Eveliina Peltola: Mathematics is beautiful because in it everything falls into place in a natural way

What are you researching and why?

I have started my assistant professorship of mathematics at Aalto, and I am also continuing in my post as professor in Bonn, Germany.

I research theories linked with mathematical physics. Physicists try to create models of the surrounding world and to confirm them through experimentation. Their mathematical foundations contain problems that mathematicians solve. The models also raise plenty of new questions that are interesting from the point of view of mathematics itself. I want to understand the surrounding world as a whole from a mathematical point of view and to ensure that the models describe what we want them to describe. This means that the theories have a strong foundation, and that the logical inferences are solid.

For example, my research includes a model describing small particles of magnetic materials with the help of statistical physics and quantum field theory. I am not yet involved in cooperation with Aalto's quantum physicists, but I hope that we might find topics of mutual interest in our research in the future.

I also study symmetries in the models, which are transformations that retain the relevant features of a system. Symmetry can be intuitive, such as rotation, or a more complicated mathematical structure. Preserving the characteristics can be literal, or it can mean statistical symmetry, in which the average features of a system are retained. In research I often start from things that are better understood through symmetry, and then I move to more difficult wholes, while building a theory. Symmetries are my friends!

How did you become a researcher?

It was partly coincidental. In upper secondary school I had no clear plan for my future and I was interested in everything.  I did not actually care much about mathematics, which seemed mechanical to me. However, being interested in sciences, I ended up at the Department of Mathematics at the University of Helsinki. My major was mathematics and I also studied physics and chemistry. I found that mathematics pleased me the most. I saw beauty in it - something greater than in the other subjects. On the other hand, my relationship with mathematics warmed up slowly, and I studied it for several years.

I feel that mathematics is beautiful because in it, everything gets arranged in a natural way. Ever since my days at university, the beauty of mathematics for me has been linked with symmetries: things started falling into place for me at a course in advanced algebra. Now I can see how this naturalness is also present in other areas of mathematics. My concept of the beauty of mathematics is abstract and intuitive.

Mathematics is a universal language that we need to get into before we can understand it. Although nowadays I make calculations that might look boring, the outcome can nevertheless be tidy, and there can even be sense in complexity. Mathematicians often get an internal intuition that guides things such as what kind of a result can be expected in a complex calculation. If the result is something different, I start to suspect that there may have been an error in the calculations, or in the reasoning.  I can also apply this intuition when checking the reasoning of the students.

Consequently, internal beauty and intuition concretely guide mathematical research, which is abstract in other respects.

What are the high points of your career?

A turning point in my studies was when I took part in a special course headed by Professor Kalle Kytölä at the University of Helsinki, where he worked at the time as an Academy Research Fellow. First, I ended up writing a master's thesis under Kytölä's guidance and from there I went on to a doctoral thesis. Both topics were linked with quantum field theory and random models.

I defended my doctoral thesis at the University of Helsinki in 2016 and getting a doctorate was naturally an important milestone. I liked working on my thesis and while I was doing it, I understood that research was a suitable field for me. My supervisor Kalle Kytölä was an academic role model for me, as nobody in my family had achieved a doctorate. He also managed to find a good opponent for me – Dmitry Chelkak. My thesis was mathematically interdisciplinary and Chelkak was very interested and capable of asking good questions.

After defending my dissertation, I went to the University of Geneva where I was a postdoctoral researcher from 2016 to 2019. It was a wonderful opportunity to get to know international research, as well as top researchers in the field, such as my mentor Stanislav Smirnov. I had time to find my own place in the research community. After this I was given a professorship in Bonn in 2019 at a somewhat unexpectedly early stage. Now, in the spring of 2021 I started as a professor also at Aalto University. Growing to be a member of the researcher community and as an academic leader has been a significant step in my career, bringing with it its own challenges and new interesting tasks.

What is the most important characteristic of a researcher?

Curiosity.  For me it means not looking at just one area, and instead being interested in many different matters. The best ideas, surprising results, and the creation of new things happen when researching something that is slightly outside one's own area.

I feel that it is not good for researchers to stay inside their own bubbles, examining only abstract theories or a very narrow field.

What are your expectations for the future?

I want to build my own research group, grow as a researcher, and follow the development of my students. It is wonderful to see students graduate and to get good jobs. I also want to inspire young school pupils and university students to go into mathematics and the natural sciences.

I hope I also get to develop Finnish mathematics, science, and academics, and to maintain international contacts. Finland is a small country, but it has done well in international competition. I would like to increase cooperation among universities, for example, by holding common seminars and courses, and in that way, strengthening the development of science in Finland. For example, I am one of those working on behalf of setting up an Academy of Finland Centre of Excellence in the fields of mathematics and artificial intelligence.