Abstract: Some 30 years ago examples were given of holomorphic functions on the unit disc with the property that their graph is precisely the $-\infty$ set of a plurisubharmonic function on $\CC^2$. This led to the conjecture that this happens for every nowhere extendable holomorphic function on the disc. While some support for this conjecture was found, it turned out that the conjecture is false. There are counterexamples with seemingly strange behavior. This behavior can be understood by considering finely holomorphic functions instead of the usual ones. In the talk I will discuss all this and explain the relevant notions.
