An invitation to homological mirror symmetry

We will give a gentle introduction to some recent developments in the area of homological mirror symmetry. We will use simple examples to illustrate Kontsevich’s conjecture and its extension beyond the Calabi-Yau setting in which it was first formulated. We will mostly focus on a one-dimensional example, the pair of pants, to give a flavor of the geometric concepts involved in a general formulation of homological mirror symmetry. If time permits, we will then describe a program to prove homological mirror symmetry for essentially arbitrary complete intersections in toric varieties (joint work in progress with Mohammed Abouzaid).