This article comes from the journal Science et Avenir – Les Indispensables n° 206, dated July/September 2021.

Olga Paris-Romaskevich is a mathematician and researcher at CNRS. he answered the questions of science and the future.

Science et Avenir: Can you tell us what dynamic systems are in your area?

Olga Paris-Romaskevich: Dynamics is motion, as opposed to statics. The study of a dynamical system is equivalent to defining the instantaneous change of an evolutionary event. For example, we can associate the bloom of a flower with a sequence of pictures taken every second: from one photograph to the next, we observe the movement of the flower. My goal as a dynamicist is to understand what will happen in the very long run if we continue with these photo suits. In this example, the flower eventually withers and stops moving! But for dynamical systems, the interesting moment is “at infinity”. The goal is to understand infinite motion in its totality, and without examining all developments, all trajectories.

How long has this domain existed?

It is much smaller than number theory, which appeared millennia ago with mathematical thought. Dynamic systems developed in the mid-19th century around questions related to the understanding of planetary motion. Since then, the area has exploded, and this question is only one of many. For my part, I am interested in paving in the subfield of billiards.

Trajectory, billiards, paving … Do these simple and concrete words have the same meaning here as in everyday life?

Mathematics is done by man, it has its roots in our language. They only borrow some everyday words from him to have a very precise meaning. Mathematical billiards are not billiards in the usual sense, that is, a rectangular table with holes and balls. Our mathematicians’ tables can be any size and have no holes. Our balls are reduced to points, they do not undergo any friction and therefore roll indefinitely. On the other hand, the law describing his rebellions remains like an intuition at first, the naive image emanating from the physical world: I have chosen the term billiards very well.

In this case, the concept is inspired by physical reality, but it seems to live in a mathematical universe… who would be a little outside of it?

One of the founders of dynamical systems in the early 20th century, Henri Poincaré, wanted to understand the potential stability of the Solar System, with nothing more real than the future of our planet! So mathematicians started with physical motivations, then built models. As he studied them, he found mathematics interesting in itself, which no longer needed physics to feed him. However, the connection with physics remains strong. I began to study billiards on the floor, touched by their mathematical beauty. Then they came out to relate the conductivity properties of metals. This example is unique.

If mathematical objects are separated from physics, is the demonstration of a new theorem or the introduction of a new object an invention or a discovery?

This is a question for a philosopher of science rather than a mathematician! In my opinion, answering it one way or another doesn’t change the practice of research. Personally, I care more about the relationship between objects and their properties than I need to know whether they are real or not. The only thing that matters to me is to get a mental picture of it. If I can work with him, that’s enough for me. The history of understanding complex numbers is a good example of this approach.

How does the creative process take place in Mathematica?

I don’t know! But I’ve seen mathematicians be interested, even obsessed, with this question, and by the way of thinking itself. Contemporary algebraist Claire Voisin says that doing mathematics is watching yourself think. I love this description. But where do ideas come from? If I knew a precise algorithm to get good algorithms, I would implement it day and night! Timing is important, of course some ideas should boil well in the mind first. And if I want to understand something substantial, I’m doing it. A little bit combinatorics, a little geometry, a little bit of ergodic theory (statistical analysis of deterministic systems, editor’s note) and a lot of enthusiasm, that’s how I do math. The other person will tell you something completely different.

Can we make a connection between artistic activity and mathematical research?

In mathematics, the choice of direction is guided, among other things, by personal aesthetics. It is associated with artistic approach. Another analogy, the management of the constraint. For clay models, the sculptor must deal with the ductility of the earth which enforces a gesture and the distribution of masses. Even in mathematics, we interact with the constraints of objects. Is a sculptor more independent in his works than a mathematician? This is a good topic for debate! In math, we can cut an orange into two oranges of the same size, but 2 does not equal 1 and we can live with these obvious contradictions. Art also plays with contradictions. But it is more accessible: everyone can appreciate a beautiful sculpture without wasting time creating it. I am sure that the time will come when mathematics museums will be found in all the cities of France.

Do these many constraints not cause interest in mathematics?

There are almost too many of them, and this is not pleasant. We must manage to get out of this! Interest comes from the path we take, and the ideas we discover as we walk it. Constraints never impose an argument. Proceeding step by step, very different from the final performance, the discovery process is chaotic and uncertain. It is he who forms the heart of the profession.

Interview by Roger Mansuyu