(Interview by Marieke van den Wegen and Mireia Martínez i Sellarès)
Utrecht, September 2019.
We asked Karma to explain the field she works on and the sort of research she does, in words that can be understood by high school students.
K: I am sitting in the Probability group but I am really an ergodic theorist. In Ergodic Theory we take a space, which can be anything, it can be “numbers” for example, and in this space we see things moving. This movement is formalised with a transformation T: “If x is here today, its position will be T(x) in the future”. In ergodic theory, we try to predict what will happen in the future, we study what happens to x in the long run. We will make probabilistic statements like “With probability 1, this or that will happen.” But in the last ten years, my concentration was on mixing ergodic theory with number theory a little bit. Well, “number theory” maybe is a big word, but I investigate different ways to look at numbers. For example, look at a decimal number. I see the decimal expansion really as a movement, which is given simply by the transformation: T(x) = 10x modulo 1, so “multiply the number by 10, and then only take the decimal part.” Then I check whether this new number is between 0 and 0.1 or between 0.1 and 0.2 or between 0.2 and 0.3, and so on. This tells me the next digit of the number. So for example if we have a number 0.7386…, we see that it is between 0.7 and 0.8, so the first digit is a 7. Then T(0.7386…) = 0.386…, which is between 0.3 and 0.4, so the next digit is a 3. So that’s how I see numbers: as moving, really, and where they land gives me the digit. Now, if I pick a number at random and perform this procedure, the probability of landing between 0.3 and 0.4 is 0.1. So in the long run, the frequency of seeing a 3 is 1/10. So in this way you can make statements about the behaviour of something in the future, and you can predict what the frequency of the digits is.
Mi: So did you start with this idea? Or is this something you jumped on (using Ergodic theory to study numbers)?
K: No, I mean, my thesis was on Ergodic theory but it was really very theoretical, it had a lot of algebra and topology in it and stuff like this. But as time went by it evolved, at one point… You don’t want to do the same thing all your life, huh? So yeah, then I moved to the interaction of Ergodic theory and Number theory and now I am still working on it. Sometimes, every once in a while I have projects on fractals.
Mi: I think that’s cool too, because I feel, when you’re still studying… To me at least it feels like, “Be careful because whatever you pick for your thesis is going to be the field you will work on, it’s going to be the topic!” But by talking to professional mathematicians I realise that’s not really true, because your interests evolve with time…
Mi: …or maybe you discover a whole new area you didn’t know about.
K: Yeah, absolutely. You know what? At the beginning, after your PhD, I think the first five years at least you will still be working on what you did in your thesis. You will probably be generalising or doing the things that you didn’t do in your PhD because you didn’t have time. I think you will still work on it. But you go to conferences and workshops and you listen to people and you meet people, and questions come in. And it’s very nice to collaborate with other people: you learn from them and you widen your scope of interests too. And it’s fun! It’s very nice to collaborate.
Mi: I agree, it’s half the fun if you can discuss mathematics with someone else.
K: Makes it also more human, you know? You have somebody to talk to.
Mi: Then I am curious to know what your experience was like coming to Utrecht University, because you were the only female professor here in Utrecht for 25 years. You mentioned the other day that you can see that things are changing now, so I am very curious to know what your experience was like and in which ways you see that things are changing.
K: Uhmm, that’s a big question… I mean, mathematics doesn’t have a lot of females, but when I first arrived in the States there were a few females in the department, so there I didn’t feel, like, lonely or anything. And when I came here, first at TU Delft there were also no women and I felt like the people didn’t know how to communicate with me. I felt like they were a bit nervous to deal with a different creature. (Laughs.) But I mean, they were very welcoming! It’s not that I felt lonely or anything, I didn’t have a problem with the fact that I was the only woman, but now I realise that in meetings and things like these I kept my mouth shut a lot of times. And it might be because my language was not perfect, so I couldn’t communicate, but also I was afraid to talk because I felt like “Maybe if I say something it will be stupid.” But then I realised: “Other people are asking the same questions that I am thinking about!” But still I kept silent for a long time. I mean, only maybe in the last ten years or something like this I’ve been more outspoken! But before I felt maybe a little bit intimidated, yeah. And now having female colleagues the atmosphere is also different. I don’t know, women are… I mean, we are a little bit different. It’s easy for us, among each other, to say what our weaknesses are, what we are afraid of or whatever. But if you are alone you can’t talk to people, you know? You compare yourself to men, who are very competitive usually. And I’ve learned through my life that I shouldn’t compare myself to other people, because otherwise you feel insecure and you feel always as if you are less than others. I really never compare myself, I just do what I like, regardless of whether people think this is important or not. If you put your heart into something, you succeed. Really. I do it in a “zen” way, like, every day I see what I have to do and I do it, and I don’t hesitate. The point is, you have to enjoy what you are doing, you have to have fun with it. Because this is your life, huh? And I do have fun, I love mathematics, I really do.
K: Yeah. Sometimes, you know, probably you also feel it: you get stuck every once in a while in your research and you don’t feel like you are up to it. But anyway I keep doing it, I don’t care what people think. I know that it will be resolved. And things have been going okay, actually, so I’m happy. (Smiles.)
Mi: So does your position as a professor at the university allow you to share these thoughts and experiences with your students? Do you feel that students are open to advice like this or is it just mathematics?
K: Nooo, I have very good relationships with the students. I talk to them quite a bit about mathematics and the way they should see things. I really believe everyone has potential inside, even the people that think they are weak. And I feel it is very important to keep encouraging, giving positive signals. This way you get the best out of everybody! I mean, the students that do theses with me, most of them do really very well, more than I expected them at the beginning. Because you get the best out of them. So it’s very important to be encouraging, and that it’s okay if once in a while things don’t go right: everybody passes through this. And just… Determination is very important, don’t give up and enjoy what you are doing! If it is something that you hate, nothing will come out, you know, you are forcing yourself. Don’t force yourself: play the game, go along with it, enjoy the journey.