a long standing problem in gromov-witten theory is to compute higher genus gromov-witten invariants of compact calabi-yau manifold such as quintic 3-folds. the defining equation of these calabi-yau manifold has a natural interpretation in landau-ginzburg/singularity theory. more than 15 years ago, witten proposed a pde as a replacement of familiar cauchy-riemann equation in the laudau-ginzburg/singularity setting. furthermore, he proposed two remarkable conjectures for his conjectural theory for ade-singularity. in the talk, we will present a moduli theory of the solution spaces of witten equation. as a consequence, we solve witten's conjectures for quantum theory of ade-singularities. at the end of the talk, we will sketch a plan to compute higher genus gw-invariants of calabi-yau manifolds.