Mirror symmetry is a collection of conjectural relationships between symplectic and algebraic geometry, inspired by the physics of superstrings. Though it is far from being fully understood, it has already had deep impact in geometry and related fields including representation theory.
According to Kontsevich’s homological mirror symmetry proposal, it may be formulated by equivalences between certain symplectic and algebraic categories. I will introduce these objects, and do some case studies: in particular, I will develop toric geometry of varieties and orbifolds to give a source of examples, and discuss recent progress in understanding mirror symmetry via constructible sheaves.
Some familiarity with symplectic or algebraic geometry.