Name: Ana Reguera Title: Divisorial valuations dominating a normal surface singularity. Abstract: >From results by G{\"o}hner and Cutkosky it follows that a complete normal local ring $R$ of dimension 2 has a rational singularity if and only if all graded algebras $gr_\nu R$ defined by divisorial valuations $\nu$ dominating $R$ are noetherian. We will show a constructive way of proving the noetherianness of $gr_\nu R$ when $R$ has a rational singularity, and describe how to get a generating sequence for $\nu$. We will also study the Hilbert function defined by $\nu$.