N E WS L E T T ER
OF THE EUROPEAN MATHEMATICAL SOCIETY

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S E
EuropeanM
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MathematicalE S 25 years Society

25th EMS Anniversary
The first three ECMs
The EMS from 1999 to 2006
EMIS
Interview
Abel Laureate
John F. Nash Jr.
Abel Science Lecture
Soap Bubbles and Mathematics

September 2015
Issue 97
ISSN 1027-488X

FeatureDiagonals of Rational
Fractions
Discussion
Mathematics between
Research, Application,
and Communication

Interview

Interview with Abel Laureate
John F. Nash Jr.
Martin Raussen (Aalborg University, Denmark) and Christian Skau (Norwegian University of Science andTechnology, Trondheim, Norway)
This interview took place in Oslo on 18 May 2015, the day
before the prize ceremony and only five days before thetragic accident that led to the death of John Nash and hiswife Alicia.
Nash’s untimely death made it impossible to follow theusual procedure for Abel interviews where interviewees
are asked to proof-read and to edit first drafts. All possible misunderstandings are thus the sole responsibility of
the interviewers.

The prize
Professor Nash, we would like to congratulate you as theAbel laureate in mathematics for 2015, a prize you sharewith Louis Nirenberg. What was your reaction when you
first learned that you had won the Abel Prize?
I did not learn about it like I did with the Nobel Prize. I
got a telephone call late on the day before the announcement, which was confusing. However, I wasn’t entirely
surprised. I had been thinking about the Abel Prize. It is
an interesting example of a newer category of prizes thatare quite large and yet not entirely predictable. I was given sort of a pre-notification. I was told on the telephonethat the Abel Prize would be announced on the morning
the next day. Just so I was prepared.
But it came unexpected?
It was unexpected, yes. I didn’t even know when the Abel
Prize decisions were announced. I had been reading
about them in the newspapers but not following closely.
I could see that there were quite respectable persons being selected.

Youth and Education
When did you realise that you had an exceptional talent for mathematics? Were there people that encouraged
you to pursue mathematics in your formative years?
Well, my mother had been a school teacher, but she
taught English and Latin. My father was an electrical engineer. He was also a schoolteacher immediately before
World War I.
While at the grade school I was attending, I would
typically do arithmetic – addition and multiplication –with multi-digit numbers instead of what was given at
the school, namely multiplying two-digit numbers. So I
got to work with four- and five-digit numbers. I just got
pleasure in trying those out and finding the correct procedure. But the fact that I could figure this out was a sign,of course, of mathematical talent.

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John F. Nash jr. and his wife Alicia were received by His Majesty
King Harald V. at the Royal Palace. (Photo: Håkon Mosvold Larsen/
NTB Scanpix.)

Then there were other signs also. I had the book by
E.T. Bell, “Men of Mathematics”, at an early age. I could
read that. I guess Abel is mentioned in that book?
Yes, he is. In 1948, when you were 20 years of age, youwere admitted as a graduate student in mathematics
at Princeton University, an elite institution that handpicked their students. How did you like the atmosphereat Princeton? Was it very competitive?
It was stimulating. Of course it was competitive also – aquiet competition of graduate students. They were notcompeting directly with each other like tennis players.
They were all chasing the possibility of some special appreciation. Nobody said anything about that but it wassort of implicitly understood.

Games and game theory
You were interested in game theory from an early stage.
In fact, you invented an ingenious game of a topological nature that was widely played, by both faculty members and students, in the Common Room at Fine Hall,
the mathematics building at Princeton. The game wascalled “Nash” at Princeton but today it is commonly
known as “Hex”. Actually, a Danish inventor and designer Piet Hein independently discovered this game.
Why were you interested in games and game theory?

