Abstract. The Landau-Ginzburg Mirror Symmetry Conjecture states that for a quasi-homogeneous singularity W and a group G of symmetries of W , there is a dual singularity W T such that the orbifold A-model of W/G is isomorphic to the B-model of W T . The Landau-Ginzburg A-model is the Frobenius algebra H W,G constructed by Fan, Jarvis, and Ruan, and the B-model is the orbifold Milnor ring of W T . We verify the Landau-Ginzburg Mirror Symmetry Conjecture for Arnol'd's list of unimodal and bimodal quasi-homogeneous singularities with G the maximal diagonal symmetry group, and include a discussion of eight axioms which facilitate the computation of FJRW-rings.

We establish a genus zero correspondence between the equivariant Gromov-Witten theory of the Deligne-Mumford stack [C N /G] and its blowup at the origin. The relationship generalizes the crepant transformation conjecture of Coates-Iritani-Tseng and Coates-Ruan to the discrepant (non-crepant) setting using asymptotic expansion. Using this result together with quantum Serre duality and the MLK correspondence we prove LG/Fano and LG/general type correspondences for hypersurfaces.

We investigate the effect of a general toric wall crossing on genus zero Gromov-Witten theory. Given two complete toric orbifolds X + and X − related by wall crossing under variation of GIT, we prove that their respective I-functions are related by linear transformation and asymptotic expansion. We use this comparison to deduce a similar result for birational complete intersections in X + and X − . This extends the work of the previous authors in [2] to the case of complete intersections in toric varieties, and generalizes some of the results of Coates-Iritani-Jiang [14] on the crepant transformation conjecture to the setting of non-zero discrepancy.