## Going beyond the boundaries: an interview with Giulia Di Nunno

### Interview by Francesca Arici and Anna Maria Cherubini

- Born inItaly
- Studied inItaly
- Lives inNorway

### Interview

**Giulia Di Nunno is a professor of mathematics at the University of Oslo and an adjunct professor at the NHH Norwegian School of Economics. In addition to her research, Giulia is active in promoting and increasing the visibility of mathematics in Africa. In this interview, we talked with Giulia about her research and her work in international cooperation.**

**How would you explain your field of research to a specialist? **

My research is stochastic analysis. In general, stochastic analysis uses both analytical and probabilistic methods. To explain, while analysis is typically dealing with deterministic structures, stochastic analysis is dealing with randomness and is then using structures and statistics. It is the continuous interactions of the two perspectives that makes the field very exciting.

I am specifically dealing with time-space dynamics, stochastic calculus and control. Stochastic calculus is a relatively young area if you think that the Itô integral was developed in the 40’s. With this, Itô has brilliantly put together the stochastic element with the analytic one to be able to define an integral with respect to the Brownian motion and overcome the non-differentiability of its paths. Since then, various frameworks for integration and calculus have been developed and there is still need for more, basically, this means that I’m developing tools to make the calculus available. I’ve been working in stochastic differentiation, which is really much younger than integration. The question, simply said, is the following: “If you have the integral, how do you find the integrand?” Otherwise said, how can one compute the dual of a stochastic integral operator?

For a non-specialist, it is easier to explain its use in an application. For example, suppose you want to hedge a risk: how do you find the best hedging strategy? That is the point where you need stochastic differentiation: you want to say something that goes beyond just the existence of an integrand process.

**How did you decide to work in this field?**

As usual, you don’t start with deciding about things you don’t know. It’s a progressive interest. I started by being interested in analysis already quite early during my studies. Why this, I have no idea. Most likely, it had to do with how interesting it was for me to go to classes. I like to work with approximations, different dimensionalities, movement. Then, stochastic analysis became the field that appealed the most to me.

You can easily imagine a number of everyday problems that can be tackled by means of stochastic analysis and control, for example, “When is the best time to cross a street?”. The question is quite easy to pose, but the answer is not trivial, because you have to analyse the present effect of future possible events. And this type of challenge is exactly what I found, and I still find, fascinating.

**How did you decide to study mathematics and to become a mathematician?**

I have to say that it was somewhat by chance, in the sense that when I was in high school, I really did not know what direction to choose. I had a better idea about what I was not going to study, and honestly, nowadays, I don’t even remember well what the rationale for that was. I had very little ideas of what was right for me because I was interested in many things and quite eclectic.

The decision to go for mathematics was made, while I was on a train travelling to Paris on vacation. I had just completed my A-level exam at that time. I was reading through the brochures and pamphlets provided by many universities, when I came across the one for mathematics. I remember that it was so thin and so dry, not very market-oriented in any way… And it was actually quite cryptic. It presented study programmes with very general subject titles. Somehow this mixture of being cryptic and being not exactly market-oriented was actually standing out and attracted my attention. Aside from that, I think what played a role was the fact that I could put mathematics at the foundations of everything I was actually interested in. Anyway, when the decision was made, it settled.

So, I started my first year in Mathematics. An interesting episode happened during the first class of analysis. To our surprise, the professor was shocked by the large number of students in the lecture hall. To our even bigger surprise, he assured that this would not be the same in a months time. His prediction was correct. The high drop-out rate, which was a real pity, created a strong group feeling among the students: we had the feeling of doing something together, even though we had not the same interests. Indeed, we mathematicians are linked to each other, even though we are all from different countries and in different fields, we have similar experiences everywhere.

… we mathematicians are linked to each other, even though we are all from different countries and in different fields, we have similar experiences everywhere.

**Can you recall any source of encouragement and inspiration during those years? **

My parents played a crucial role. My mother is a linguist, and even though she is not actually interested in mathematics, she considered mathematics a language that one has to study and learn, like every other language. My father is an economist, very interested in history, archaeology, classical studies and rational thinking. Both of them have always kept very much behind the scenes and have always encouraged me to do/study whatever I liked.

