My father loved teaching as much as I do, and he taught me many things: sometimes mathematics, but also the names of wild flowers. We played music and examined pond water. If there was a direction in which he pointed me, it was toward chemistry. He never suggested that I should follow in his footsteps, and I never made a conscious decision to become a mathematician.

I originally thought I might do topology. When I was an undergraduate, my advisor was Ralph Fox, who was a topologist, and he got me doing things. In Princeton - in those days, at least - you had to write a junior paper, and a senior paper. I did what was thought to be original research in my senior thesis, but I never published it. I think Ralph mentions the result in one of his books, but I didn't think it was good enough. Biology was my other field when I was an undergraduate. But in those days, mathematical biology was zero. ... But now its a fascinating subject. Maybe I chose the wrong field.

Those were exciting times for algebraic geometry. The crowning achievement of the Italian school, the classification of algebraic surfaces, was just entering the mainstream of mathematics. The sheaf theoretic methods introduced by Jean-Pierre Serre were being absorbed, and Grothendieck's language of schemes was being developed. Zariski's dynamic personality, and the explosion of activity in the field, persuaded me to work there. I became his student along with Peter Falb, Heisuke Hironaka, and David Mumford.

Michael Artin has helped to weave the fabric of modern algebraic geometry. His notion of an algebraic space extends Grothendieck's notion of scheme. The point of the extension is that Artin's theorem on approximating formal power series solutions allows one to show that many moduli spaces are actually algebraic spaces and so can be studied by the methods of algebraic geometry. He showed also how to apply the same ideas to the algebraic stacks of Deligne and Mumford. Algebraic stacks and algebraic spaces appear everywhere in modern algebraic geometry, and Artin's methods are used constantly in studying them. He has contributed spectacular results in classical algebraic geometry, such as his resolution (with Swinnerton-Dyer in 1973) of the Shafarevich-Tate conjecture for elliptic K3 surfaces. With Mazur, he applied ideas from algebraic geometry (and the Nash approximation theorem) to the study of diffeomorphisms of compact manifolds having periodic points of a specified behaviour.

Since André Weil pointed out the need for invariants, analogous to topological ones, of varieties over fields of characteristic p, several proposals to define such invariants have been made, notably by Jean-Pierre Serre, Alexander Grothendieck, and Paul Monsky and Gerard Washnitzer. I would like to describe some of the recent work on one of these approaches, that of the 'etale cohomology' of Grothendieck.

My interest in noncommutative algebra began with a talk by Shimshon Amitsur and a visit to Chicago, where I met Claudio Procesi and Lance Small. They prompted my first foray into ring theory, and in subsequent years noncommutative algebra gradually attracted more of my attention. I changed fields for good in the mid-1980s, when Bill Schelter and I did experimental work on quantum planes using his algebra package, 'Affine'.

I didn't start out to write a textbook at all. I just wanted to teach a class, and to do some topics that weren't traditional. So I handed out notes for those, and eventually I started using them instead of the textbook. And then I revised them every year.

This is a remarkable text designed for highly motivated undergraduates having some preparation in linear algebra and some other post-calculus mathematics. It is noteworthy for its contents and the style of presentation. In the preface, the author lists three principles that he followed (briefly: examples should motivate definitions, technical points are presented only if needed later in the book, topics should be important for the average mathematician) and takes pains to point out that "Do it the way you were taught'' is not one of them. The style throughout the text is to present basic concepts, give many nontrivial examples and present brief and understandable discussions of advanced material.

I cannot imagine bothering to search a book so deliberately and thoroughly written to make the reader ask and answer their own questions.

It's not easy for the traditional algebraists to use it as a textbook, because it has other stuff in it.

... there's the volunteer side, the president and the council and committees; and the executive director's side, which runs the Providence and Washington offices and does things like arrange meetings and publish journals. They're both very complicated. And often at the council meeting, there are political issues brought up, which concern the Mathematical Society only very peripherally, and which are very divisive.

... an architect of the modern approach to algebraic geometry.