Moment maps and surjectivity in various geometries

August 9 to August 13, 2004

at the

American Institute of Mathematics, Palo Alto, California

organized by

Tara Holm, Eugene Lerman, and Susan Tolman

This workshop, sponsored by AIM and the NSF, will be devoted to the question of surjectivity of the Kirwan map for quotients in contact, hyper-Kaehler, and 3-Sasakian geometries.

If a group G acts on a manifold M with a moment map (symplectic, contact, hyper-Kaehler, 3-Sasakian) and the action on the zero level set Z of the moment map is free, there is a well defined map from the G-equivariant cohomology of M to the ordinary cohomology of the quotient Z/G. If the manifold is compact and symplectic, then by a theorem of Kirwan, the map in question ("the Kirwan map") is surjective.

In the past thirty years, tremendous progress has been made in the study of moment maps, symplectic quotients, and the question of surjectivity. In recent years, similar questions have arisen in fields other than symplectic geometry: contact, hyper-Kaehler, and 3-Sasakian geometries. While some headway has been made in understanding moment maps and surjectivity in these fields, there remain many open questions. We wish to explore phenomena that are well understood in symplectic geometry but are more puzzling in these new settings.

Thus, the goal of this workshop is to obtain a better understanding of moment maps and the question of surjectivity. Well understood in the context of symplectic geometry, these have many applications. For example these results give an explicit description of the cohomology ring of moduli spaces of stable bundles over Riemann surfaces. In closely related fields, including contact, hyper-Kaehler, and 3-Sasakian geometries, the theory of moment maps and the question of surjectivity are not as well studied. We want to determine what techniques developed in symplectic geometry for understanding the topology of symplectic quotients, specifically, their cohomology rings, carry over to the settings of hyper-Kaehler, contact and 3-Sasakian geometries.

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 working sessions.

Participants include J. Agapito Ruiz, C. Boyer, D. Burns, C. Chan, G. Daskalopoulos, B. Goetz, R. Goldin, M. Harada, T. Hausel, N. Ho, T. Holm, L. Jeffrey, H. Konno, G. Landweber, E. Lerman, H. Li, M. Libine, Y. Lin, L. Mare, J. Munn, M. Pflaum, N. Proudfoot, R. Sjamaar, J. Talvacchia, S. Tolman, A. Wade, J. Weitsman, G. Wilkin, S. Wu, and C. Zara.

The deadline to apply for support to participate in this workshop has passed.

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