When I came back from my trip to Paris, I told my family about my decision of studying mathematics, which was a surprise to them, possibly because they were expecting a different choice. But they supported my decision and gave me all the freedom I needed.

Also, I have to say that I was lucky to have met good teachers in mathematics: they were not hiding the difficult points in doing maths and they were preparing us to be careful and precise when working on challenging questions.

**Do you think being a woman had any impact, positive or negative, on your career?**

I have started thinking about being a woman in mathematics only very recently compared to any other choice I have taken before. I have never thought that there was something I could not do just because I was a woman. When I was growing up, all the women in my family, including my grandmother and grand grandmother, were working or had worked. So, it was normal to see working women around me and I have never associated what they were doing with the fact of being a woman. In their particular case, they were doing something they liked.

Of course, at university, you start realising that when you take certain directions in mathematics, women start to be fewer and fewer. Finally, when you move to the academic working environment and you see that women are less than 10 percent of the whole department, then you cannot but start wondering: what is going on?

**Paradoxically, women’s under-representation in mathematics seems to hit the most in Nordic countries, which are the countries that perform better with respect to the gender equality index****. Do you have any insight on this?**

This is something I have also often mentioned on different occasions. It is extremely surprising, and I think it has deeper motivations.

The “Nordic” countries are the countries where equal opportunities are pushed. In reality, these are exactly the countries where you find very, very few women professor, in mathematics in particular, and also in physics and chemistry, in general. On the other hand, in fields like biology or medicine, women are better represented. So, the issue is not science per se, it must be something that goes deeper.

**Let’s go back to mathematics: is there a particular result in your work that you like most?**

One result that I like -it is an earlier work so I can now look at it with some distance- is about the stochastic derivative of a non-anticipating type. It is a computational formula that then turned out to be the dual of the Itô integral. I like this because it is a clean self-contained result. It is strange how when you present the result it looks easy and you have forgotten how many attempts it took to get you there…

It is strange how when you present the result it looks easy and you have forgotten how many attempts it took to get you there.

I related the formula to some applications to hedging in finance. The non-anticipating derivative is designed for a very large class of integrators, and this is something I like because many problems in stochastic analysis are often tackled with different techniques depending on the type of noise. For example, the Hölder continuity of the paths of the Brownian motion is critically exploited, this technique cannot be applied directly to noises represented by discontinuous process. In stochastic calculus this means that the framework depends on the integrator and these frameworks may be dramatically different from each other.

**Is there a mathematical problem you would particularly like to solve?**

To be honest, there are many things I would like to do. I am now interested in statistical dependencies, particularly within time-space dynamic modelling. Time-space models include stochastic partial differential equations and random fields. The literature related is large, particularly with respect to the analysis of solutions and properties and typically relates to Gaussian driving noises. But what about the statistical dependences within these models in time and in space and, also, the dependency among these? In particular, one would like to depart from the classical description of statistical dependency via correlation, which is well suited for Gaussian distributions, but not beyond. Then we work on the development of concepts that would fit better various distributions (also not continuous) and that would capture at the same time the dynamics of the dependency structure in time and space, both finite and infinite in dimension.

**You have received a number of recognitions for your work, prestigious fellowships and awards. In 2019 you were the only woman to be awarded an ICIAM Prize in 2019, the Su Buchin, for your work in international cooperation. Can you tell us about the projects you supervised, in Africa and in general?**

Although I am participating in the committee for developing countries of the European Mathematical Society and I am involved with CIMPA, both with a worldwide perspective, my personal experience in developing countries is mostly in Africa. In general, most of my work has been focused in various ways on activities increasing visibility and promoting excellence in developing countries. For example, the work around the Emerging Regional Centres of Excellence (ERCE). This is a label, a recognition, which is awarded by the European Mathematical Society, to mathematics centres in a developing country providing education at a high level, meaning Masters and PhD programs with an international perspective, and that have a proactive role in the development of education and research in their own region. The label provides visibility and international recognition, strengthening the importance that these centres play and reinforcing their work. Also, it promotes a way of working among institutions in the spirit of cooperation between countries, and across mathematical disciplines.

