at the

American Institute of Mathematics, Palo Alto, California

organized by

Steven Bradlow, Oscar Garcia-Prada, William M. Goldman, and Anna Wienhard

This workshop, sponsored by AIM and the NSF, will bring together researchers studying representations of fundamental groups of Riemann surfaces into real semsimple Lie groups. Such representations form multi-component algebraic sets. Recent progress in understanding these components has come from quite different approaches:

- Higgs bundle techniques, which combine (twisted) harmonic maps into symmetric spaces and holomorphic vector bundles; Morse theory methods on the moduli spaces of Higgs bundles have led to the calculation of the cohomology of these components;
- The bounded cohomological, the dynamical and the algebro-geometric approach relate the geometry of flag varieties at the boundary of symmetric spaces to the dynamics of surface group actions. This has led to a geometric understanding of surface group actions in certain components of the moduli space of representations.

- split $\mathbb{R}$-forms of semisimple groups;
- automorphism groups of Hermitian symmetric spaces.

A central focus of this workshop will be the real symplectic group $Sp(2n,\mathbb{R})$, which is the unique $\mathbb{R}$-split Lie group of Hermitian type. Specific topics to be addressed include:

- Relation of the different Lagrangian subbundles constructed using the harmonic map (in the Higgs bundle approach) and the boundary map;
- Coordinates and parametrizations of the Hitchin component and of the space of maximal representations.
- Structures on the moduli space: symplectic, holomorphic, Kahler;
- Mapping class group action and the energy function on Teichmuller space.

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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