Dynamics of the Weil-Petersson geodesic flow

This workshop, sponsored by AIM and the NSF, will be devoted to the study of recent advances in understanding deformations of Riemann surfaces via the Weil-Petersson metric on Teichmuller and moduli spaces. The works of Brock, Liu-Sun-Yau, McMullen, Mirzakhani, and Wolpert have opened new exploration on two fronts: establishing novel connections to topics in geometry, topology and dynamics; and understanding fine properties of the metric itself, such as comparison estimates and curvature expansions. Further advances have come from developing the relation of the metric to problems in 2 and 3-dimensional hyperbolic geometry and in understanding the dynamics of the Weil-Petersson geodesic flow.

This work has brought together a variety of ideas from disparate areas of mathematics. One of the main goals of this workshop will be to bring researchers from different cultures together to study problems of common interest. A secondary goal is to explore the possibility that ideas from classical Teichmuller theory might be relevant in the study of representations of surface groups into higher rank Lie groups, the so-called "higher Teichmuller theory". The major topics of the workshop will be:

1. higher statistical properties of the Weil-Petersson flow such as rates of mixing, central limit theorem, and invariant measures;
2. the relationship of the Weil-Petersson metric to hyperbolic 3-manifolds and ending laminations;
3. possible connections of the Weil-Petersson and Teichmuller metrics to higher Teichmuller theory.

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.

For more information email workshops@aimath.org

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