Professor Federica Dragoni

Publication

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Research

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.

Some of the topics I am interested in are listed below.

• Nonlinear PDEs, in particular degenerate ellipitc and parabolic PDEs.
• PDEs in the Heisenberg groups and Carnot groups
• PDEs in sub-Riemannian manifolds and in general related to Hoermander vector fields
• Geometrical properties for PDEs
• Convexity and starshapedness in non Euclidean spaces
• Stochastic methods for deterministc PDEs
• Stochastic Homogenization
• Hamilton-Jacobi equations and Hopf-Lax formulas
• Absolutely Minimizing Lipschitz extensions
• Infinite-Laplacian
• Mean Field Games
• Tug-of-war, more in general deterministic and stochastic differential games

Research group

• Mathematical Analysis Research Group
• Analysis, Probablity and Stochastic Process Group

External funding and grants

• 2023-2024: LMS Scheme 4; Cardiff University Research Leave
• 2020-2024: Welsh Node for EPSRC Network on Generalised and Low-regularity Solutions for Nonlinear PDEs 
• 2019-2020: ERASMUS Mobility Grant; CU Funds to visit Campinas
• 2018-2019: LMS Scheme 5 (Collaborations with developing countries)
• 2017-2018: Cardiff University Research Leave; LMS Scheme 3; LMS Educational Grant
• 2016-2017: LMS Grant Scheme 4; LMS Grant Scheme 3
• 2015-2016: EPSRC First Grant; LMS Grant Scheme 3 
• 2012-2013: LMS grant Scheme 1; OxPDE grant for conference; WIMCS grant for conference
• 2010: LMS collaborative small grant 
• 2007: INDAM research grant

Teaching

I am currently teaching the following modules:

• MA20006 Real Analysis (Spring Term)

Former teaching experience

• Real Analysis. year 2, Cardiff (2021-present)
• Further topics in Analysis with applications to PDEs, Postgraduate and MMath course, Cardiff University (2016-2023)

• Foundation II, year 1, Cardiff (2016-2020)

• Analysis 2, year 1, Cardiff (2012, 2014, 2015)

• Theoretical and computational PDEs, year 3, Cardiff (2012, 2013)

• Linear Algebra, year 1, University of Bristol (2010)

• Calculus 2, year 1, University of Bristol (2010)

• Introduction to viscosity theory for Nonlinear PDEs, Postgraduate and MMath course, Imperial College London (2009)

• Calculus I and Geometry and Linear Algebra (teaching assistant), year 1, Department of Engineering, University of Florence (2007)

• Preparatory class of Mathematics, University of Florence (2006)

• Mathematics and Physics at Secondary Schools, Florence (2006)

Biography

Education and Qualification

• 2013: Fellow of the Higher Education Academy (FHEA), Cardiff University.
• 2006: PhD in Mathematics, Scuola Normale Superiore di Pisa, Italy. Mark: 70/70 cum Laude. Title: Carnot-Carathéodory metrics and viscosity solutions. Advisor: Prof. Italo Capuzzo Dolcetta.
• 2002: Laurea (equivalent of Master Degree) in Mathematics, University of Florence, Italy. Mark: 110/110 cum Laude. Title: Photon transport in an interstellar cloud: direct and inverse problems. Advisor: Prof. Luigi Barletti

Employment

• 2021-present: Chain in Mathematics at  Cardiff School of Mathematics, Cardiff University
• 2016-2021: Senior Lecturer and the Reader at Cardiff School of Mathematics, Cardiff University
• 2011-2016: Lecturer at Cardiff School of Mathematics, (including two breaks of maternity leave in 2011 and 2013)
• 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

Supervisions

• Nonlinear PDEs degenerate elliptic and parabolic.
• PDEs in Carnot groups, subRiemannian manifolds and related to Hoermander vector fields.
• Geometrical properties for PDEs.
• Convexity in non Euclidean spaces.
• Periodic Homogenization
• Stochastic Homogenizitaion
• Mean Field Games

Current supervision

Prachi Sahjwani

Graduate Tutor