Academic history

• 2006: PhD in Mathematics, Scuola Normale Superiore di Pisa, Italy
• 2002: Laurea (equivalent of Master Degree) in Mathematics, University of Florence, Italy

Employment

• 2011: Lecturer at Cardiff School of Mathematics, Cardiff University
• 2010: Research associate at University of Padova, Italy and temporary position, University of Bristol
• 2009: Research associate at Imperial College London
• 2008-2009: Research associate at University of Padova, Italy
• 2007-2008: Post-doc position at Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany
• 2007: INDAM research position, at University of Pittsburgh, USA

• Date
• Type
• Selected

I teach the following modules:

• MA1006 Foundations of Mathematics II
• MA4013 Advanced topics in Analysis with application to PDEs

Postgraduate students

• Doaa Filali
• Ahmed Jama

Research students

• Ermal Feleqi

Research interests

My research is motivated by a broad range of interrelated problems in the area of analysis in sub-Riemannian manifolds and degenerate nonlinear PDEs. In this settings I have dealt with very different questions, making use of many interdisciplinary methods and techniques from probability, analysis, differential geometry, Lie algebras, metric spaces, calculus of variations and measure theory.

Sub-Riemannian geometries and related PDEs (as subelliptic/ultraparabolic PDEs) turn out to be extremely useful to create mathematical models to describe many different phenomena from applications. An example are the use of the Rototranslation geometry for modelling the first layer of the visual cortex and problems in finance related to pricing so-called Asian options.

Unlike Riemannian manifolds (where the structure looks locally always like the Euclidean R n), sub-Riemannian spaces are never, at any scale, isomorphic to the Euclidean space. In particular they are highly anisotropic in the sense that at any point some directions for the motion on the manifold turn out to be forbidden, making the metric and geometric structure much more complicated than in the non-degenerate case (Euclidean space and Riemannian manifolds). The admissible directions for the motion are described by vector fields which do not span at any point the whole tangent space. PDEs on these geometries are defined by replacing the standard partial derivatives by the vector fields.

Recently I got interested in stochastic homogenization problems for first-order and second-order PDEs associated to H¨ormander vector fields. 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 H¨ormander-type PDEs, the rescaling becomes usually anisotropic.

In the past years I have worked on first-order equations (non coercive HamiltonJacobi equations) as well as several nonlinear second order degenerate subelliptic/ultraparabolic equations (e.g. infinite-Laplacian and evolution by horizontal mean curvature flow). In this setting I (together with Martino Bardi from Padova) have also developed a notion of convexity along vector fields which has several important applications to PDEs associated with H¨ormander vector fields.

Research group

• Mathematical Analysis Research Group

External funding

• 2015-2016: EPSRC First Grant
• 2012: London Mathematical Society grant for conference £5000; OxPDE grant for conference £3500; WIMCS grant for conference £2000
• 2010: LMS collaborative small grant £600
• 2007: INDAM research grant Euro 6000 (Italian grant)

Major conference talks since 2010

• 001/2010: Imperial College London, UK
• 03/2010: University of Bristol, UK
• 04/2010: University of Padova, Italy
• 04/2010: University of Bath, UK
• 08/2010: University of Basel, Switzerland
• 10/2011: at EPCR workshop Geometric measure theory in non-Euclidean spaces, University of Pisa, Italy
• 11/2011: OxPDE Center, Oxford, UK
• 12/2011: Heriot-Watt University, Edinburgh, UK
• 04/2012: Universit´a La Sapienza, Rome, Italy
• 10/2012: Swansea University, UK
• 02/2014: Recent Advances in Nonlinear PDE and Calculus of Variations, University of Reading, UK
• 04/2014: University of Sussex, UK
• 05/2014: Young Applied Analysts, University of Glasgow, UK
• 11/2014: GW4 consortium meeting, University of Bristol, UK
• 11/2014: University of Bologna, Italy
• 12/2014: University of Florence, Italy
• 01/2015: University of Birmingham, UK
• 03/2015: LMS Bath-WIMCS Analysis Day, Swansea University, UK

Goruchwyliaeth gyfredol

Raffaele Grande

Research student