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The Complex Structure of Attracting Sets
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.