Almost twenty years ago, the celebrated theorem of Witten-Kontsevich asserted that the intersection theory of Deligne-Mumford space is governed by KdV-hierarchies. Around the same time, Witten proposed a sweeping generalization which leads to the representation theory of infinite dimensional Lie algebras and singularity theory. In the talk, we will sketch the recent resolution of Witten's vision by Fan-Jarvis and the author and the appearance of new phenomena such as mirror symmetry in the integrable hierarchy problem.