30 Years later…

Looking back over these lectures, given at Duke University in 1979, I can say with some pride that they contain early hints of a number of important themes in modern arithmetic geometry. Of course, the flip side of that coin is that they are now, thirty years later, seriously out of date. To bring them up to date would involve writing several more monographs, a task best left to mathematicians thirty years younger than me. What I propose instead is to comment fairly briefly on several of the lectures in an attempt to put the reader in touch with what I believe are the most important modern ideas in these areas. The section on motives just below is intended as a brief introduction to the modern viewpoint on that subject. The remaining sections until the last follow roughly the content of the original book, though the titles have changed slightly to reflect my current emphasis. The last section, motives in physics, represents my recent research.

In the original volume I included a quote from Charlie Chan, the great Chinese detective, who told his bumbling number one son “answer simple, but question very very hard.” It seemed to me an appropriate comment on the subject of algebraic cycles. Given the amazing deep new ideas introduced into the subject in recent years, however, I think now that the question remains very very hard, but the answer is perhaps no longer so simple…

At the end of this essay I include a brief bibliography, which is by no means complete.