Professor Cornelia Druţu discusses her deep love for Mathematics
How and why did you become a mathematician?
I was very lucky with my teachers in school. They helped me understand early in life that Mathematics is about both imagination and clarity of thought. My initial plan was that having benefitted from the commitment and generosity of my teachers, I would pay it forward and become a Mathematics teacher myself.
In my final high school years, I started to read more on my own and was struck by the incredible power of Mathematics as a tool in the advancement of knowledge. Stories like the one of Eratosthenes, circa 230 BC, who managed to measure the circumference of the Earth with an error of just 50 miles, using nothing but a well, sundial, and geometry, showed me that Mathematics could push the boundaries of knowledge further than any other science, because it depended less on the technical sophistication of the age, and more on the depth of thought of mathematicians.
I wanted to be part of this effort to understand the world so, as an undergraduate, I began to think about a career involving research.
After I graduated, I was offered a scholarship for a DPhil at the University of Paris XI. My supervisor was the truly inspiring mathematician Pierre Pansu. The Institut des Hautes Études Scientifiques (IHES) was only 20 minutes away and it was the chance to be in the proximity of some of the greatest mathematicians of all time: Pansu’s supervisor (my ‘mathematical grandfather’, in a sense), Mikael Gromov, was working there, along with Alain Connes, David Ruelle etc, all of them at the height of their powers. Everything fell into place from there on.
How did you end up here at Oxford?
After my DPhil, I became Maitre de Conferences (the equivalent of Assistant Professor) at the University of Lille, France and passed my Habilitation – a degree higher than a DPhil that allows one to become full professor and to start supervising DPhil students. My plan was to apply for full Professorships in France, but to my surprise I was also invited by colleagues to apply to Imperial and Oxford. I was offered positions in both places, and Oxford was the more appealing to me, with its combination of top-level research and great architectural beauty.
You were recently named this year’s Emmy Noether Guest Professor within the Faculty of Mathematics at the University of Göttingen. What did this position mean to you?
To have been awarded this Professorship meant a lot to me. Göttingen is a sacred place for Mathematics, especially for geometry. An incredible number of great mathematicians studied or lived there and, through their work, changed the face of science: Gauss, Hilbert, Riemann, Klein, Hermann Weyl, Minkowski, Caratheodory, Dedekind, von Neumann and, last but not least, Emmy Noether, the first female professor of Mathematics in Göttingen.
Emmy Noether is in fact much more than just the ‘first female professor’. Her breakthrough work in Algebra changed the face of the subject and is now taught in every university around the world. There even exists an adjective derived from her name, ‘Noetherian’. I admire her greatly. For me and, I think, for every woman working in Mathematics, she is a very inspiring figure.
What are you currently looking at in your research?
Several areas in Mathematics, including my own, share a general technique, which consists of embedding the structure under scrutiny in a well-understood space and in drawing from this embedding strategies of problem-solving. My current research follows this trend: I am studying infinite groups and looking for connections between their algebraic properties and their equivariant embeddings into spaces endowed with various structures, in particular Banach spaces and non-positively curved spaces. In layman’s terms, you could say that at the core of my current research lies the idea that if you want to understand an object better, the best plan is to draw a good picture of it, and therefore first to choose the right board to draw it on. In my case, the object is an infinite group, which can be identified via a standard procedure with an infinite graph. At first, you would expect that an infinite dimensional board would suffice to draw such a graph on it well enough (the usual boards are two-dimensional, the space we live in is three dimensional, and if we keep adding dimensions the limit object is infinite dimensional). It turns out that this is not always the case. Moreover, the groups that cannot be drawn on infinite dimensional boards have close and explicit connections with expander graphs (or, in the language of telecommunications, ‘robust networks’ or ‘networks that are difficult to disconnect’).
At the opposite end, groups that can be drawn well on an infinite dimensional board are closely related to median graphs (‘economic networks’, attempting to minimise the cost of servicing a number of requests). Both cases are also relevant for several key open questions in modern Mathematics, for instance to the Baum-Connes conjecture.
You recently were offered a Visiting Researcher position at the Max Planck Institute for Mathematics in Bonn, Germany. What did you work on while there?
I have been working with a group of collaborators located at the MPIM, at the University of Bonn, and at the Hausdorff Institute. Our project is part of a larger effort to reformulate results related to the classical notion of curvature in the context of graphs and groups. The concept of curvature describes a fundamental spatial attribute, therefore it has found a place at the core of science since its inception. Aristotle identified three types of curvature: ‘straight’, ‘circular’ and ‘mixed’. These three types of curvature are relevant, respectively, to: classical geometry, taught at school and used the most often, from everyday life to engineering; positive curvature geometry, which appears for example in large-scale measurements in Earth Sciences; and non-positive curvature, which occupies a central place in modern geometry, and is considered the most plausible large-scale model of our universe.