We have recently applied for funding to evolve this network of ERCE further, providing some possibility for further activities including a visiting program for young and established researchers.

Another project particularly targeting mathematics in Africa is carried out as a collaboration of the committee of developing countries and the Simons Foundation. This is a program of visitors grants targeting different career levels, starting from PhD students, and reaching up to established researchers. We give particular weight to the young researchers, because, as we know, in mathematics, getting a PhD is surely an achievement but it is also the beginning of a research career which has to be continuously fostered for a more stable and long lasting impact.

Within this project, we have also implemented some top-up grants for female researchers, that could be used freely, for example for childcare, with the goal of giving means, visibility, and recognition to the work of women mathematicians. These top-up grants are very important, but they turn out difficult to implement, for the simple reason that we do not have that many women mathematicians that apply.

The general response to this program is very varied. Some countries are more receptive than others, and the big challenge is to have a more uniform distribution in applications. For example, we had an extremely good response from Northern and West African countries, but little in Southern African countries and almost none from those in the central region. Naturally, there are very many differences among these countries, including a difference in the possibility of obtaining national funding and general structure of academia. All these incur in the motivations for the different responses to the program.

**Which countries are involved in the project?**

Since the start of the program, we have had centres in Africa, Benin, Botswana, Marocco, in Asia, Indonesia, Iran, Malaysia, Vietnam, and also in the Americas, Mexico. Some new centres will be announced soon. This is still a young program, but you can find all information about it on the European Mathematical Society CDC webpage.

**Why do you choose the approach of promoting excellence rather than starting at basic education?**

It’s a complex question. The problem of education is very wide, the range of application goes from kindergarten to high levels of academia. There are many projects focusing on the early stage, by very competent and powerful institutions, why should we do any better at these levels?

The choice of going for excellence is motivated by long-term thinking and by being researchers in mathematics. As for long-term thinking, we can say that if a country does not possess enough capacity of people with a certain level of know-how, then it can easily fall into a state of under-development. Take the problem of clean water, for example: the lack of safe water is a crucial problem in many places. There have been many national and international programs for the development of water pipes, reservoirs etc. This appears clearly as a much more imminent problem than possibly the development of a new theorem, at least in terms of immediate impact. But, if the country does not have enough people that know how to handle the water pipes, optimize the distribution of water, the interplay of the reservoirs, then it is easy that the long lasting effects of the initial investment maybe shorter than expected. The development of high education in all fields provides the national expertise necessary for the development of any country. As researchers in mathematics, this is something we can work on and with. We can promote the development of the local academic capacity and excellency and by this improve on the higher education.

**What was your motivation for starting this kind of activities?**

When I started working in Oslo, my colleague Bernt Øksendal was already involved in a project funded by the Norwegian ministry for the development of a Master’s programme in pure and applied mathematics, based in Zimbabwe. It was an international programme in the sense that the network included more Southern African countries. The Southern African Mathematical Sciences Association, SAMSA, hosted an international conference every year in one of the countries in the region, that was my first time in Southern Africa. A whole new world emerged for me in terms of mathematics, education, the experience of local realities, network and connections. And that was when I got involved in the Norwegian project and started to supervise some students remotely. I learnt a lot from this experience. Then one project led to another one based in Tanzania, and finally the cooperation, with the committee for developing countries and CIMPA.

… in mathematics, the world is much closer than it looks. As we all try to push the frontiers of knowledge, working and playing with mathematics so to reach beyond its present boundaries, then we can do it also geographically.

**And you are still involved in cooperation?**

I am still involved, in different ways. For example, I’ve recently been happy to participate in some round tables on different developing programs. Different countries are promoting various actions out of the national developing programs and it is good to compare and collaborate on the actions that are proposed.

**Do you have a dream for the future, or for your future, you would like to share with us?**

I think it would be nice if people were aware that in mathematics, the world is much closer than it looks. Students and researchers that are keen in mathematics are so everywhere. As we all try to push the frontiers of knowledge, working and playing with mathematics so to reach beyond its present boundaries, then we can do it also geographically. I mean not only travelling but going further, being open to other societies and other opportunities.