The talk explains singular Coxeter combinatorics, i.e., combinatorics of parabolic double cosets in a Coxeter group. In particular, we give a generators and relations presentation (or two) of the double cosets. Here appears a singular analogue of the simple reflections, called atoms. Atoms generate a new combinatorial structure which, by a work of Iyama and Wemyss, describes the tilting theory of contracted (i.e. idempotent subalgebras of) preprojective algebras.
Based on a joint project with Ben Elias, Nico Libedinsky, Leonardo Patimo.