Fano varieties are geometric shapes of positive curvature defined by algebraic equations. Whether they admit a Kähler-Einstein metric or not is determined by an algebraic condition, known as K-polystability. This condition can be verified on a given object, if we have enough information about the geometry of the Fano variety in hand. The aim of this project is to work out the existence of Kähler-Einstein metrics for a range of Fano varieties, and possibly extend the results into singular setting.
Loughborough University is a top-ten rated university in England for research intensity (REF2014). In choosing Loughborough for your research, you’ll work alongside academics who are leaders in their field. You will benefit from comprehensive support and guidance from our Graduate School, including tailored careers advice, to help you succeed in your research and future career.
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Fano varieties are geometric objects generalising spheres in higher dimensions. Their study is central to modern geometry, with applications ranging from arithmetic geometry and theoretical physics to geometric design. They have attracted much focus in recent years. However, many fundamental problems remain open. Notable areas of research on Fano varieties are the "classification problem" and the "study of moduli spaces". Both problems require invention and calculation of various numerical invariants attached to Fano varieties. The student will develop effective methods to compute these invariants in specific classes. An in-depth use of explicit methods in birational geometry will be required for a successful thesis. Specific research directions will depend on progress and timing but will most likely contribute to verification of K-stability for families of Fano varieties, or to their birational classification.
The successful applicant will join the Centre for Geometry and Applications at Loughborough. Established in 2017, this Centre supports research in geometry and related areas such as mathematical physics. At present there are 11 permanent academic staff, together with 6 postdoctoral researchers and 10 PhD students. The Centre has a vibrant research culture, with activities include weekly research seminar, regular meetings, and excellent links with groups in neighbouring institutions.
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Entry requirements for United Kingdom
Applicants should have, or expect to achieve, at least a 2:1 honours degree (or equivalent) in mathematics. A relevant master’s degree and/or experience in one or more of the following will be an advantage: birational geometry, computer algebra, toric geometry.
Please see the programme website for international entry requirements by country.
English language requirements
Applicants must meet the minimum English language requirements. Further details are available on the International website.
HOW TO APPLY
All applications should be made online. Under programme name, select Mathematical Sciences. Please quote reference number: HA/MA/2022