x FOREWORD

Another key feature of the AWS is that the participants come from a wide

variety of programs from all over the country, with very different levels of prepa-

ration, pressure and expectations. The organizers try to preserve this diversity in

each of the four working groups. Typically a group will have an advanced student

working in an area close to the group’s project, it will have students with a very

strong, but also not so strong background, it will have students from many dif-

ferent universities, some elite, some not, it will have students of both sexes, and

at least one student from the southwest, the region represented by the organizers.

Preferences for projects expressed by students on their applications are taken into

account, but not all can be satisfied. In fact, the AWS has become so popular that

not all students are oﬃcially assigned to groups. But all are encouraged to hang

out and informally participate anyway. The whole scheme is flexible.

The interactions between students from different backgrounds that the AWS

makes possible has a positive impact on all attending. For all students it is an

opportunity to feel the exhilaration, drive and power that the students from the

top programs show. On the other hand, the very nature of the AWS is to create

an environment where students with lower level backgrounds are encouraged to

ask basic questions. Discussion of these can be very valuable to all; it may reveal

that what seemed clear was not quite so clear after all. The cross-pollination of

mathematical cultures which takes place at the AWS is of benefit to everybody.

Ultimately participants get a clear picture of what it means to do mathematics in

the real world and this can be a significant learning experience for students and

postdocs of any background.

Almost all of the participants are housed in the same hotel, and the evening

sessions are held there too. There is no escape. Everyone is constantly involved in

small and large discussion groups, on the lectures, on the project topics and toward

the joint presentations to be given on the last day.

The school is five days of very hard work — 16 hour days — for all participants.

In spite of that, or perhaps just because of the intellectual intensity, most seem to

thrive. As an informal participant in several of the schools I am aware of many

testimonials, oral and written, that it is a rewarding experience, both scientifically

and interpersonally. The topic changes every year and the frequency with which

many students return year after year is another indication of the value of the school.

Work at the school has been the germ of many Ph.D. theses.

The first Winter School was held in 1998, and it has been going strong since

then. The organizers (which by now include a few ex-students from early schools)

deserve great credit, for the original conception, for improving it in various ways

over the years, and for the mostly excellent choices of topics and speakers they have

made each year. The 2007 lectures were outstanding and unusually closely related.

Hence the idea to collect them in a book.

Now I would like to switch gears and discuss some old (pre-1965) history.

The basic problem in creating a global theory of analytic manifolds over a non-

archimedean local field K is that analytic continuation in the usual sense does not

work. We can agree that a function in a “closed” disc |x − a| ≤ r (which is also

open), or in an “open” disc |x− a| r (which is also closed), is analytic if and only

if it has a power series expansion convergent in the disc, and, in fact, this turns out

ultimately to be the right idea. But this is a much stricter condition than to have

a power series expansion in a neighborhood of every point of the disc, because in