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
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