or study secular subjects at a university. For a while I did both. I used to

get up in the morning at 6:15, go to the university in uptown New York

from Brooklyn”an hour and a quarter on the subway”then study calcu-

lus for an hour, then go back to the yeshiva on the lower east side for

most of the morning, then go back up to City College at 139th Street

and study there until 10 p.m., then go home and do some homework or

whatever, and then I would get up again at 6:15. I did this for one

semester, and then it became too much for me and I made the hard

decision to quit the yeshiva and study mathematics.

Hart: How did you make the decision?

Aumann: 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 17 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 scienti¬c 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 char-

acter. I took just one course from him and that was functions of real

variables”measure, integration, et cetera. 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

330 Sergiu Hart

mathematics table. Between classes we would sit there and have ice

cream and”

Hart: Discuss the topology of bagels?

Aumann: 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 prom-

inent 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 undergraduate, 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 sim-

ple to formulate. A schoolchild can understand Fermat™s last theorem,

but it took extremely deep methods to prove it. A schoolchild can under-

stand 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 dif¬cult 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 Ph.D. with you, please give me a problem. But not just any problem;

please, give me an open problem in knot theory.” And he did; he gave

me a famous, very dif¬cult problem”the “asphericity” of knots”that

had been open for 25 years and had de¬ed 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.

An Interview with Robert Aumann 331

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,

dif¬cult 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 [Aumann (1956)]; 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”50

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 char-

acteristics 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 med-

ical school, and my grandson is studying it. I invited Yakov to come over,

and told him about knots, and linking numbers, and my thesis.

Hart: This is indeed fascinating. Incidentally, has the “big, famous”

problem ever been solved?

Aumann: Yes. About a year after my thesis was published, a mathem-

atician by the name of Papakyriakopoulos solved the general problem of

asphericity. He had been working on it for 18 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 18 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 ¬‚ower once in 18 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 come apart. So that aspect of my

thesis”which is the easily formulated part”did survive.

332 Sergiu Hart

A little later, but independently, Dick Crowell also proved that altern-

ating knots do not come apart, using a totally different method, not

related to asphericity.

Hart: Okay, now that we are all tied up in knots, let™s untangle them and

go on. You did your Ph.D. at MIT in algebraic topology, and then what?

Aumann: Then for my postdoc, I joined an operations research

group at Princeton. This was a rather sharp turn because algebraic topo-

logy is just about the purest of pure mathematics and operations research

is very applied. It was a small group of about 10 people at the Forrestal

Research Center, which is attached to Princeton University.

Hart: In those days operations research and game theory were quite

connected. I guess that™s how you”

Aumann: ”became interested in game theory, exactly. There was

a problem about defending a city from a squadron of aircraft most of

which are decoys”do not carry any weapons”but a small percentage do

carry nuclear weapons. The project was sponsored by Bell Labs, who

were developing a defense missile.

At MIT I had met John Nash, who came there in ™53 after doing his

doctorate at Princeton. I was a senior graduate student and he was

a Moore instructor, which was a prestigious instructorship for young

mathematicians. So he was a little older than me, scienti¬cally and also

chronologically. We got to know each other fairly well and I heard from

him about game theory. One of the problems that we kicked around was

that of dueling”silent duels, noisy duels, and so on. So when I came to

Princeton, although I didn™t know much about game theory at all, I had

heard about it; and when we were given this problem by Bell Labs, I was

able to say, “This sounds a little bit like what Nash was telling us; let™s

examine it from that point of view.” So I started studying game theory;