There has been significant progress recently in the birational geometry of algebraic varieties over the complex numbers. Apart from proving the outstanding conjectures in the Minimal Model Program, one of the main practical problems in the field is the lack of examples to which the theory can be applied and which can, in turn, be used as a testing ground for various unresolved conjectures in higher dimensional geometry. This project aims to remedy this situation by constructing new examples of projective algebraic varieties in dimensions at least 3 and using them to test several important conjectures in birational geometry. Furthermore, we will investigate the birational geometry of symplectic singularities from the point of view of representation theory as well as via polyhedral decompositions of cones which occur naturally in Mori theory.
