Lagrangian Floer theory on compact toric manifolds: survey

Abstract: This article is a survey of a series of papers [FOOO3,FOOO4,FOOO5] in which we developed the method of calculation of Floer cohomology of Lagrangian torus orbits in compact toric manifolds, and its applications to symplectic topology and to mirror symmetry. In this article we summarize the main ingredients of calculation and illustrate them by examples. The second half of the survey is devoted to discussion of the most recent result from FOOO5 where the mirror symmetry between the two Frobenius manifolds arising from the big quantum cohomology and from the K. Saito theory of singularities was established.