The study of derived categories is a subject that attracts increasingly many mathematicians from various fields of mathematics, including abstract algebra, algebraic geometry, representation theory, and mathematical physics.

The concept of the derived category of sheaves was invented by Grothendieck and Verdier in the 1960s as a tool to express important results in algebraic geometry such as the duality theorem. In the 1970s, Beilinson, Gelfand, and Gelfand discovered that a derived category of an algebraic variety may be equivalent to that of a finite-dimensional non-commutative algebra, and Mukai found that there are non-isomorphic algebraic varieties that have equivalent derived categories. In this way, the derived category provides a new concept that has many incarnations. In the 1990s, Bondal and Orlov uncovered an unexpected parallelism between the derived categories and the birational geometry. Kontsevich's homological mirror symmetry provided further motivation for the study of derived categories.

This book contains the proceedings of a conference held at the University of Tokyo in January 2011 on the current status of the research on derived categories related to algebraic geometry. Most articles are survey papers on this rapidly developing field.

The book is suitable for mathematicians who want to enter this exciting field. Some basic knowledge of algebraic geometry is assumed.