EMS Newsletter September 2015

Interview
Well, I studied economics at my previous institution, theCarnegie Institute of Technology in Pittsburgh (today
Carnegie Mellon University). I observed people who
were studying the linkage between games and mathematical programming at Princeton. I had some ideas:some related to economics, some related to games likeyou play as speculators at the stock market – which is really a game. I can’t pin it down exactly but it turned outthat von Neumann1 and Morgenstern2 at Princeton had
a proof of the solution to a two-person game that wasa special case of a general theorem for the equilibriumof n-person games, which is what I found. I associatedit with the natural idea of equilibrium and of the topological idea of the Brouwer fixed-point theorem, whichis good material.
Exactly when and why I started, or when von Neumann and Morgenstern thought of that, that is something
I am uncertain of. Later on, I found out about the Kakutani fixed-point theorem, a generalisation of Brouwer’s
theorem. I did not realise that von Neumann had inspired
it and that he had influenced Kakutani.3 Kakutani was astudent at Princeton, so von Neumann wasn’t surprisedwith the idea that a topological argument could yield
equilibrium in general. I developed a theory to study a
few other aspects of games at this time.
You are a little ahead of us now. A lot of people outside
the mathematical community know that you won theNobel Memorial Prize in Economic Sciences in 1994.
That was much later.
Yes. Due to the film “A Beautiful Mind”, in which youwere played by Russell Crowe, it became known to avery wide audience that you received the Nobel Prizein economics. But not everyone is aware that the Nobel Prize idea was contained in your PhD thesis, which
was submitted at Princeton in 1950, when you were21-years-old. The title of the thesis was “Non-cooperative games”.
Did you have any idea how revolutionary this would
turn out to be? That it was going to have impact, notonly in economics but also in fields as diverse as political science and evolutionary biology?
It is hard to say. It is true that it can be used wherever
there is some sort of equilibrium and there are competing or interacting parties. The idea of evolutionists isnaturally parallel to some of this. I am getting off on ascientific track here.
But you realised that your thesis was good?
Yes. I had a longer version of it but it was reduced by mythesis advisor. I also had material for cooperative games
but that was published separately.
Did you find the topic yourself when you wrote yourthesis or did your thesis advisor help to find it?
1

1903–1957.
1902–1977.
3 1911–2004.

Well, I had more or less found the topic myself and then
the thesis advisor was selected by the nature of my topic.
Albert Tucker 4 was your thesis advisor, right?
Yes. He had been collaborating with von Neumann andMorgenstern.

Princeton
We would like to ask you about your study and work
habits. You rarely attended lectures at Princeton.
Why?
It is true. Princeton was quite liberal. They had introduced, not long before I arrived, the concept of an Ngrade. So, for example, a professor giving a course would
give a standard grade of N, which means “no grade”.
But this changed the style of working. I think that Harvard was not operating on that basis at that time. I don’tknow if they have operated like that since. Princeton has
continued to work with the N-grade, so that the numberof people actually taking the courses (formally taking
courses where grades are given) is less in Princeton thanmight be the case at other schools.
Is it true that you took the attitude that learning too
much second-hand would stifle creativity and originality?
Well, it seems to make sense. But what is second-hand?
Yes, what does second-hand mean?
Second-hand means, for example, that you do not learn
from Abel but from someone who is a student of abelian
integrals.
In fact, Abel wrote in his mathematical diary that one
should study the masters and not their pupils.
Yes, that’s somewhat the idea. Yes, that’s very parallel.
While at Princeton you contacted Albert Einstein and
von Neumann, on separate occasions. They were at theInstitute for Advanced Study in Princeton, which is located close to the campus of Princeton University. It
was very audacious for a young student to contact such
famous people, was it not?
Well, it could be done. It fits into the idea of intellectual
functions. Concerning von Neumann, I had achieved myproof of the equilibrium theorem for game theory using
the Brouwer fixed-point theorem, while von Neumann
and Morgenstern used other things in their book. But
when I got to von Neumann, and I was at the blackboard,
he asked: “Did you use the fixed-point theorem?”“Yes,” I said. “I used Brouwer’s fixed-point theorem.”
I had already, for some time, realised that there wasa proof version using Kakutani’s fixed-point theorem,
which is convenient in applications in economics since
the mapping is not required to be quite continuous. It
has certain continuity properties, so-called generalised

2

EMS Newsletter September 2015

4

1905–1995.

