the Airy equation, we construct almost periodic initial data whose KdV evolution remains bounded, but loses almost periodicity at a later time. This text is based on a Laurent Schwartz seminar given by the first author in November 2024, which in turn is based on joint work with Rowan Killip and Monica Vişan.