Geometric and Numerical Invariants of Derived Categories of Sheaves

In algebraic Geometry derived categories of sheaves on projective varieties are objects of homological nature that encode several properties of the underlying variety. One of the main conjectures relating derived categories to birational geometry is the so-called DK-hypothesis. This asks whether K-equivalent varieties have the same derived category. In this course we will survey this problem. We will first introduce the general background regarding triangulaterd categories, derived functors, Fourier-Mukai transforms and semiortogonal decompositions, and then discuss Bridgeland's result for the 3-dimensional case. Time permitting we will also discuss the problem of the invariance of the number of linearly independent holomorphic one-forms.