The classical theory of curvature is formulated in spaces that are ‘continuous’, as is the space we live in. In mathematical terminology, such spaces are called Riemannian (after Riemann, who is also known for the ‘Riemann hypothesis’, one of the most famous problems in Mathematics, and who is, incidentally, also a former head of the Mathematics department at
the University of Göttingen). Nowadays, researchers try to translate the main results from Riemannian geometry to discrete geometry, and any advance in this direction is very useful.
How does your position in Bonn differ to your work at Exeter College?
It is a position entirely devoted to research, and it is part of a research project. It is in fact part of a series: I have been a Visiting Researcher at MPIM Bonn for the past three years, as this is an ongoing long-term effort that we hope we brought close to completion this summer.
You are the principal organiser of an upcoming half-year research programme at the Isaac Newton Institute in Cambridge. Could you describe the programme and your own position there?
I am very excited about this programme. It is called ‘Operators, Graphs and Groups’ and, as the title suggests, it has a strongly multidisciplinary nature, with several areas of research involved, each represented by an organiser. The three types of objects mentioned in the title are ubiquitous, and play key roles in most areas of Mathematics. Some of the most groundbreaking research has taken place recently at the interface between these areas, and the hope is that this programme will intensify the knowledge exchange and advance some of its most striking approaches and novel insights. I felt honoured that my co-organisers, all mathematicians that I admire, chose me as the leading organiser. It is a huge responsibility. I am hopeful that it sends an encouraging message to young female mathematicians.
How has your field of research evolved recently?
My initial area of study was algebraic structures, in particular infinite groups, by means of geometric methods, thus allowing us to visualise abstract objects that are otherwise difficult to understand. This approach has been particularly successful, allowing us not only to solve open problems, but also to begin to introduce some form of clarity in a world otherwise notoriously difficult to understand, that of infinite groups. Recently, more non-algebraic methods have been added to the toolkit: measure theoretical, combinatorial, analytical and so on. It is very gratifying to see the way in which these new methods complete the picture and allow for more rapid advances.
Where would you like to see the field of Mathematics go in the Future?
While the assistance that mathematicians now have from computers and AI increases in extent and quality every year, and that will continue to be the case with the further development of Quantum Technologies and AI, what I find even more exciting is the rapid and intense knowledge exchange between different fields and the fact that a significant part of the current research in Mathematics is multidisciplinary. The frontiers between areas of research have been blurred, and all mathematicians have to be aware of techniques from a wide range of different fields. This makes our work harder but also considerably more exciting. What I would like is to see
the multidisciplinary trend becoming stronger and wider, perhaps even predominant, as the time passes.
What advice would you give to Mathematics students at Exeter College?
To never lose sight of what made them love Mathematics in the first place. Even students who say that they chose Mathematics ‘just because I was good at it, and did not know where else to go’ usually mean that they found working in Mathematics gratifying and fun. The beginning of one’s studies in Oxford might make students forget this, as it requires a huge adjustment, both socially and academically, and a lot of fast-paced hard work. The best plan is to climb the steep learning slope as diligently and as selflessly as possible from day one and try to be as well organised and economical with one’s time, but at the same time to never lose sight of the fun of Mathematics, of what they found inspiring and congenial about it, whether that was the beauty of arguments, the irresistible logical consistency, or the imagination and inventiveness that so obviously lie behind some core concepts.
Another piece of advice to students is to keep themselves in the best possible shape, with healthy eating, enough sleep and regular exercise. Studying Mathematics is very demanding and you cannot cope without paying attention to your health. The most recent medical research on the functioning of the brain confirms that sleep, exercise and food have a significant impact. Although there may be times when you must work the entire night for an assignment, this should be an exception, not a rule.
Added to that, I would advise students to never stop believing in the power of hard work and breaking things into small steps. There will always be someone who will claim that Mathematics is not about hard work, but about some innate talent that only a chosen few are born with, that they solved this and that difficult problem in five minutes and then they lazed around the rest of the week. This may or may not be true, but what is certain is that all the great mathematicians that I have had the honour to meet work extremely hard, and sometimes experienced low points in their research but they bounced back. It is very effective to break a difficult task into small steps, that look less intimidating and more achievable.
It may be though that our Mathematics students do not need most of this advice: the mere fact that they managed to get into Oxford, sometimes from an environment that was the opposite of nurturing for their talent, means that they have already devised a working strategy. Thus, my advice above should be seen more as a set of suggestions to improve our students already successful working strategy.
