PhD in Mathematical Sciences

Calabi-Yau manifolds are an important class of objects in algebraic geometry, yet many of their properties are still poorly understood. Evidence from theoretical physics, and in particular string theory, suggests that Calabi-Yau manifolds seem to come in pairs, with related properties. This observed phenomenon is called mirror symmetry. However, whilst we now have thousands of examples of such 'mirror pairs' of Calabi-Yau manifolds, there is still no underlying mathematical theory to explain how and why they arise. This project aims to use algebraic geometry to study the relationship between 'fibrations' on a Calabi-Yau manifold and 'degenerations' of its mirror partner. Fibrations can be thought of as ways to slice a Calabi-Yau manifold into thin slices, whilst degenerations can be thought of as ways to bend a Calabi-Yau manifold until it breaks into smaller pieces. It is conjectured that fibrations and degenerations are mirror to one another; this project aims to gather evidence for such a conjecture. The successful applicant will join the Geometry and Mathematical Physics Research Group, which currently includes 12 permanent academic staff along with several research associates and PhD students. The group has a vibrant research culture, with activities including weekly research seminars, weekly meetings between students and supervisors, and excellent links with groups in neighbouring institutions.