Title: On the conjectures of Campana, Lang, and Vojta.

Abstract: Why do some polynomial equations have only finitely many solutions in the integers? Lang-Vojta's conjecture provides a conjectural answer and relates this number-theoretic question to complex geometry. I will start out this talk explaining the Lang-Vojta conjectures and provide a survey of currently known results. I will then present two new results:

- If a projective variety has only finitely many rational points over every number field, then it has only finitely many birational automorphisms. (Joint with Junyi Xie.)
- If a projective variety X is a ramified cover of an abelian variety A over a number field K with A(K) dense, then the complement of (the image of ) X(K) in A(K) is still dense. (Joint with Pietro Corvaja, Julian Lawrence Demeio, Davide Lombardo, and Umberto Zannier.)

These results are motivated by the Lang-Vojta conjectures (I will explain how), and also provide evidence for these conjectures.

I will then move on to Lang-Vojta's conjectures over function fields in characteristic zero and explain how to verify a version of Lang-Vojta's conjecture for the moduli space of canonically polarized varieties (joint with Ruiran Sun and Kang Zuo). If time permits, I will discuss the conjecture "opposite" to Lang, as formulated by Campana, and some recent progress here (joint with Erwan Rousseau).