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Interview
continuity properties, and there is a fixed-point theorem
in that case as well. I did not realise that Kakutani provedthat after being inspired by von Neumann, who was using
a fixed-point theorem approach to an economic problem
with interacting parties in an economy (however, he was
not using it in game theory).
What was von Neumann’s reaction when you talked
with him?
Well, as I told you, I was in his office and he just mentioned some general things. I can imagine now what he
may have thought, since he knew the Kakutani fixedpoint theorem and I did not mention that (which I could
have done). He said some general things, like: “Of course,
this works.” He did not say too much about how wonderful it was.
When you met Einstein and talked with him, explainingsome of your ideas in physics, how did Einstein react?
He had one of his student assistants there with him. I was
not quite expecting that. I talked about my idea, which
related to photons losing energy on long travels throughthe Universe and as a result getting a red-shift. Other
people have had this idea. I saw much later that someone
in Germany wrote a paper about it but I can’t give youa direct reference. If this phenomenon existed then the
popular opinion at the time of the expanding Universe
would be undermined because what would appear to bean effect of the expansion of the Universe (sort of a Doppler red-shift) could not be validly interpreted in thatway because there could be a red-shift of another origin.
I developed a mathematical theory about this later on. I
will present this here as a possible interpretation, in my
Abel lecture tomorrow.
There is an interesting equation that could describe
different types of space-times. There are some singularities that could be related to ideas about dark matter anddark energy. People who really promote it are promoting
the idea that most of the mass in the Universe derives
from dark energy. But maybe there is none. There could
be alternative theories.
John Milnor, who was awarded the Abel Prize in 2011,
entered Princeton as a freshman the same year as you
became a graduate student. He made the observationthat you were very much aware of unsolved problems,
often cross-examining people about these.
Were you on the lookout for famous open problems
while at Princeton?
Well, I was. I have been in general. Milnor may have noticed at that time that I was looking at some particular
problems to study.
Milnor made various spectacular discoveries himself.
For example, the non-standard differentiable structureson the seven-sphere. He also proved that any knot has acertain amount of curvature although this was not reallya new theorem, since someone else5 had – unknown to
Milnor – proved that.
5

István Fáry.

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John F. Nash jr. at the Common Room, Institute of Advanced Study,
Princeton. (Courtesy of the Institute for Advanced Study. Photo by
Serge J.-F. Levy.)

A series of famous results
While you wrote your thesis on game theory at Princeton
University, you were already working on problems of avery different nature, of a rather geometric flavour. And
you continued this work while you were on the staff at
MIT in Boston, where you worked from 1951 to 1959.
You came up with a range of really stunning results. In
fact, the results that you obtained in this period are themain motivation for awarding you the Abel Prize thisyear.
Before we get closer to your results from this period, we would like to give some perspective by quoting
Mikhail Gromov, who received the Abel Prize in 2009.
He told us, in the interview we had with him six years
ago, that your methods showed “incredible originality”. And moreover: “What Nash has done in geometryis from my point of view incomparably greater thanwhat he has done in economics, by many orders of magnitude.”
Do you agree with Gromov’s assessment?
It’s simply a question of taste, I say. It was quite a struggle.
There was something I did in algebraic geometry, whichis related to differential geometry with some subtleties in
it. I made a breakthrough there. One could actually gain
control of the geometric shape of an algebraic variety.
That will be the subject of our next question. You submitted a paper on real algebraic manifolds when you
started at MIT, in October 1951. We would like to quote
Michael Artin at MIT, who later made use of your result. He commented: “Just to conceive such a theorem
was remarkable.”
Could you tell us a little of what you dealt with andwhat you proved in that paper, and how you got started?
I was really influenced by space-time and Einstein, andthe idea of distributions of stars, and I thought: ‘Suppose
some pattern of distributions of stars could be selected;
could it be that there would be a manifold, something
curving around and coming in on itself that would be insome equilibrium position with those distributions ofstars?’ This is the idea I was considering. Ultimately, I de-

