Given a homogeneous degree five polynomial W in the variables X_1, . . . , X_5, we may view W as defining a quintic hypersurface in P^4 or alternatively, as defining a singularity in [C^5/Z_5], where the group action is diagonal. In the first case, one may consider the Gromov-Witten invariants of {W=0}. In the second case, there is a way to construct analogous invariants, called FJRW invariants, of the singularity. The LG/CY correspondence conjectures that these two sets of invariants are related. In this talk I will explain this correspondence, and its relation to a much older conjecture, the crepant resolution conjecture (CRC). I will sketch a proof that the CRC is equivalent to the LG/CY correspondence in certain cases using a generalization of the ``quantum Serre-duality'' of Coates-Givental. This work is joint with Y.-P. Lee and Nathan Priddis.
