Cardiff University conduct Ph.D Studentship OutsideIndia

Reference Number: R2739

PhD Studentship Mathematics conducts Cardiff University

Project title : Homogenisation for degenerate nonlinear Partial Differential Equations.

The project is in the area of stochastic homogenization for nonlinear PDEs (Partial Differential Equations) associated to a low regularity condition called the Hormander condition. In particular I am interested in those cases where, even starting from a stochastic microscopic model, the effective problem (= PDE modelling the macroscopic behaviour) is deterministic. Where the microscopic stochastic model is related to Hormander-type PDEs, the rescaling becomes usually anisotropic. The regularity theory of such PDEs mimics, at least in the linear case, the standard one (H ¨ormander’s Theorem ‘69), but deriving finer properties like asymptotics in the presence of randomness requires an intricate combination of ideas from analysis, probability and geometry. The influence of the geometry on the associated analysis problems, is particularly evident in the case of homogenisation where the rescaling of the microscopic model is strictly connected to the underlying geometry. In fact, in the standard uniformly elliptic/parabolic case (respectively coercive for the first order case), one takes the limit as e tends to zero of an equation depending on e.g. x/e (where x is a point in Rn) i.e. isotropic rescaling. When considering a degen- erate PDE related to H ¨ormander vector fields, the rescaling becomes anisotropic: for example in the first Heisenberg group a point (x, y, z) scales as (x/e, y/e, z/e2). Moreover points (elements on the manifold) and velocities of curves (elements in the tangent space) scale now in a very different way and any identification between objects with different natures (which is often a well-hidden key point in the stan- dard elliptic/parabolic case) is no longer allowed.

Eligibility : Applicants for a studentship must have obtained, or be about to obtain, a Master Degree or Math in Mathematics. A strong background in analysis is also required.

Number of Studentships : 01

Funding : This studentship consists of full UK/EU tuition fees, as well as a Doctoral Stipend matching UK Research Council National Minimum (£14,057 p.a. for 2015/16, updated each year). One studentship is available.

Regions : EU (Non UK), International (Non EU), UK

Start date : 1 October 2016