EMS Newsletter September 2015

Interview
veloped some mathematical ideas so that the distributionof points (interesting points) could be chosen, and thenthere would be some manifold that would go around in
a desired geometrical and topological way. So I did thatand developed some additional general theory for doingthat at the same time, and that was published.
Later on, people began working on making the representation more precise because I think what I proved
may have allowed some geometrically less beautiful
things in the manifold that is represented, and it mightcome close to other things. It might not be strictly finite.
There might be some part of it lying out at infinity.
Ultimately, someone else, A. H. Wallace,6 appeared tohave fixed it, but he hadn’t – he had a flaw. But later it
was fixed by a mathematician in Italy, in Trento, named
Alberto Tognoli.7
We would like to ask you about another result, concerning the realisation of Riemannian manifolds. Riemannian manifolds are, loosely speaking, abstract smooth
structures on which distances and angles are only locally defined in a quite abstract manner. You showed thatthese abstract entities can be realised very concretely as
sub-manifolds in sufficiently high-dimensional Euclidean spaces.
Yes, if the metric was given, as you say, in an abstract
manner but was considered as sufficient to define a metric structure then that could also be achieved by an embedding, the metric being induced by the embedding.
There I got on a side-track. I first proved it for manifolds
with a lower level of smoothness, the C1-case. Some otherpeople have followed up on that. I published a paper onthat. Then there was a Dutch mathematician, Nicolaas
Kuiper,8 who managed to reduce the dimension of theembedding space by one.
Apart from the results you obtained, many people havetold us that the methods you applied were ingenious.
Let us, for example, quote Gromov and John Conway.
Gromov said, when he first read about your result: “Ithought it was nonsense, it couldn’t be true. But it wastrue, it was incredible.” And later on: “He completelychanged the perspective on partial differential equations.” And Conway said: “What he did was one of themost important pieces of mathematical analysis in the20th century.” Well, that is quite something!
Yes.
Is it true, as rumours have it, that you started to workon the embedding problem as a result of a bet?
There was something like a bet. There was a discussion
in the Common Room, which is the meeting place forfaculty at MIT. I discussed the idea of an embedding with
one of the senior faculty members in geometry, Professor
Warren Ambrose.9 I got from him the idea of the reali6

1926–2008.
1937–2008.
8 1920–1994.
9 1914–1995.
7

EMS Newsletter September 2015

sation of the metric by an embedding. At the time, thiswas a completely open problem; there was nothing there
beforehand.
I began to work on it. Then I got shifted onto the C1case. It turned out that one could do it in this case withvery few excess dimensions of the embedding space compared with the manifold. I did it with two but then Kuiper
did it with only one. But he did not do it smoothly, which
seemed to be the right thing – since you are given something smooth, it should have a smooth answer.
But a few years later, I made the generalisation tosmooth. I published it in a paper with four parts. Thereis an error, I can confess now. Some 40 years after thepaper was published, the logician Robert M. Solovay
from the University of California sent me a communication pointing out the error. I thought: “How could itbe?” I started to look at it and finally I realised the error in that if you want to do a smooth embedding andyou have an infinite manifold, you divide it up into portions and you have embeddings for a certain amount of
metric on each portion. So you are dividing it up into a
number of things: smaller, finite manifolds. But what I
had done was a failure in logic. I had proved that – how
can I express it? – that points local enough to any point
where it was spread out and differentiated perfectly if
you take points close enough to one point; but for two
different points it could happen that they were mapped
onto the same point. So the mapping, strictly speaking,wasn’t properly embedded; there was a chance it had
self-intersections.
But the proof was fixed? The mistake was fixed?
Well, it was many years from the publication that I
learned about it. It may have been known without being
officially noticed, or it may have been noticed but people
may have kept the knowledge of it secret.10
May we interject the following to highlight how surprising your result was? One of your colleagues at MIT,
Gian-Carlo Rota,11 professor of mathematics and also
philosophy at MIT, said: “One of the great experts on
the subject told me that if one of his graduate students
had proposed such an outlandish idea, he would throw
him out of his office.”That’s not a proper liberal, progressive attitude.

