Applications are invited from strongly motivated and academically excellent candidates for fully funded PhD study in Geometry and Analysis. The Geometry and Analysis group at Leeds is large and vibrant, comprising 8 permanent members, 2 postdocs and 9 PhD students, with wide interests and expertise in differential geometry and mathematical analysis. This allows us to offer a PhD project with several possible distinct but related emphases:
Geometric flows (Dr. Ben Lambert). These are powerful tools which have settled hard open conjectures, most famously, the Poincaré conjecture, and provided beautiful proofs of important results such as the differentiable sphere theorem and the Penrose inequality. Work in this area would investigate the properties of an extrinsic geometric flow such as mean curvature flow, inverse mean curvature flow, Gauss curvature flow or symmetric curvature polynomial flows.
Dualities in convex geometry (Dr. Kasia Wyczesany). Duality is an influential concept that manifests itself across many different areas of mathematics. In particular, duality of finite dimensional normed spaces, which can be represented via the duality of their unit balls, has been central in convex geometry. The aim here would be to develop a parallel theory for other order-reversing dualities on sets with particular focus on phenomena such as concentration of measure.
Spectral geometry (Dr. Gerasim Kokarev). The study of how the spectrum of a linear operator depends on the geometric properties of its domain is, to a large extent, motivated by questions regarding real-life phenomena, such as vibration, heat propagation and quantum mechanical effects. Work in this area has many possible starting points: isoperimetric inequalities, eigenvalue problems and spectral invariants in Riemannian geometry, eigenvalue problems in minimal surface theory, and extremal eigenvalue problems.
SubRiemannian geometry (Dr. Francesca Tripaldi). SubRiemannian manifolds are a specific geometric setting where motions are only allowed along certain prescribed directions. They represent a vast generalisation of Riemannian manifolds that naturally appears in several areas of pure and applied mathematics, such as control theory, thermodynamics, and robotics. The noncommutativity of the local geometry of such manifolds has hindered the development of a “subRiemannian” tensor calculus, and so geometric and analytic tools such as the curvature tensor, elliptic Hodge-Laplacian operators, Stoke’s theorem, and currents, are currently missing in this more general setting. Work here would focus on bridging the technical gaps that currently exist towards the resolution of such problems.
Conformal geometry of infinite-dimensional spaces (Dr. Vladimir Kisil). Conformal and inversive geometries are elegant classic theories. We may look for analogous constructions in infinite dimensional Hilbert spaces. This gives an extended treatment of operator spectral theory.
Minimal surfaces (Dr. Ben Sharp). These constitute a central area of research in mathematics, straddling analysis, geometry and theoretical physics. Possible entry points for PhD study here include the analytical study of geometric objects as solutions to nonlinear elliptic PDE (e.g. abstract existence and regularity theory, spectral analysis of Schrödinger operators) and the geometric study of constrained submanifolds (e.g. harmonic maps, prescribed curvature submanifolds, Willmore surfaces).
Topological solitons (Dr. Derek Harland, Professor Martin Speight) Originating in theoretical physics, these are structures on manifolds that minimize some natural measure of energy, and are stable for topological reasons. Work in this area could focus on constructing examples on spaces of high dimension and special geometry, or analyzing the geometric properties of spaces of solitons.
This studentship is funded by the UK Engineering and Physical Sciences Research Council and has a duration of 42 months, subject to satisfactory performance. It pays all tuition fees and provides a stipend of £19,237 per annum. It also provides a grant for research related travel of £3,000 for the duration of the studentship.
You should have a 2:1 or first-class undergraduate degree, or an MMath, MSci or Master’s degree with Merit or Distinction (or the equivalent awarded by an overseas institution), in Mathematics, or a closely relate discipline.
Candidates are encouraged to contact potential supervisors (as named above) informally before applying. Formal applications, which must be completed by 28 February 2025, should be made online: https://eps.leeds.ac.uk/maths-research-degrees/doc/apply
Please clearly note the name of the lead supervisor and the project title "Geometry and analysis" on the application form. Your application must include a personal statement, an academic CV, and copies of your degree certificates and transcripts. A research proposal is not required.
After a pre-selection based on documents, suitably qualified candidates will be invited for interview either in person or online.
Decisions will be based on academic merit. We encourage applications from all backgrounds and are committed to having a diverse, inclusive team.
The application process is described in more detail here. If you require any additional assistance in submitting your application or have any queries about the application process, please contact maps.pgr.admissions@leeds.ac.uk
One of the Russell Group of research intensive Universities, Leeds has a distinguished track record in mathematics research. 98% of our work was classed as ‘world leading’ or ‘internationally excellent’, in the most recent Research Excellence Framework, ranking us 9th in the UK in research power. The University is located in the heart of Leeds, the UK's third largest city, a financial and cultural hub on the edge of the Yorkshire Dales.