I happen to have seen this post only a couple of hours after stumbling on a recollection by Bill Howard, from an article in the Mathematical Intelligencer written by Howard's student Amy Shell-Gellasch. (Howard went on to a quite distinguished career and is a fellow of the AMS.)

By the summer of 1951, I had been at Chicago for two
years, and it was time that I got to work on a thesis. That
summer Weil lived in a little monastic cell in International
House, while his family went to France; so I encountered him
almost daily at lunch or supper. One day he put down his tray
and said, totally out of the blue, that he had a thesis problem
for me. I suspect that there had been a faculty meeting, and
they thought it was time Bill Howard was put to work.

This was a problem of Elie Cartan. Given one complex
variable on a bounded domain, the group of one-to-one analytic mappings of that domain onto itself is a finite-dimensional Lie group. This group of all isometries in the
space of constant negative curvature is the Poincare group.
The elder Cartan generalized this to two and three complex
variables. Namely, he showed that the underlying metric is
"symmetric" in the sense that locally, reflection through a
point by means of geodesics is an isometry. Weil said I
should try to prove the result for the general case of n complex variables. And moreover, try to get a simple general
proof for the case of two and three complex variables, so
you wouldn't have to look at it case by case.

The way he told it to me, I assumed that Cartan had
thought the theorem was true, and I think Weil thought the
theorem to be true in general for n dimensions. I spent several months just acquiring the necessary background. I had
to learn the theory of several complex variables, Lie groups,
Riemannian geometry; it was too much. I was out of my
depth but didn't know it. I had a good teacher for the geometry, Professor Chern.

By the spring of 1952 I thought I had solved the problem. I told Weil my proof and he thought it was okay. I presented it in a seminar; several high-powered Visiting mathematicians were there. They thought what I had done was
reasonable; they couldn't check every detail in seminar, but
they thought I was using the proper methods. Weil told me
to write it up and submit it with a proof of Lemma 2, which
lie was sure was true.

I went home, and as I was falling asleep, I was thinking of how I would write out the proof of Lemma 2. Then
I realized that I couldn't prove Lemma 2. Then I noticed
that there was a counterexample. Lemma 2 was false, so
the whole thing imploded. I remember feeling that the bottom had physically fallen out of the bed. That was a real
sinking feeling. The next day, I told Weil. He said, "What
do you mean Lemma 2 is false!" I showed him my counterexample. He stormed out of the office. Ten minutes
later he stormed back in and gave me his own counterexample.

He saw that I didn't look very happy, so he took me for
a walk in Jackson Park. We walked along in silence for
about ten minutes, and then he said, "Well, you've got
enough results, bits and pieces, it's not what we had hoped.
But you have enough there for a Ph.D. thesis, just write the
bits and pieces up."

I guess he must have been serious, thinking about what
I had already done, and there was enough there. I said I
wanted it all or nothing. No bits and pieces for me.

We walked along in silence a little longer, and he said,
"You have experience climbing mountains. You must know
that you don't have to go to the top, to the peak. You can
enjoy yourself going partway up. Just play around, you can
see the other people going up."

You can imagine what I felt like at that point. He was
trying to make it better. He was trying to put himself in my
place, and he realized that he wouldn't have accepted it either. So what was coming out was sounding kind of strange.

Suddenly out of the blue, he said, "Have you ever thought
of becoming an administrator?" That was a real inspiration
to him. I told him that knowing what he thought about administrators, I considered that an insult. He said, "Take Walter Bartky" (who was the Dean of Physical Sciences). "Of
course his mathematics is weak. But he's done some very
worthwhile things."

I asked what he had done that was worthwhile. And Weil
responded, "Well, he brought me here!"

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