## Dr Federica Dragoni

### Overview

Telephone: +44(0)29 208 75529

Fax: +44(0)29 208 74199

Extension: 75529

Location: M/2.32

#### 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

#### Teaching

**Spring Semester**

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

#### External Duties

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

#### Personal Website

### Publications

### Research

#### 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).

2004: Research grant to at University La Sapienza, Rome.

2007: INDAM (Istituto Nazionale di Alta Matematica) research grant for research abroad. 2010: LMS Grant (Scheme 4)

#### Major Conference Talks

09/2008: "Stochastic representation for evolution by horizontal mean curvature flow", Conference on Viscosity, metric and control theoretic methods in nonlinear PDEs, University La Sapienza, Rome.

07/2009: "Convexity along vector fields and applications to equation of Monge-Ampère type" ISAAC Conference 2009. Imperial College London.

### Biography

#### Academic history

09/2006: PhD in Mathematics, Scuola Normale Superiore di Pisa, Thesis: Carnot-Carathéodory metrics and viscosity solutions. Advisor: Prof. Italo Capuzzo Dolcetta.

09/2002: Laurea (equivalent of Master Degree) in Mathematics, University of Florence, Thesis: : Photon transport in an interstellar cloud: direct and inverse problems. Advisor: Prof. Luigi Barletti.

#### Employment

2011: Lecturer at Cardiff School of Mathematics, Cardiff University.

2010: Research associate at University of Padova, Italy and teaching position, University of Bristol

2009: Research associate at Imperial College London.

11/2008-02/2009: Research associate at University of Padova, Italy.

09/2007-10/2008: Post-doc position at Max Planck Institute for Mathematics in the Sciences, Leipzig.

02-06/2007: INDAM (Istituto Nazionale di Alta Matematica) research position, at University of Pittsburgh.

### Postgraduate Students

#### Current Students

Doaa Filali

Ahmed Jama