Suggested plan: Define Teichmüller space in terms of both complex and hyperbolic structures and describe their tangent and cotangent spaces in terms of Beltrami differentials and holomorphic quadratic differentials, respectively. Introduce the Teichmüller distance, the complex structure (briefly), and the Weil-Petersson metric on Teichmüller space, and briefly mention how the last two structures relate to the Goldman symplectic form. Define the Thurston asymmetric distance, both in terms of length spectra and Lipschitz maps. The rest of the time can be dedicated to the description of some of the properties of these metrics, e.g. the Weil-Petersson metric is not complete, it is negatively curved and admits a combinatorial description via the Pants Graph; why the Thurston distance is asymmetric, and which analogies/differences with the Teichmüller distance can be observed (completeness, description via hyperbolic vs extremal lengths, etc.).

Suggested plan: Briefly introduce the concept of higher Teichmüller spaces and define the Hitchin component for PSL(n, R). Define convex projective structures on surfaces and hyperbolic affine spheres in R^3, and explain the equivalence between these two classes of objects and PSL(3, R)-Hitchin representations. Then present the paper of Qiongling Li in which she constructs a Riemannian metric on the space of convex projective structures, and prove that this metric, as well as the metric of Darvishzadeh-Goldman, restricts to be the Weil-Petersson metric on the Teichmüller locus.

Suggested plan: Introduce the non-abelian Hodge correspondence as well as the existence theory for harmonic maps and minimal surfaces (without saying much about the proofs). Then explain Labourie’s mapping class group invariant parametrization of the rank 2 Hitchin components as holomorphic vector bundles over Teichmüller space (without going into details about the uniqueness of minimal surfaces). In the remainder of the talk, present the construction of the Kähler metrics on the rank 2 Hitchin components due to Kim-Zhang for PSL(3, R) and Labourie in general.

Suggested plan: First introduce the Atiyah-Bott-Goldman symplectic form on the moduli space of local systems (or, equivalently, the moduli space of representations). Then sketch the construction of the hyperKähler metric on the moduli space of SL(n, C)-Higgs bundles, which goes through symplectic geometry and moment maps. To make things more digestible, one can focus on the SL(2, C)-case originally considered by Hitchin. For the second part of the talk, the author should read about the hyperKähler geometry of the moduli space of Higgs bundles and share something they’ve learned. There are a few directions to go, so we suggest either 1) Hitchin’s work on L^2-cohomology, or 2) the asymptotic geometry of the moduli space and comparisons to the semi-flat metric studied by Dumas-Neitske and/or (more ambitiously) Frederickson. Other ideas are welcome.

Suggested plan: Introduce globally hyperbolic maximal Cauchy compact (in short, GHMC) anti-de Sitter 3-manifolds, and describe how they relate to maximal representations in PSL(2, R) × PSL(2, R) by the work of Mess. Explain how the existence and uniqueness of maximal surfaces determine a natural diffeomorphism between the deformation space of GHMC anti-de Sitter 3-manifolds and the cotangent bundle to Teichmüller space T^* Teich. Using various descriptions of such deformation space, introduce the complex structure inherited from T^* Teich and the para-complex structures coming from the character variety and from the parametrization via constant Gaussian curvature surfaces, the complex cotangent bundle symplectic form, and the (para-complex valued) Goldman symplectic structure. Define the notion of para-hyperKähler structure and describe the relations between the aforementioned ingredients, in particular how their combination describes a natural pseudo-Riemannian metric on the space of GHMC anti-de Sitter 3-manifolds. The remaining time can be dedicated to the presentation of an outline of the proof, possibly highlighting similarities and differences between Donaldson’s infinite-dimensional hyperKähler reduction and the space of almost-Fuchsian manifolds.

Suggested plan: First, present the essential aspects of Thermodynamic formalism used in constructing the metric. Show that the construction of the Pressure metric on Teichmüller space provides a multiple of the Weil-Petersson metric. Describe the Pressure metric on the Hitchin component of SL(n, R) and show why it is non-degenerate. Time permitting and according to your taste, you might elaborate on how to generalize the Pressure metric to Anosov representations in SL(n, R).

Suggested plan: Starting from McMullen’s description of the Weil-Petersson metric in terms of the Hessian of the Hausdorff dimension of the limit set (time permitting, the speaker should elaborate on the proof of this result), introduce and describe the semidefinite metric on the quasi-Fuchsian space defined by Bridgeman and Taylor. Then (as shown by Bridgeman-Canary-Labourie-Sambarino), prove that this metric is the pull-back of the Pressure metric through the Plücker embedding, concluding that it is Riemannian and that it can be extended to an open subset of convex cocompact representations.

Suggested plan: Give a brief overview of the preliminary notions on pressures, Anosov flows, Markov codings, and Anosov representations (see in particular Sections 2 and 4 from Carvajales-Dai-Pozzetti-Wienhard [CDPW22], and compare with the comment below). Introduce the definition of Thurston’s distance and Finsler norm on the space of projectivized H ̈older reparametrizations of a topologically transitive flow admitting a Markov coding (see Section 3 from [CDPW22]). Describe how these notions restrict to the study of Anosov representations (see Section 6 from [CDPW22]) and discuss their main properties (see Sections 7 and 8 from from [CDPW22]). We encourage the speaker to focus their presentation of Thurston’s distances for Θ-Anosov representations inside G = PSL(n+1, R), with particular emphasis on Hitchin and Benoist representations, and to specialize the discussion to these settings whenever convenient for the sake of clarity. 

Suggested plan: Start introducing the general work of Bonahon on transverse cocycles for a maximal geodesic lamination, and introduce the shear coordinates, which give an embedding of Teichmüller space onto an open cone of the space of transverse cocycles. Define the Thurston intersection form on the space of transverse cocycles, and the two descriptions (provided in the main paper) of the Weil-Petersson symplectic form. Then, conclude by showing that the shear coordinates determine an identification up to a constant between the two forms. Time permitting, remark that this can be seen as an extension of Wolpert’s magic formula to maximal geodesic laminations.