Bernd Schober: Known cases in Resolution of Singularities

Nowadays, there are several quite accessible accounts for Hironaka's theorem of resolution of singularities for varieties over a field of characteristic zero. In contrast to this, there exist only results in small dimensions for the case of positive or mixed characteristic. In my talk I will briefly recall some of the key techniques which lead to a proof for resolution of singularities in characteristic zero. After this I will point out why these can not always be applied in positive characteristic. Finally, I will mention some of the known results in low dimensions and talk about special cases in higher dimensions.