Princeton graduate school in 1956.

That was when I started graduate study in mathematics.

The math department had invented the grade of "N", which meant "No grade is given in this course". We had to take courses, but we didn't have to attend or understand anything. I understood Ralph Fox's courses a bit, and that is why he became my advisor.

The Graduate College at Princeton was a dormitory for graduate students, and it had a nice antique appearance. At dinner time, we would put on our black robes and go to the Great Hall, listen to the Master's invocation or benediction, and then eat, socially. We math students used to eat together and discuss things like Spectral Sequences, trying to impress each other.

One of that group of graduate students was Barry Mazur. He was especially impressive since he had gotten into graduate school, as a teenager, without managing to get a bachelor's degree.

After a while, I began to have silly math ideas, trivialities that became my thesis after the obligatory three years of hanging around Fine Hall. Fox did a course in which he went through parts of Kurosh's book on Group Theory, word by word, symbol by symbol. The bit about Grushko's theorem was completely incomprehensible; but somehow, I meditated on it and came up with a topological idea for proving it.

But Barry disappeared for several months and then came back to Princeton. He had been to France. When he returned, he had a simple little theorem.

Adrien Douady, a French student who obeyed all the rules, was visiting from France. The rule was that one had to wear the black gown to dinner. Now, I did not check this out in detail, but it appeared that when he went to dinner, he wore his black robe. With nothing else underneath it. Douady had a room with a blackboard in it, and Barry took me up there and explained to me his great result.

It was marvellous! It was totally trivial; well, except for a minor detail or two. And it was new! And interesting. Let me explain it like this:

1 = 1 + 0 + 0 + ...

= 1 + (-1+1) + (-1+1) + ...

= (1-1) + (1-1) + ...

= 0 + 0 + 0 + ...

= 0.

Now, this might not be absolutely true in all cases. But there are situations in algebra and in topology where it makes sense. Barry applied this to a situation where "+" meant connected sum, and "..." was a sort of one-point compactification.

I'm sure you all know what I'm referring to. The theorem about how a smoothly embedded (n-1)-sphere in the n-sphere is topologically unique.

There are some lessons to be learned from this story. One that appeals to me is that Barry broke the rules; he didn't have a Bachelor's degree, he didn't stay quietly in Princeton for three years. Even Douady broke the unspoken rule that one is supposed to wear clothes under the gown. All those rules, BA diploma, residence in Princeton, wearing shoes, being nice to chairmen and deans, being a team player --- basically I hate those rules, even though I give in and adjust to them. It makes me feel good to see a rule-breaker succeed!

And, teenagers are creative and amazing. Barry's theorem could be compared to Opus One of some great composer. For instance, Gustav Mahler, at the age of 17, wrote a long poem, and set it to music by the age of 19. This cantata, "Das klagende Lied", is rarely performed in its original form, because Mahler "improved" it later on. I recently heard the original work in Berkeley, played by the Berkeley Symphony Orchestra under Kent Nagano, with a large chorus, six vocal soloists, two of which were from the San Francisco Girls Choir, and an offstage band from Hayward State University that played in a different key and time-signature. It had its adolescent moments, but it is truly a great work of art. Just as Barry's sphere theorem was.

Perhaps the most important lesson of the story is that simple and easy mathematical ideas exist. They can be discovered, they are new, they are often important. I am a very lazy person. I have never been able to read thousands of pages of mathematics, or attend talk after talk for weeks on end, understanding every word and building on them myself; I respect people who are not lazy like I am, but I don't have that kind of ability. Barry's simple, trivial, great result gave me hope.

Now, I have recently had a trivial idea, related to my thesis, and I will tell you all about it...