The classical conjectures of Green—Griffiths—Lang—Vojta predict the precise interplay between different notions of hyperbolicity: Brody hyperbolic, arithmetically hyperbolic, Kobayashi hyperbolic, algebraically hyperbolic, and groupless.
In his thesis, Cherry defined a notion of non-Archimedean hyperbolicity; however, his definition does not seem to be the "correct" version, as it does not mirror complex hyperbolicity. In recent work, Javanpeykar and Vezzani introduced a new non-Archimedean notion of hyperbolicity, which fixed this issue and also stated a non-Archimedean version of the Green-Griffiths-Lang-Vojta conjecture.
In this talk, I will discuss algebraic, complex, and non-Archimedean notions of hyperbolicity and a proof of the non-Archimedean Green—Griffiths—Lang--Vojta conjecture for closed subvarieties of semi-abelian varieties and projective surfaces admitting a dominant morphism to an elliptic curve.