This project is in algebraic geometry, a thriving research area at the heart of modern mathematics. The successful applicant will investigate the geometry of Calabi--Yau varieties geometric objects of central importance, both in algebraic geometry itself and in related areas such as differential geometry and high-enrgy physics. Theoretical understanding of Calabi--Yau varieties remains very incomplete, but in recent years exciting progress has been made in generating extensive databases of examples of Calabi--Yau varieties in dimensions 3 and 4. These databases will provide a rich testing ground in which to develop our understanding. A major focus of the project will be the Morrison--Kawamata conjecture, an important open problem in the field that aims to describe the geometry of algebraic varieties in a uniform way. The student will develop effective methods to verify the conjecture for large families of Calabi--Yau varieties from the databases described above. Successful progress will require the student to use a combination of explicit computational methods together with powerful tools from birational geometry. The project has scope for expansion in several directions, for example the development of a more theoretical approach to the conjecture for Calabi--Yaus.