Partial differential equations
But nevertheless it seems that the result you proved wasperceived as something that was out of the scope of thetechniques that one had at the time.
Yes, the techniques led to new methods to study PDEs
in general.
10 The result in Nash’s paper is correct; it has been reproved byseveral researchers (notably Mikhail Gromov) using the general strategy devised by Nash. Nash gave his own account on
this error in the case of embeddings of non-compact manifolds in the book The essential John Nash (eds. Harold W.
Kuhn and Sylvia Nasar), Princeton University Press, 2002.
11 1932–1999.

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Interview
Let us continue with work of yours purely within thetheory of PDEs. If we are not mistaken, this came about
as a result of a conversation you had with Louis Nirenberg, with whom you are sharing this year’s Abel Prize,at the Courant Institute in New York in 1956. He told
you about a major unsolved problem within non-linear
partial differential equations.
He told me about this problem, yes. There was some workthat had been done previously by a professor in California, C. B. Morrey,12 in two dimensions. The continuity
property of the solution of a partial differential equation
was found to be intrinsic in two dimensions by Morrey.
The question was what happened beyond two dimensions. That was what I got to work on, and de Giorgi13, an
Italian mathematician, got to work on it also.
But you didn’t know of each other’s work at that time?
No, I didn’t know of de Giorgi’s work on this, but he didsolve it first.
Only in the elliptic case though.
Yes, well, it was really the elliptic case originally but I sortof generalised it to include parabolic equations, which
turned out to be very favourable. With parabolic equations, the method of getting an argument relating to an
entropy concept came up.
I don’t know; I am not trying to argue about precedents but a similar entropy method was used by Professor Hamilton in New York and then by Perelman. Theyuse an entropy which they can control in order to control
various improvements that they need.
And that was what finally led to the proof of the Poincaré Conjecture?
Their use of entropy is quite essential. Hamilton used it
first and then Perelman took it up from there. Of course,
it’s hard to foresee success.
It’s a funny thing that Perelman hasn’t accepted any
prizes. He rejected the Fields Prize and also the Clay
Millennium Prize, which comes with a cash award of one
million dollars.
Coming back to the time when you and de Giorgi worked
more or less on the same problem. When you first foundout that de Giorgi had solved the problem before you,were you very disappointed?
Of course I was disappointed but one tends to find someother way to think about it. Like water building up andthe lake flowing over, and then the outflow stream backing up, so it comes out another way.
Some people have been speculating that you might havereceived the Fields Medal if there had not been the coincidence with the work of de Giorgi.
Yes, that seems likely; that seems a natural thing. De Giorgi did not get the Fields Medal either, though he did getsome other recognition. But this is not mathematics, think-ing about how some sort of selecting body may function. It
is better to be thought about by people who are sure they
are not in the category of possible targets of selection.
When you made your major and really stunning discoveries in the 1950s, did you have anybody that you could
discuss with, who would act as some sort of sounding
board for you?
For the proofs? Well, for the proof in game theory there
is not so much to discuss. Von Neumann knew that there
could be such a proof as soon as the issue was raised.
What about the geometric results and also your otherresults? Did you have anyone you could discuss theproofs with?
Well, there were people who were interested in geometry
in general, like Professor Ambrose. But they were not somuch help with the details of the proof.
What about Spencer 14 at Princeton? Did you discusswith him?
He was at Princeton and he was on my General Examcommittee. He seemed to appreciate me. He worked incomplex analysis.
Were there any particular mathematicians that you met
either at Princeton or MIT that you really admired, thatyou held in high esteem?
Well, of course, there is Professor Levinson15 at MIT. I
admired him. I talked with Norman Steenrod16 at Princeton and I knew Solomon Lefschetz,17 who was Department Chairman at Princeton. He was a good mathematician. I did not have such a good rapport with the algebra
professor at Princeton, Emil Artin.18

The Riemann Hypothesis
Let us move forward to a turning point in your life. You
decided to attack arguably the most famous of all open
problems in mathematics, the Riemann Hypothesis,
which is still wide open. It is one of the Clay Millennium Prize problems that we talked about. Could youtell us how you experienced mental exhaustion as a result of your endeavour?
Well, I think it is sort of a rumour or a myth that I actually made a frontal attack on the hypothesis. I was cautious. I am a little cautious about my efforts when I try
to attack some problem because the problem can attack
back, so to say. Concerning the Riemann Hypothesis,
I don’t think of myself as an actual student but maybe
some casual – whatever – where I could see some beautiful and interesting new aspect.
Professor Selberg,19 a Norwegian mathematician who
was at the Institute for Advanced Study, proved back in
14 1912–2001.
15 1912–1975.
16 1910–1971.
17 1884–1972.

