Derived Categories and Fourier-Mukai Transforms

The body of this thesis unfolds in detail the theory of derived categories in order to exploit and explore the geometry of the derived category of coherent sheaves on a projective variety, and inherently of the variety itself. By tracing the development of Fourier-Mukai transforms, we elucidate some of the different modalities and criteria for establishing derived equivalences between two varieties. In particular we delve into some of the results that bridge the geometry of a variety with that of its derived category---among them---we discuss the Bondal-Orlov reconstruction theorem and the derived equivalence between an abelian variety and its dual.