Strong Cosmic Censorship Conjecture and Black Hole Dynamics

In the wake of major breakthroughs in General Relativity during the 1960s, Roger Penrose introduced Strong Cosmic Censorship, a profound conjecture regarding the deterministic nature of the theory. Nearly six decades later, Penrose's conjecture remains one of the most captivating unsolved problems in general relativity. Perhaps the most critical arena for testing the conjecture is the spacetime region hidden behind the event horizon: the interior of black holes. The key challenge is to prove the dynamical formation of singularities for solutions to the Einstein equations in this context. We will discuss recent results on Strong Cosmic Censorship leveraging modern techniques in the theory of quasilinear waves equations at the crossroad of PDEs and Lorentzian geometry.