Funded PhD Studentship in Geometry and Analysis

Summary

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.

We are delighted to offer a fully funded PhD project and applications are invited from strongly motivated and academically excellent candidates for fully funded PhD study in Geometry and Analysis, within these strategic priority Research areas:

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. Please contact Dr Ben Lambert by email to b.s.lambert@leeds.ac.uk.

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. Please contact Dr Kasia Wyczesany by email to K.B.Wyczesany@leeds.ac.uk.

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. Please contact Dr Gerasim Kokarev by email to g.kokarev@leeds.ac.uk.

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. Please contact Dr Francesca Tripaldi by email to f.tripaldi@leeds.ac.uk.

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. Please contact Dr Vladimir Kisil by email to v.kisil@leeds.ac.uk.

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). Please contact Dr Ben Sharp by email to b.g.sharp@leeds.ac.uk.

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. Please contact Dr Derek Harland by email to d.g.harland@leeds.ac.uk or Professor Martin Speight by email to j.m.speight@leeds.ac.uk.

Applicants to research degree programmes should normally have at least a first class or an upper second class British Bachelors Honours degree (or equivalent) in an appropriate discipline. The criteria for entry for some research degrees may be higher, for example, several faculties, also require a Masters degree. Applicants are advised to check with the relevant School prior to making an application. Applicants who are uncertain about the requirements for a particular research degree are advised to contact the School or Graduate School prior to making an application.

The minimum English language entry requirement for research postgraduate research study is an IELTS of 6.0 overall with at least 5.5 in each component (reading, writing, listening and speaking) or equivalent. The test must be dated within two years of the start date of the course in order to be valid. Some schools and faculties have a higher requirement.