The notion of stability in algebraic geometry was introduced 40 years ago by David Mumford in order to classify vector bundles on (complex) curves. More recently physicists developed the notion of stability of B-branes in string theory. It was Tom Bridgeland who found the appropriate language to unify both approaches of stability. He was able to show that the space Stab(X) of all stability conditions (with some properties) on the derived category of a complex manifold X has itself a natural structure of a complex manifold. This leads immediately to the following questions?

1. How does the geometry of X affects the geometry of Stab(X)?
2. Can we extract some properties of the derived category of X from the geometry of Stab(X)?

I will give a brief introduction to the theory of stability conditions. After a short discussion of the category of coherent sheaves, we will develop the notion of its derived category and of a triangulated category. Finally, I will explain you the notion of t-structures and stability conditions.

I will try to make the talk as elementary as possible, assuming that you already know what a category and a complex manifold is.