The moduli space $\overline{M}_{0,n}$ of stable curves of genus 0 with $n$ marked points is a compactification of the set

$M_{0,n}=(\mathbb{P}^1)^n-\Delta/\mathrm{Aut}(\mathbb{P}^1), \quad \Delta=\bigcup_{i\ne j}\{p_i=p_j\}$

of $n$ distinct ordered points on $\mathbb{P}^1$, obtained by allowing the curve to degenerate into a tree of rational curves while keeping the $n$ marked points distinct. It is one of the most important varieties in algebraic geometry and has been much studied. It comes with a natural action of the symmetric group $S_n$ which permutes the marked points. In this talk, I will report on recent progress about the cohomology of $\overline{M}_{0,n}$ as a graded $S_n$-representation and its characteristic polynomial as well as the log-concavity conjecture. Based on joint works with Jinwon Choi and Donggun Lee (arxiv:2203.05883, 2408.10728, 2505.01087).