January 10-18, 2016

We will study the role of Higgs bundles, Anosov representations, Hermitian symmetric spaces and harmonic maps in maximal $\mathsf{Sp}(4,\mathbb{R})$ representations. After preliminary talks on symmetric spaces, Anosov representations and Higgs bundles, further topics will include: boundaries of symmetric spaces for groups of Hermitian type, domains of discontinuity, and Hitchin representations. Subsequent talks will be based on current developments in the theory. In particular, Geometric structures associated to $\mathsf{Sp}(4,\mathbb{R})$-Hitchin representations, Zariski closures of maximal $\mathsf{Sp}(4, \mathbb{R})$-representations, the Higgs bundle parameterization and connected component count of maximal $\mathsf{Sp}(4,\mathbb{R})$-representations and the existence of minimal surfaces associated to maximal $\mathsf{Sp}(4,\mathbb{R})$-representations.

The overarching theme may be summarized as follows: how can the tools of Anosov representations, Higgs bundles, Hermitian symmetric spaces and harmonic maps be used together to understand questions concerning surface group representations and the geometric structures associated to them. We hope to elucidate connections between the above techniques through a detailed of the specific case of $\mathsf{Sp}(4,\mathbb{R})$.

This workshop is inspired by similar successful workshops:

The workshop will consist of whiteboard talks by the participants on the following topics. Speakers will be alotted 2.5 hours per talk. Speakers will be asked to submit a 5-6 page summary, clicking on the each title below will lead you to that talks summary.

In this talk, the speaker will introduce the background material from Lie theory that will be required for the following talks. In particular, after presenting some decompositions of the semi-simple group G, the speaker will introduce (simple) roots, Weyl group, and Weyl chambers, parabolic subgroups. Then symmetric spaces will be introduced, as well as their boundaries (visual, Tits, Furstenberg). Of course, all the definition will be made concrete with examples, and with focus on the case of $\mathsf{Sp}(4,\mathbb{R})$.

In this talk, the speaker will introduce the notion of convex-cocompact actions on real rank one symmetric spaces. This talk will be an introduction to the following two talks about Anosov representations since these coincide with convex-cocompact actions in the real rank one case. Exemples coming from hyper bolic geometry (in dimension 2 and 3) and from complex hyperbolic geometry will be presenated.

The speaker will recall general definitions for harmonic maps with Riemann surface domain, sketch of Wolf's harmonic map parameterization of Teichmuller space by holomorphic quadratic differentials and generalizations to higher rank, namely Corlette's theorem. The Energy functional on Teichmüller space will be defined and the fact that critical points are weakly conformal maps (equivalently branched minimal immersions) will be discussed.

The speaker will definition of Higgs bundles for real groups with special attention to the groups $\mathsf{Sp}(4,\mathbb{R})$ and maximal $\mathsf{SL}(2,\mathbb{R})$. A general sketch of nonabelian Hodge correspondence and Hitchin's construction of the Hitchin component (Hitchin fibration and Hitchin section) will be given with the explicit construction for $\mathsf{Sp}(4,\mathbb{R})$.

In this talk the speaker will introduce the notion of Anosov representations, following the recent characterizations in terms of a Cartan projection of F. Gueritaud, O. Guichard, F. Kassel, A. Wienhard. In addition, the speaker will present the construction of domain of discontinuity of O. Guichard and A. Wienhard. If time permits, the talk can finish with an overview of the applications to proper actions on homogeneous spaces.

In this talk the speaker will introduce the notion of Anosov representations, following the recent characterizations of M. Kapovich, B. Leeb, J. Porti. This approach looks at Anosov actions on the symmetric spaces, and so it generalises the well-known theory of Kleinian groups. The speaker is invited to follow the notes from the minicourse given by Kapovich and Leeb at MSRI.

The speaker will explain how to identify the deformation space of convex foliated $\mathbb{RP}^3$ structures on the unit tangent bundle of the surface to the $\mathsf{SL}(4,\mathbb{R})$ Hitchin component. Special attention should be given to the additional structure induced when one specializes to the case of $\mathsf{Sp}(4,\mathbb{R})$ Hitchin representations instead. For $\mathsf{Sp}(4,\mathbb{R}),$ the ideas of the Higgs bundle approach of Baraglia should be sketched.

The speaker will explain what are maximal representations, and focus on examples pertaining to the case of $\mathsf{Sp}(4,\mathbb{R})$. In particular the possible Zariski closures of a maximal $\mathsf{Sp}(4,\mathbb{R})$ should be discussed.

The speaker will describe the connected component count of maximal $\mathsf{Sp}(4,\mathbb{R})$ representations and describe the Higgs bundle parmameterization of these components and possible Zariski closures of representations in each component. Special attention should be put on the $2g-3$ smooth components which contain only Zariski dense representations.

Following the paper Topological invariants of Anosov representations, the speaker should describe how one associates invariants to connected components of Anosov representations with focus on the case of $\mathsf{Sp}(4,\mathbb{R}).$ Also, the model representations (hybrid representations) for the $2g-3$ smooth components which contain only Zariski dense representations should be described.

Discuss Labourie's conjectured mapping class group invariant parameterization for the Hitchin component and explain how it is equivalent to the existence of a unique conformal structure in which the harmonic map to the symmetric space is a minimal immersion. Sketch how the properness of the mapping class group action and properness of the energy functional follow from maximal representations being well displacing and imply the existence of such a conformal structure for all maximal representations. Briefly describe Labourie's proof of uniqueness for $\mathsf{Sp}(4,\mathbb{R})$.

Explain the low dimensional isomorphism $\mathsf{PSp}(4,\mathbb{R})\cong \mathsf{SO}_0(2,3)$ and discuss the Lorentzian geometry on the flag manifolds of the form $\mathsf{Sp}(4,\mathbb{R})/\mathsf{P}$, for a maximal parabolic subgroup $\mathsf{P}\subset\mathsf{Sp}(4,\mathbb{R})$ .

Organizers: Brian Collier, Sara Maloni, Tengren Zhang

#### Location:

The workshop will take place January 10-18, 2016 in a large cabin in the mountains of Colorado near Granby.

#### Transportation:

Fly into Denver airport then take shuttle to Snow Mountain YMCA. The shuttle will drop you off at the reception where you should say you are with the math conference.

#### Funding:

Full funding for the workshop is provided by the GEAR network. Invited participants will be reimbursed for travel (see your email).