Prof. Elon Lindenstrauss, 2010 Fields Medal Laureate, in an interview for the Polish Mathematical Society and Ericpol

On September 13th 2011, as part of the joint initiative of the Polish Mathematical Society and Ericpol, an interview was held in the Biedermann Palace in Łódź with Elon Lindenstrauss, an outstanding mathematician, professor at the Einstein Mathematics Institute at the Hebrew University in Jerusalem and winner of the 2010 Fields Medal.

Elon Lindenstrauss was awarded the prestigious Fields Medal for his far-reaching advances in ergodic theory concerning the measure rigidity of higher rank diagonal actions in homogeneous spaces and its application in the analytical numbers theory. The Fields Medal has been awarded every four years since 1936 by the International Mathematical Union (IMU) during the International Congress of Mathematicians (ICM), and is viewed as the greatest honor which can be bestowed upon mathematicians under forty years old. The Medal is sometimes referred to as the equivalent of the Nobel Prize, which – as is commonly known – is not granted for achievements in mathematics.

Elon Lindenstrauss was one of the key plenary speakers at the 2011 Israeli-Polish Mathematical Congress which took place in Łódź, Poland, on September 11-15th 2011. Below you will find a transcript from the interview with Professor Lindenstrauss conducted by Aleksandra Cholewińska, a representative of Ericpol. This interview is part of a wider conversation, in which Mariusz Lemańczyk and Krzysztof Pawałowski from the Polish Mathematical Society also took part. The complete interview with Elon Lindenstrauss will be published in the Wiadomości Matematyczne (Mathematical News) journal in spring 2012.

Aleksandra Cholewinska, Ericpol: I’d like to ask a few questions more about how to promote mathematics rather than about the specifics of your work. This year, the International Banach Prize was awarded for the third time and I’d like to start with its patron. 18-year old Banach thought that everything there was to be discovered in the mathematics, had already been discovered, so there is no point studying it. A few years later he changed his mind completely and set new directions for this field. What do you think, how much it is there left to discover, will we ever be able to say we know everything about mathematics?

Elon Lindenstrauss: Actually, I didn’t know this about Banach. The belief, that everything there is to be discovered in mathematics, had already been discovered is a common misconception, that somehow I have encountered many times among people, and, of course, it is completely false. There are no limits to what we can know, we can always push the frontier more and more, and furthermore, build on what has already been discovered. I have heard a story about a colleague interviewed by some news organization. He told the interviewer about Euclid’s proof, given about 2500 years ago, that there exist infinitely many prime numbers, and was asked by the interviewer if the result is still true today…

That is a nice thing about mathematics – it never becomes obsolete. A theorem remains a theorem. The fact, that it is always much more to know, is not so surprising. You could always ask more questions, and – what’s important – more interesting questions. Somehow, the most wonderful thing is, that we can still manage this growth of information. All the knowledge we have gained since Euclid (realistically, in the last couple of hundreds of years) we have built on mathematics, we can still manage and use this knowledge effectively. Right now, I think it is a very good time for mathematics, there are lots of interesting ideas, new connections between different fields. We also see, that some conjectures, open for hundreds of years, now suddenly become solved, like the Fermat and Poincare Conjecture.

A.C.: Last year, at the Polish Mathematical Meeting there was a thesis, that biology is currently at the same level of development as physics was in the 19th century. Thanks to mathematics, biology could become even more advanced than physics is today. You’ve worked at the famous Institute for Advanced Study, where also Jon von Neumann, a good friend of Banach worked. Given your experiences from Princeton, how do you judge the possible uses of mathematics in other sciences, other fields?

E.L.: Mathematics and physics were in a way twins for a long time. They share a hundred years of fruitful interaction. Biology and mathematics – it is a much newer interaction. I am not very knowledgeable about the extent to which mathematics can be used in the context of biology. I am sure, that people with mathematical training could pick a good approach to do a lot of things. Mathematical training is a very healthy training, it is a sort of ability how to think in a clear and rigorous way, and apply criticism to what you do. So, this can be useful anywhere, including biology. It is also true, that nowadays in biology we have, in a sense, many things that you could translate into sequences of digits. Eg. DNA translates into sequences of digits. One might think that, potentially, you could use mathematics to analyze these things, but I have never worked in this direction, so I can’t give you an honest answer.

A.C.: In Poland, a country with good mathematics traditions, for a period of time maths was not compulsory for matriculation exams. Do you think it should be a subject only for those, who choose it, or for everyone? How should we teach mathematics in order to encourage young people? Stefan Banach said once, that maths is a sharp instrument, not suitable for children.

