【续上部。摘录自可轻松阅读的部分，省略了所有数学问题的推导细节】

infinity and Limits (1991)

With mortality on our minds, maybe we were both seeking refuge in the one place where infinity becomes real.

Chaos (1992-95)

At present your research does not constitute adquately deep mathematics and, as a result, your present prospects for tenure at MIT are low. Of course, it is possible that future outstanding research work could change this assessment. However, on the basis of your work to-date it is necessary to advise you that the tenure outlook at this time is not positive.

The Path of Quickest Descent (2000-2003)

The next letter shows how hard it was for Joff to let go of his earlier life. He recounts how he buttonholed a good-natured waitress and taught her about the path of quickest descent, a classic problem in the calculus of variations.

Bifurcation (2004)

The contrast here is meant to striking: the conditions (the parameters) chage continously, yet the resulting behavior changes discontinuously. You gradually turn up the heat and nothing happens until you reach the bifurcation point. Then the pot begins to boil.

I didn’t write back to Joff about his stroke. Or call him either.

Maybe it was a kind of exhaustion. My dad had died a few months earlier, in October 2003.

But eventually I did contact him. It took something else to precipitate it.

In April 2004 my brother Ian, 57 years old, died suddenly one night.

【part of note of condolence】 In my dreams (NIGHTMARES?) I struggle to find my classroom as the time for the start bell to ring approaches! X@%X!

He was nervous about seeing me. Unbelievable. I was nervous about seeing him. What was he nervous about?

Hero’s Formula (2005-Present)

While writing this book, I’ve been thinking a lot about what I learned from Joff. For years I would have said, not much, meaning not much math. That was true even in high school. But I’m starting to realize what it was that he gave me.

He let me teach him.

Before I had any students, he was my student.

Somehow he knew that’s what I needed most. And he let me, and encouraged me, and helpd me. Like all great teachers do.

But now I also see that I did learn something from him, something profoundly mathematical, about how to live. From his hobbies to the way he faces the ups and downs in his life, Joff is brave about change.

These old letters make me smile. Not so much because of the math they contain but because of the Joff I see once again on the pages – his cartoon drawings, his self-flagellation and outbursts of happy exasperation, and his pure pleasure in the logic of the argument.

【摘录完】