Strong generation, approximability theorems, representability and applications

In a 2003 paper Bondal and Van den Bergh proved that, if $X$ is a scheme smooth over a field $k$, then the category $D^b(Coh X)$ is strongly generated. In 2008 Rouquier developed and expanded this considerably, introducing what is now called the Rouquier dimension of a triangulated category. I will review the known results, discuss applications and explain some recent improvements.

At the end of the talk I would like to discuss a very recent development. Motivated by beautiful work of Jack Hall, I have found a major improvement to another theorem in the 2003 paper by Bondal and Van den Bergh. I will say something about Hall's new proof of GAGA and the direction it has led me.