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SERGIU HART
H: How did you make the decision?
A: I really can’t remember. I know the decision was mine. My parents put a lot
of responsibility on us children. I was all of seventeen at the time, but there was no
overt pressure from my parents. Probably math just attracted me more, although I
was very attracted by Talmudic studies.
At City College, there was a very active group of mathematics students. The
most prominent of the mathematicians on the staff was Emil Post, a famous
logician. He was in the scientific school of Turing and Church—mathematical
logic, computability—which was very much the “in” thing at the time. This was
the late forties. Post was a very interesting character. I took just one course
from him and that was functions of real variables—measure, integration, etc. The
entire course consisted of his assigning exercises and then calling on the students
to present the solutions on the blackboard. It’s called the Moore method—no
lectures, only exercises. It was a very good course. There were also other excellent
teachers there, and there was a very active group of mathematics students. A lot
of socializing went on. There was a table in the cafeteria called the mathematics
table. Between classes we would sit there and have ice cream and—
H: Discuss the topology of bagels?
A: Right, that kind of thing. A lot of chess playing, a lot of math talk. We
ran our own seminars, had a math club. Some very prominent mathematicians
came out of there—Jack Schwartz of Dunford–Schwartz fame, Leon Ehrenpreis,
Alan Shields, Leo Flatto, Martin Davis, D. J. Newman. That was a very intense
experience. From there I went on to graduate work at MIT, where I did a doctorate
in algebraic topology with George Whitehead.
Let me tell you something very moving relating to my thesis. As an under-
graduate, I read a lot of analytic and algebraic number theory. What is fascinating
about number theory is that it uses very deep methods to attack problems that
are in some sense very “natural” and also simple to formulate. A schoolchild
can understand Fermat’s last theorem, but it took extremely deep methods to
prove it. A schoolchild can understand what a prime number is, but understanding
the distribution of prime numbers requires the theory of functions of a complex
variable; it is closely related to the Riemann hypothesis, whose very formulation
requires at least two or three years of university mathematics, and which remains
unproved to this day. Another interesting aspect of number theory was that it was
absolutely useless—pure mathematics at its purest.
In graduate school, I heard George Whitehead’s excellent lectures on algebraic
topology. Whitehead did not talk much about knots, but I had heard about them,
and they fascinated me. Knots are like number theory: the problems are very simple
to formulate, a schoolchild can understand them; and they are very natural, they
have a simplicity and immediacy that is even greater than that of prime numbers or
Fermat’s last theorem. But it is very difficult to prove anything at all about them;
it requires really deep methods of algebraic topology. And, like number theory,
knot theory was totally, totally useless.
So, I was attracted to knots. I went to Whitehead and said, I want to do a PhD
with you, please give me a problem. But not just any problem; please, give me
INTERVIEW WITH ROBERT AUMANN
687
an open problem in knot theory. And he did; he gave me a famous, very difficult
problem—the “asphericity” of knots—that had been open for twenty-five years
and had defied the most concerted attempts to solve.
Though I did not solve that problem, I did solve a special case. The complete
statement of my result is not easy to formulate for a layman, but it does have an
interesting implication that even a schoolchild can understand and that had not
been known before my work: alternating knots do not “come apart,” cannot be
separated.
So, I had accomplished my objective—done something that (i) is the answer to
a “natural” question, (ii) is easy to formulate, (iii) has a deep, difficult proof, and
(iv) is absolutely useless, the purest of pure mathematics.
It was in the fall of 1954 that I got the crucial idea that was the key to proving
my result. The thesis was published in the Annals of Mathematics in 1956 [1];
but the proof was essentially in place in the fall of 1954. Shortly thereafter, my
research interests turned from knot theory to the areas that have occupied me to
this day.
That’s Act I of the story. And now, the curtain rises on Act II—fifty years later,
almost to the day. It’s 10
P
.
M
., and the phone rings in my home. My grandson Yakov
Rosen is on the line. Yakov is in his second year of medical school. “Grandpa,”
he says, “can I pick your brain? We are studying knots. I don’t understand the
material, and think that our lecturer doesn’t understand it either. For example, could
you explain to me what, exactly, are ‘linking numbers’?” “Why are you studying
knots?” I ask “What do knots have to do with medicine?” “Well,” says Yakov,
“sometimes the DNA in a cell gets knotted up. Depending on the characteristics
of the knot, this may lead to cancer. So, we have to understand knots.”
I was completely bowled over. Fifty years later, the “absolutely useless”—the
“purest of the pure”—is taught in the second year of medical school, and my
grandson is studying it. I invited Yakov to come over, and told him about knots,
and linking numbers, and my thesis.
H: This is indeed fascinating. Incidentally, has the “big, famous” problem ever
been solved?
A: Yes. About a year after my thesis was published, a mathematician by the
name of Papakyriakopoulos solved the general problem of asphericity. He had
been working on it for eighteen years. He was at Princeton, but didn’t have a
job there; they gave him some kind of stipend. He sat in the library and worked
away on this for eighteen years! During that whole time he published almost
nothing—a few related papers, a year or two before solving the big problem. Then
he solved this big problem, with an amazingly deep and beautiful proof. And then,
he disappeared from sight, and was never heard from again. He did nothing else.
It’s like these cactuses that flower once in eighteen years. Naturally that swamped
my result; fortunately mine came before his. It swamped it, except for one thing.
Papakyriakopoulos’s result does not imply that alternating knots will not come
apart. What he proved is that a knot that does not come apart is aspheric. What
I proved is that all alternating knots are aspheric. It’s easy to see that a knot
that comes apart is not aspheric, so it follows that an alternating knot will not