12 1907–1984.
13 1928–1996.

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18 1898–1962.
19 1917–2007.

EMS Newsletter September 2015

Interview
What might be favourable for me to try? So I don’t thinkof trying anything of the latest popular nonsense.
You have said in an interview (you may correct us)
something like: “I wouldn’t have had good scientificideas if I had thought more normally.” You had a different way of looking at things.
Well, it’s easy to think that. I think that is true for me justas a mathematician. It wouldn’t be worth it to think likea good student doing a thesis. Most mathematical theses
are pretty routine. It’s a lot of work but sort of set up by
the thesis advisor; you work until you have enough andthen the thesis is recognised.
John F. Nash jr. with last year’s Abel laureate Yakov Sinai (right) andMichael Th. Rassias (left). (Photo: Danielle Alio, Princeton University, Office of Communications.)

the time of World War II that there was at least somefinite measure of these zeros that were actually on thecritical line. They come as different types of zeros; it’s
like a double zero that appears as a single zero. Selberg
proved that a very small fraction of zeros were on thecritical line. That was some years before he came to theInstitute. He did some good work at that time.
And then, later on, in 1974, Professor Levinson at
MIT, where I had been, proved that a good fraction –around 1/3 – of the zeros were actually on the critical line.
At that time he was suffering from brain cancer, which
he died from. Such things can happen; your brain can beunder attack and yet you can do some good reasoning
for a while.

A very special mathematician?
Mathematicians who know you describe your attitude
toward working on mathematical problems as very different from that of most other people. Can you tell us alittle about your approach? What are your sources ofinspiration?
Well, I can’t argue that at the present time I am working
in such and such a way, which is different from a more
standard way. In other words, I try to think of what I cando with my mind and my experiences and connections.

Interests and hobbies
Can we finally ask you a question that we have asked
all the previous Abel Prize laureates? What are yourmain interests or hobbies outside of mathematics?
Well, there are various things. Of course, I do watch thefinancial markets. This is not entirely outside of the proper range of the economics Nobel Prize but there is a lot
there you can do if you think about things. Concerning thegreat depression, the crisis that came soon after Obama
was elected, you can make one decision or another decision which will have quite different consequences. The
economy started on a recovery in 2009, I think.
It is known that when you were a student at Princeton
you were biking around campus whistling Bach’s “Little Fugue”. Do you like classical music?
Yes, I do like Bach.
Other favourite composers than Bach?
Well, there are lot of classical composers that can be quite
pleasing to listen to, for instance when you hear a goodpiece by Mozart. They are so much better than composers like Ketèlbey and others.
We would like to thank you very much for a very interesting interview. Apart from the two of us, this is onbehalf of the Danish, Norwegian and European Mathematical Societies.
After the end of the interview proper, there was an informal chat about John Nash’s main current interests. He
mentioned again his reflections about cosmology. Concerning publications, Nash told us about a book entitledOpen Problems in Mathematics that he was editing withthe young Greek mathematician Michael Th. Rassias,
who was conducting postdoctoral research at Princeton
University during that academic year.

From left to right: John F. Nash Jr., Christian Skau, Martin Raussen.
(Photo: Eirik F. Baardsen, DNVA.)EMS Newsletter September 2015

Martin Raussen is professor with special responsibilities(mathematics) at Aalborg University, Denmark. Christian
Skau is professor of mathematics at the Norwegian University of Science and Technology at Trondheim. They
have together taken interviews with all Abel laureatessince 2003.

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