Lectures on mirror symmetry, derived categories, and $D$-branes

This paper, mainly intended for a mathematical audience, is an introduction to homological mirror symmetry, derived categories, and topological $D$-branes. Mirror symmetry from the point of view of physics is explained, along with the relationship between symmetry and derived categories, and the reason why the Fukaya category must be extended by using co-isotropic $A$-branes. There is also a discussion of how to extend the definition of the Floer homology to these objects and a description of mirror symmetry for flat tori. The paper consists of four lectures given at the Institute of Pure and Applied Mathematics (Los Angeles) in March 2003, as a part of the programme “Symplectic Geometry and Physics”.