This
workshop focused on a well-known open problem in holomorphic dynamical systems,
the Bedford Conjecture, which considers the complex structure of stable
manifolds. A classical result states that for an invertible holomorphic map,
the attracting basin of an attractive fixed point is always equivalent to
complex Euclidean space. Whether the same holds for more general stable
manifolds is not known. The goal of this workshop was to bring together many
researchers working on the Bedford conjecture, for the purpose of exchanging thoughts
and starting new collaborations. Solving the main conjecture during this
workshop was not to be expected, instead we aimed at finding new approaches and
related open problems that might be more accessible. Both in terms of
establishing new collaborations and in finding new ways to attack the main
conjecture, the workshop was a tremendous success, and will certainly lead to
several new publications.

The
organization of the workshop was unusual. First of all, there were no talks on
recent research. Instead, we opened each day with a presentation by one of the
participants on an area of research related to the main problem of the
workshop. In each of these talks new open problems were highlighted, which
opened new directions for discussion.

The
opening lecture on Monday was followed up by a long interactive problem
session. Participants were asked to explain their suggestion to a moderator,
who would ask further questions until the suggestion was completely clear, and
then the moderator would write the suggestion on the blackboard. Besides
listing some of the well-known open problems in the field, new questions were
thought up on the spot, often in reaction to problems suggested by others. Some
of these questions could be answered immediately, others led to very
interesting discussions later in the week. After the problem session we split
into smaller groups that were going to attack the different problems.

Every
following day we held a session in which the different groups reported on their
progress. Some problems were solved, some other problems turned out to be too
difficult to approach. Often groups thought they had no progress to report, but
when urged they were able to report on interesting new thoughts. Then groups
would switch to a different problem, or mix with other groups. The atmosphere
at the workshop was very open, and often researchers from one group would be
sharing thoughts with other groups before going back to their own group.

In
our experience problem sessions at conferences usually do not work very well,
so it is worthwhile to analyze why this workshop was such a success. First of
all, it was made very clear to all the participants that this workshop was
aimed at sharing, and that there would be no research talks. Many of the participants
had already visited each other prior to the workshop, and had discussed
possible new approaches to the problem. These discussions had led to the
writing of a survey article, which was posted on the online archive shortly
before the workshop and contained many open problems. As a result, many
participants came prepared to the workshop and knew what to expect. Finally, it
seems to us that the Lorentz Center@Snellius venue
was absolutely perfect and greatly contributed to the open atmosphere at the
workshop.