Derived categories in algebraic geometry

The first talk will be a colloquium-style introduction to homological
methods in algebraic geometry. I’ll briefly recall the derived category, and
give some examples of equivalences of derived categories ( a.k.a Fourier-Mukai
transforms ) as well as explaining how such examples are constructed. Examples
relate to moduli of bundles, the McKay correspondence and birational geometry
of threefolds. In the later talks I’ll introduce the space of stability
conditions, a construction which allows one to associate complex manifolds to
derived categories. Again, I’ll focus on examples, but I’ll also mention the
relationship with D-branes in string theory.