Question from Rene - Byzantine cultural transfer

We know about the scientific and cultural transfers between Islam and mediaeval Christianity, especially in the days of the Crusader states. Was there any significant scientific and/or cultural transfer from Byzantium into Catholic Europe in the 13th and 14th centuries too, or did the Papacy discourage study of the ways of the Greek "apostates" too?

The reason why I ask is that far too often the links between Western Catholicism and Islam are frequently touched upon by mainstream Western media (read: Hollywood) but hardly anything is mentioned about the Byzantines who would've been more advanced and prosperous ....