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

Analysis

Teaching

Spring Semester

MA0126 Analysis II

MA4013 Advanced topics in Analysis with application to PDEs

Administrative Duties

• Work-Life Balance Coordinator
• Year 1, Year 2, Year 3 and Year 4 Senior Tutor
• Member of the Athena Swan Panel
• Member of Learning and Teaching Committee
• Coordinator of Review Panel on Tutoring

Conference Organisation

Stochastic methods and nonlinear PDEs, Cardiff, 2012

Extended local Organising Committee for the 7th ISAAC congress, Imperial College London 2009

External Duties

• London Mathematical Society Representative for Cardiff
• Member of GW4 network 'Functional Materials Far From Equilibrium'

Personal Website

Dr Federica Dragoni's Personal Website.