Let $k$ be a perfect field of characteristic $p>0$. A. Petrov has recently constructed a projective, smooth scheme $X/W(k)$, of relative dimension $p+1$, such that the Hodge to de Rham spectral sequence of $X_0/k$, where $X_0 = X \otimes_{W(k}k$, does not degenerate at $E_1$, thus solving a question of Deligne-Illusie left open since 1987. I will explain the main ideas of the proof, which uses inputs from Bhatt-Lurie's theory of diffracted Hodge complexes and combines  techniques of homotopical algebra and representation theory of algebraic groups.