# Dr Federica Dragoni

Senior Lecturer

*Email:*- dragonif@cardiff.ac.uk
*Telephone:*- +44 (0)29 2087 5529
*Location:*- M/2.32, 2nd Floor, Mathematics Institute, Senghennydd Road, Cardiff, CF24 4AG

### 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

- New Trends in nonlinear PDEs: from theory to applications, Cardiff 2016
- Stochastic methods and nonlinear PDEs, Cardiff, 2012
- Extended local Organising Committee for the 7th ISAAC congress, Imperial College London 2009

### Meetings

### External duties

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

### 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

### 2018

- Dragoni, F. and Feleqi, E. 2018. Ergodic mean field games with Hörmander diffusions. Calculus of Variations and Partial Differential Equations 57(5), article number: 116. (10.1007/s00526-018-1391-1)
- Dirr, N.et al. 2018. Stochastic homogenization for functionals with anisotropic rescaling and non-coercive Hamilton-Jacobi equations. SIAM Journal on Mathematical Analysis

### 2014

- Bardi, M. and Dragoni, F. 2014. Subdifferential and properties of convex functions with respect to vector fields. Journal of Convex Analysis 21(3), pp. 785-810.

### 2013

- Dragoni, F. 2013. Partial differential equations. Cardiff University. - teaching_resource
- Dragoni, F., Manfredi, J. J. and Vittone, D. 2013. Weak Fubini property and infinity harmonic functions in Riemannian and sub-Riemannian manifolds. Transactions of the American Mathematical Society 365, pp. 837-859. (10.1090/S0002-9947-2012-05612-1)

### 2012

- Dragoni, F., Kontis, V. and Zegarliński, B. 2012. Ergodicity of Markov semigroups with Hörmander type generators in infinite dimensions. Potential Analysis 37(3), pp. 199-227. (10.1007/s11118-011-9253-x)
- Dragoni, F. 2012. Analysis 2. Cardiff University. - teaching_resource

### 2011

- Bardi, M. and Dragoni, F. 2011. Convexity and semiconvexity along vector fields. Calculus of Variations and Partial Differential Equations 42(3-4), pp. 405-427. (10.1007/s00526-011-0392-0)
- Gentil, I.et al. Zegarlinski, B. ed. 2011. Aspects of analysis curvature criterion, isoperimetry, evolution equations. Mathematical Notebooks, Vol. 3. Matrix Press.

### 2010

- Bardi, M. and Dragoni, F. 2010. Convexity along vector fields and applications to equations of Monge-Ampére type. In: Ruzhansky, M. and Wirth, J. eds. Progress in Analysis and Its Applications. Proceedings of the 7th international ISAAC Congress.. World Scientific, (10.1142/9789814313179_0059)
- Dragoni, F. 2010. Linear Algebra. Cardiff University. - teaching_resource
- Dragoni, F. 2010. Calculus in several variables: remarks for non mathematicians. Cardiff University. - teaching_resource

### 2009

- Dragoni, F. 2009. Introduction to viscosity solutions for nonlinear partial differential equations. Cardiff University. - teaching_resource
- Dirr, N. P. and Dragoni, F. 2009. Evolution by mean curvature flow in sub-Riemannian geometries: a stochastic approach. Communications on Pure and Applied Analysis 9(2), pp. 307-326. (10.3934/cpaa.2010.9.307)
- Bieske, T., Dragoni, F. and Manfredi, J. J. 2009. The Carnot-Carathéodory Distance and the Infinite Laplacian. Journal of Geometric Analysis 19(4), pp. 737-754. (10.1007/s12220-009-9087-6)

### 2007

- Dragoni, F. 2007. Metric Hopf-Lax formula with semicontinuous data. Discrete and Continuous Dynamical Systems 17(4), pp. 713-729. (10.3934/dcds.2007.17.713)
- Dragoni, F. 2007. Limiting behavior of solutions of subelliptic heat equations. Nonlinear Differential Equations and Application NoDEA 14(3-4), pp. 429-441. (10.1007/s00030-007-6013-0)

### 2006

- Dragoni, F. 2006. Carnot-Carathéodory metrics and viscosity solutions. PhD Thesis, Scuola Normale Superiore, Pisa, Italy.

### 2005

- Barletti, L. and Dragoni, F. 2005. An inverse problem for two-frequency photon transport in a slab. Mathematical Methods in the Applied Sciences 28(14), pp. 1695-1714. (10.1002/mma.633)

### 2004

- Capuzzo Dolcetta, I. and Dragoni, F. 2004. Hamilton-Jacobi equations. - teaching_resource

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

### 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