Svante Janson creates new mathematics with his imagination

In mathematics, achieving results can take a long time. This is, however, not the case for Svante Janson. Since the beginning of his career, Svante Janson has authored two doctoral dissertations, and written nearly three hundred articles.

- I write articles quickly because I choose problems that are not too difficult, which makes it easier to solve them in a shorter period of time. Another reason as to why I write quickly is that I have many ideas for the solutions, so I am usually eager to put them in print and get it done.

Svante Janson started studying mathematics at university level already at the age of twelve, although it was not obvious from the beginning that mathematics was the subject that he would devote himself to. Unlike many others who started to learn advanced mathematics at a young age, he has no early memories of his mathematical talent. Instead, it was  the natural sciences that aroused his curiosity. He read his mother's old school books, and was mostly interested in chemistry. But seeing that he was only twelve years old when he entered university, he could not perform any laboratory experiments, and so the choice fell on mathematics.

How was it to study at university as a twelve-year-old?

- I didn't spend much time at the university. I didn’t go to the lectures during my first years. Instead, I studied by myself at home in Stora Tuna. I went to the university a few times to talk to the teachers, but besides that I only went there for the exams.

Although practical reasons initially laid behind Svante Janson's career choice, he has never regretted that he became a mathematician.

- It suites me perfectly. There are many other interesting subjects, but I would never choose anything else before mathematics.

What is the best thing about your job?

- The best thing is that I can devote a large part of my time to think of fun problems that interest me.

It sounds like a great freedom.

- It is. I choose which problems I want to solve, and how and when I'll try to solve them. I also read books and journals to see what others have done. Reading is fun, especially if the results are beautiful, which they often are.

Many mathematicians can relate to the experience of mathematics as beautiful. There are also scientific studies indicating that the brain can react to mathematical reasoning in the same way it reacts to music or art. But even if the experience of beauty is real, it's hard to distinguish exactly what it is that makes mathematics beautiful.

- What is beautiful is hard to say. Short reasonings are generally more appealing than long computations over multiple pages, but all short reasonings are not elegant. It's a matter of taste. Of course, solutions that use methods I already know and understand are more appealing to me than others, either long or short, that use methods I haven’t learned. In this way the experience of beauty is subjective, says Svante Janson and continues.

- Sometimes it's the new and unexpected things that are beautiful. When someone has used a method that has no obvious connection to the problem, something that no one else has thought of before. Unexpectedness in itself is not always elegant, but it can be when it helps you to see how things can be related in new surprising ways.

Svante Janson's current research is about a probabilistic phenomenon called random graphs. In particular, he focuses on a special case of random graphs called trees. A tree is a number of points which are connected in a tree structure. In his research Svante Janson tries to find different ways to construct such trees randomly.

- I'm interested in what happens when you have very large random trees, and what it looks like on average. In probability theory chance typically evens out in the long run. Things that are random become regular if you have very large sets.

It makes me think of the overlap that exists between probability theory and analysis. Does it have to do with chance evening out in the long run?

- As I see it, the overlap is found mostly in the methods. By looking at limits ​​and asymptotics, the analysis sneaks in through the back door. It is not visible in the problems from the beginning, but it appears in the solutions. It's good and important to know the different areas of mathematics, because this type of contact often emerges.

You mentioned in another interview that one can use random graphs to make predictions of how disease spreads. Are you involved in those kinds of applications?

- No. I look at purely mathematical models, simplified models that I study in great detail. In order to make them more useful, you'd need models that are more complex and adapted to reality, which I have no knowledge of. To be able to work in applied mathematics one's required to know the applications properly, to have a knowledge of reality, and be able to make the necessary connections.

What do you like about math?

- Seeing how things fit together. It's like solving crosswords or jigsaw puzzles. Of course, in mathematics you have other kinds of problems, but the satisfaction when things fit together in the end is the same.

How do you choose problems?

- Often I get ideas when I'm attending conferences, when I talk to people, or hear talks that lead on to new problems. Either I start wondering what happens in other situations, if there's a way to generalize, or I find the problem interesting and think that it can be done in another, better way.

So it is in interaction with others that ideas arise?

- Yes.

It goes against the myth of mathematicians as loners solving problems by themselves.

- You always need to have something to start from. Of course, you can come up with problems by yourself too, or by reading others' articles, but when you meet and talk to people, new things come up.

How do you go about solving a problem?

- I try to think about whether I can succeed by using one of the methods that I've used in similar problems, or seen others do. I also try to find out if I can get a new version or a new combination of methods that could work in this case. I usually sit down and do a few of calculations and see if it leads somewhere.

- It's kind of like you did on your exam problems when you studied mathematics. I suppose you tried to remember methods you used on similar problems, and then checked whether one of them could work here too. It's roughly the same principle, except that I have a much larger and more open field of methods. And above all, I'm not even sure that any method will work at all.

- I have often felt that it's very much coincidences and chance that determine whether you come up with something. It's not like I can go through all conceivable methods one by one. It requires some imagination too.

Problem solving in mathematics and creativity in general are processes that often are surrounded by a mystique. But I have a feeling that it's possible to figure out how they work. What do you think about that?

- I think it's important to have a lot of basic knowledge and try to have a little imagination to use that knowledge on new problems in new ways. But not too new ways. Much of the work includes using old methods in a standard way, and checking where they fit, what you can do with them and where you need to come up with something new.

- You have to have new ideas. But once you got these ideas, it often requires a lot of work to really see if they fit, or modify them so that they will fit. There are no shortcuts. You have to do the work.

Alma Kirlic