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.