My main interest is the geometry of uniruled varieties. The appearence of Mori Theory first and Minimal Model Program then, brought a breaktrough in the study of both birational and biregular theory of uniruled varieties, cfr J. Kollàr, S. Mori Birational geometry of uniruled varieties, Cambridge Univ. Press 134 (1999).
Thank's to this powerful theories and techniques we have now a good philosophical understanding of those varieties.
The next step is to deepen our knowledge and to start unveil a finer classification.
biregular a better understanding of terminal Fano varieties
birational a concrete understanding of birational modification between Mori spaces