In both her personal and her professional life, Valentina Franceschi is all about discovering new places, meeting new people, and having new experiences: in short, travelling. This young researcher left her native Italy and the cities of Bologna and Padua a few years ago after a brilliant start to her academic career, during which time she earned herself a Master’s degree and then a PhD in mathematics.
She began working in France in late 2016, when she joined a scientific community at the Inria Paris Centre operating at the forefront of research into sub-Riemannian geometry, her area of research in mathematics. In late 2019, she was awarded a Marie Curie grant by the European Union. These grants enable researchers to develop their creative and innovative potential and to diversify their skillsets, supporting research programmes for periods of 12 to 24 months in an EU country, or in a non-EU country but with a return phase inside the EU. Each year, more than a thousand researchers in Europe take advantage of this funding, with several hundred thousand Euros given out in total**.
Passionate about mathematics and geometry
“I have always loved maths, ever since I was young. It was something my dad was passionate about, which no doubt inspired me”, reveals Valentina Franceschi.
I enjoy studying abstract objects, following a logical route, and getting to grips with concepts put forward by famous mathematicians, even ones who lived hundreds of years ago. More than anything, I enjoy the illumination of understanding.
More travelling, then, only this time across an intellectual space, with the only limits those of her imagination. The area in which Valentina Franceschi specialises, sub-Riemannian geometry, is a branch of mathematics with possible applications in a range of fields, including physics, automation and robotics. Geometry enables us to describe our world in an abstract way, and can be used to perform such tasks as measuring the dimensions of objects, plotting the course of a ship or building road structures. The world we know, with its three dimensions (height, width, depth) is the world of Euclidean geometry: buildings are typically built using right angles; roads and railway lines run in parallel lines, etc. The world surveyed by Franceschi is described using Riemannian and sub-Riemannian geometry...and is infinitely more complex. Comprised of convoluted surfaces, many of which are curved or folded, a range of different dimensions have to be factored in, with all of the concepts from “classical” geometry (angles, distances, etc.) taking on a new mathematical meaning. For example, although we are all taught at school that the shortest path between two points is a straight line, this is no longer the case in Riemannian and sub-Riemannian geometry, where the shortest distance between two points may be a curve.
Riemannian and sub-Riemannian geometry can be used to unpack some basic physics equations, such as those describing heat transfer or the behaviour of quantum particles. Although highly abstract, it has a range of concrete applications”, explains Valentina Franceschi.
This branch of mathematics is also useful for “control theory”, which is used to model the dynamics of systems (the trajectory of a car, changes in populations, the spread of an epidemic, etc.) in order to then act on them (optimising consumption of a resource, deciding on an investment, devising risk-management strategies, etc.).
A research project supported by the European Union
Control theory is the area of expertise of CAGE, a team at the Inria Paris Centre which Franceschi has been a member of for four years. On the advice of Mario Sigalotti, director of research at Inria and coordinator of the team, and Ugo Boscain, a member of the team and director of research at the CNRS, in autumn 2017, Valentina Franceschi filed her application for an individual grant as part of the Marie Skłodowska-Curie Actions (MSCA) initiative.
"Whether it’s thinking on your own, reading scientific publications or speaking with colleagues, mathematics is highly time-consuming”, explains Valentina Franceschi.
These Marie Curie grants enable researchers to devote themselves almost exclusively to advanced research, over the course of a full working programme, organising meet-ups with colleagues from universities and laboratories in EU countries.
Franceschi spent the spring and summer of 2017 putting together the project and drafting her application. “All applications are evaluated based on criteria of scientific excellence. The selection committee places a particular emphasis on the possible end results of research projects, whether for the wider public or the scientific community. I was fortunate enough to have support when it came to putting my application together.”
An opportunity to diversify your skills and to expand your network
MesuR, the project Franceschi is using this grant for, began its first year in autumn 2019, and is focused on applications of sub-Riemannian geometry for equations describing the distribution of physical quantities in non-Euclidean contexts.
Our aim is to develop theoretical mathematical tools capable of working out how heat spreads through a material with so-called anisotropic properties (which depend on the direction of the spread, as is the case for some industrial materials such as concrete, composite materials, etc.).
"There are also potential applications in neuroscience, where sub-Riemannian geometry tools are used to develop models for understanding sight in humans", explains Valentina Franceschi.
Marie Curie grants include a dedicated budget enabling researchers to organise work meetings or symposia with other researchers from a whole host of countries, such as Switzerland, the UK, Italy, France, or Czechia. “These scientific exchanges have enabled me to make more progress with my research than I had ever imagined. I have also found a number of useful scientific contacts for my project within CAGE itself, a team very much at the forefront of research into sub-Riemannian geometry. Working in Paris, which attracts a community of leading mathematicians, also presents many opportunities for collaborations.” Whether it’s to diversify her skills or to expand her network, Franceschi sees such opportunities as vital: “This EU project is a major addition to my CV, and will doubtless have a positive impact on my future academic career, whether that’s in France or in Italy.”
Planning and drafting her application and the time and energy this required also proved a formative experience for Franceschi, and one that will no doubt come in handy when it comes to conceiving further collaborative research projects, particularly within an EU context. “By the time I have completed the work supported by the Marie Curie grant, I will have expanded my knowledge, from both a scientific and a human perspective. If the opportunity presents itself, I would very much like to contribute to other research projects involving new partners. The fact that, like travelling, it gives me the opportunity to discover new people and places is another reason why I love mathematics.”
* CAGE (Control And GEometry) is a a joint undertaking involving Inria and Sorbonne University, which is based at the Jacques-Louis Lions laboratory (UMR 7598) on the Jussieu campus.
** In 2020, the European Commission granted individual Marie Skłodowska-Curie grants to 1,475 postdoc researchers, committing to an overall total of 296.5 million Euros to support their research into solutions to both current and future societal problems.