What I would eventually like to do in this talk is describe some recent, if fragmentary, results on intertwining numbers. But given this rather special occasion, I thought I'd indulge in a bit of reminiscence and at the same time trace some of the twine that connects my present work with the spirit of the work that Maurice and I did so many years ago.

There are many bonds that interlace all of us here today. Of course you know that Maurice spent a good part of his middle and late life on representation theory of Artin algebras, but you may not realize that Emil Artin played a fundamental role in the very early mathematical lives of Maurice and me. When Maurice and I finished our theses at Columbia in 1953, Maurice went to Chicago and I went to Princeton. At that point, Maurice was still very much interested in group cohomology (he hated categories!), and I was puzzling over the implications of homological algebra to commutative ring theory, a puzzlement brought on by the Cartan-Eilenberg proof of the Hilbert Syzygy Theorem.

Although I had been invited to Princeton largely through the efforts of Steenrod, Emil Artin was kind enough to take a very young and ignorant new instructor seriously. I can't remember when in the 1953–54 academic year the topic came up, but it was in conversations with Artin that I first became aware of the open question: If R is regular, and P a prime ideal, is Rp regular?