American Institute of Mathematics, Palo Alto, California
Ilijas Farah and David Kerr
This workshop, sponsored by AIM and the NSF, will be devoted to applications of set theory to C*-algebras. It would be hard to imagine a workshop on set theory and C*-algebras as recently as ten years ago, and this meeting will bring together researchers in these two fields. Dramatic progress has brought set theory and operator algebras closer together over the last decade, and this meeting will invigorate the collaboration between the two subjects.
The workshop will concentrate on the following topics:
- Set-theoretic analysis of Elliott's classification program for nuclear C*-algebras. This would be a continuation of the broad endeavor to study the comparative complexity of classification problems prominent in descriptive set theory in the last twenty years. Progress in C*-algebra theory over the last decade has brought into focus the role of regularity properties like Z-stability as a means for understanding the effectiveness and limitations of K-theoretic invariants for the purpose of classification. One outstanding problem is to determine the jump in Borel complexity at the boundary of this effectiveness. While descriptive techniques have been successfully applied to classification problems in ergodic theory, most notably via Hjorth's notion of turbulence, and many nuclear C*-algebras arise from dynamical systems as crossed products, the fact that some dynamical information gets lost in the crossed product construction in an often mysteriously complicated way suggests that the analysis of Borel complexity in context of C*-algebras might require the development of new tools and lead to new insights in the application of descriptive set theory to classification problems.
-Potential applications of combinatorial set theory to C*-algebras, such as the question whether Naimark's problem can have a positive answer in some model of set theory or whether the Calkin algebra admits a K1-reversing automorphism. Negative answers to both of these questions are known to be relatively consistent with ZFC.
The workshop will differ from typical conferences in some regards. Participants will be invited to suggest open problems and questions before the workshop begins, and these will be posted on the workshop website. These include specific problems on which there is hope of making some progress during the workshop, as well as more ambitious problems which may influence the future activity of the field. Lectures at the workshop will be focused on familiarizing the participants with the background material leading up to specific problems, and the schedule will include discussion and parallel working sessions.
The deadline to apply for support to participate in this workshop has passed.
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