Foreword

John Tate

This book contains the notes of the four lecture series of the 2007 Arizona

Winter School on non-archimedean geometry, together with a short account by

Berkovich of how he developed, in diﬃcult circumstances, the remarkable theory of

the spaces which bear his name.

Brian Conrad’s talks are an excellent introduction to the general theory of the

original rigid analytic spaces, of Raynaud’s view of them as the generic fibers of

formal schemes, and Berkovich spaces. The p-adic upper half plane as rigid analytic

space and several applications of it are discussed in the notes of Samit Dasgupta

and Jeremy Teitelbaum. Matt Baker’s lectures offer a detailed discussion of the

Berkovich projective line and p-adic potential theory on that and more general

Berkovich curves. Finally, Kiran Kedlaya discusses cohomologies, deRham and

rigid, the comparison between them, and how to compute them.

In this foreword I want to do two things: (1) tell more than is in the brief

preface about the institution which gave rise to this book, the AWS, for I think it’s

a great concept which would be good to try in other subjects and places, and (2)

give a bit of ancient history of the book’s subject, p-adic geometry, especially the

part I was involved in.

It’s hard to describe the AWS adequately. “Intense”, “interactive”, “infectious

enthusiasm”, “flexibility in support” give some idea of the flavor. You have to

experience it to believe it can work so well. The unique and novel feature of the

AWS is that after the usual set of lectures during the day that all such schools

have, there are working sessions until late in the night in which students work with

the help of the speakers and their assistants on projects related to the lectures. In

fact, even the lecture series part of the school is not so usual, because the speakers

produce a complete set of notes for their lectures a month ahead of time, with

background references, so that the participants can be better prepared (this book

is essentially the notes for the 2007 school). The speakers also describe ahead of time

the projects which their group of students will work on at the school. These projects

can involve elaborations of the theory, working out special examples, computing

some quantity in various cases, writing computer code to do that, background

work on foundational aspects... anything to challenge the participants and help

them learn by working with the material. The projects associated to the lectures

in this book, and also the lectures and projects from several earlier Winter Schools,

can be found on the Arizona Winter School website. On the last day of the school

students from each group make a joint presentation of the results the group has

obtained.

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