E.L.: It is another new thing I never heard about Banach. Still, it is certainly true that for many things one does, a certain amount of mathematical literacy is essential. Whether one wants to be an engineer, a biologist, any other type of scientist, a banker, or someone else from the finance industry, or software industry, you definitely need some sort of mathematical literacy. Certainly, it would be a good thing if people learn mathematics. I think it’s also an unfortunate fact, that many students view mathematics as an unpleasant subject, which it, obviously, isn’t. Of course, what they are given is not mathematics but a repetitive routine tribute, which is not so much fun and perhaps not super useful in any kind of discipline.

A.C.: It’s more about the way of teaching than the maths itself?

E.L.: I must say something that is trivial. Teachers should try to teach mathematics in an interesting way. It should be a part of what a student, who graduates from high school, should be expected to know. A little bit of mathematical literacy, as well as, some general literacy about the culture, about the government of the country, history, and other things that every knowledgeable citizen should feel comfortable with. Mathematics should be part of that , but not exclusively. I think it is right to say, that it is just one of the things people should learn.

A.C.: The mathematicians of Lwów used to meet in cafeteria, drink cognac and coffee, smoke cigarettes, pose questions and then write down solutions on tabletops, and later in the Scottish Book. Those meetings would last several hours, and as a form of teamwork would bring interesting, sometimes surprising results. What does a mathematician need in his work today and how does that work look like?

E.L.: Well, it’s a nice way to work – meet with friends over a coffee, share ideas, talk about this and that, and try to learn from each other. That’s definitely something which is very important to me. I think it’s wonderful to sit in the office, to work without any disturbance from outside, to deal with one problem after another. At the same time, I do like to work with other people, to share my ideas, to learn their point of view. To be an effective mathematician, you need time free of outside hassles, so that you can concentrate on what you do… I find, that I like to have, and I am fortunate to have, people that I enjoy talking to and trust. Likewise, they trust me and we share ideas. Basically, you don’t need much more than that – we don’t need laboratories, we don’t need anything very fancy, we need our books, we need our journals, we need…

A.C.:…a computer?

E.L.: The computer is an interesting and very powerful tool. Of course, one thing is that we use computers, like everybody else nowadays, to document what we do. We use them for email, to communicate, to access information. It is also true that sometimes one could use a computer as a research tool, just to check that something which should be true, really is. It’s even a useful tool in proofs that maybe would be very tedious or impossible to do, but are very simple to perform with use of a computer. So yes, it is an important tool but not indispensible, for some kinds of mathematical activity.

A.C.: Mathematicians are considered as one of the best jobs in the world. Is it a pleasure being a mathematician, or is it painful to always have to find a path into the unknown?

E.L.: It could be a very frustrating job. We work all the time. The moments, when we actually gain insight into something, when we suddenly understand something, that nobody understood before us, when we get these new ideas, that you can apply – these moments are very, very rare. In fact, most of the time you get very frustrated. So, it is quite a hard occupation. One of the ways to mitigate this, is that most of us teach. I believe teaching is a nice activity – it is not one where you have only a rare insight, but instead you work in a predetermined way. You get into a routine, some kind of continuous feedback from teaching. Of course, there are many good things about being a mathematician. You are really able to freely share ideas with many very smart people, it’s a beautiful thing, and I cherish being a mathematician.

A.C.: What are your goals in mathematics?

E.L.: There are some things that I really want to understand, some circles of conjectures that I would like to understand better. There are also many interesting results out there that I just would like to know.

A.C.: Mathematics is a tool for engineers. The Vice-President of Ericpol, Mr Marek Gajowniczek said, when awarding the Banach prize, that every good engineer must admire mathematics.

E.L.: Engineers use mathematics, and I think it’s extremely important that they understand the mathematics that they use. It’s very important, that when an engineer uses a formula, or an estimate, he needs to understand the meaning of it, and when it is valid. I think, one of the differences between a good engineer and a not-so-good engineer, is his ability to not just use mathematics, but to understand what he uses and why.

A.C.: What do mathematicians admire in the modern world? What do you admire?

E.L.: I am very curious about many things. I’d like to know how computers work, I’d like to know how things work in general. There are many interesting things that are happening in the world, changes taking place in societies. Knowledge is much more accessible than it was not so long ago. Definitely, this is something truly incredible!

A.C.: Prof Alex. Lubotzky says that mathematics is a most beautiful, exciting and challenging task, and when you’ve finished solving a problem it simply makes you happy. What would you wish to young mathematicians, who are just getting to know the subject in school, but will perhaps follow your footsteps in the future?

E.L.: The important thing for young mathematicians is to try to find some directions that they can be passionate about. If you aren’t passionate about a problem you can’t really work on it. You can’t focus yourself if you don’t have something you are passionate about. The best thing for people, let’s say for graduate students, is to find directions which really excite the imagination. I think it’s very important!