I will introduce higher Teichmüller spaces (e.g. Hitchin components and maximal components of character varieties). I will describe some of the techniques we can use to understand their topology, and how they are related to parameter spaces of geometric structures on manifolds.

Brian Collier: Magical Nilpotents and Higher Teichmuller components

In this talk, we will present a special class of nilpotent elements of a complex semisimple Lie algebra. For such a nilpotent element, we will describe how Higgs bundles can be used to construct components of the character variety of a closed surface of genus at least 2. Moreover, such components are deformation spaces of Teichmuller space. The classification of this class of nilpotent elements turns out to be equivalent to Guichard and Wienhard's notion of Theta-positivity, and so, this construction should describe all Higher Teichmuller spaces. 

Viveka Erlandsson: Mirzakhani’s Curve Counting

Mirzakhani proved two theorems about the asymptotic growth of the number of curves of a given type on a surface: one for simple curves and one for general curves. In this talk I will discuss a new approach to proving both these results, using different methods to those of Mirzakhani’s. This is joint work with Juan Souto.

Sourav Ghosh: Affine Anosov representations

In this talk, I will define affine Anosov representations in SO(n-1,n)⋉R^2n-1 and describe their relation with proper affine actions of hyperbolic groups on R^2n-1. This part is a joint work with Nicolaus Treib. Moreover, I will also describe how affine Anosov representations are infinitesimal versions of certain special class of Anosov representations in SO(n,n) which satisfy a stronger Anosov condition than the flag spaces of SO(n,n) can detect. This part is my independent work which has some overlap with independent work done by Jeff Danciger and Tengren Zhang.

Olivier Glorieux: Regularity of limit sets of AdS quasi-Fuchsian groups

The aim of this talk is to explain the different behaviours for limit set of  Anosov representations. Depending on the arrival group, the limit of a surface group can have very different regularity : from non rectifiable  to C¹. We will explain in details the case of quasi Fuchsian representations SO(p,2) where we showed with D. Monclair that the limit set is never C¹ except if it preserves a copy of H^p.

Qiongling Li: Higgs bundles in certain Hitchin fibers

We study Higgs bundles in the nilpotent cone and in the Hitchin fiber at (q2, 0,...0). L. Ness introduced a function on the nilpotent cone of a Lie algebra and showed that a nilpotent element is unitarily conjugate to a standard Jordan canonical form if and only it is a critical point of the function. In the first part of the talk, by relating the function to the curvature formula, we apply Ness' theorem to study the holonomy of Higgs bundles in the nilpotent cone and the Hodge metric for complex variation of Hodge structures. In the second part of the talk, we generalize Ness' theorem from nilpotent elements to the space of elements in a copy of 3-dimensional sub Lie algebra. Applying the generalized theorem, we are able to show the holonomy of Higgs bundles in the Hitchin fibers at (q2,0,...,0) are always dominated by the Fuchsian ones. Part of this work is joint with Song Dai. 

Brice Loustau: Computing equivariant harmonic maps

I will present effective methods to compute equivariant harmonic maps. The main setting will be equivariant maps from the universal cover of a surface into a nonpositively curved space. By discretizing the theory appropriately, we show that the energy functional is strongly convex and derive the convergence of the discrete heat flow to the energy minimizer, with explicit convergence rate. We also examine center of mass methods, after showing a generalized mean value property for harmonic maps. We conclude by showing the convergence of our method to a smooth harmonic map as one takes finer and finer meshes. We feature a concrete illustration of these methods with Harmony, a computer software with a graphical user interface that we developed in C++, whose main functionality is to numerically compute and display equivariant harmonic maps.

Given a pants decomposition of a surface S and a vector c with positive real entries, we describe a plumbing construction which endows S with a complex projective structure for which the associated holonomy representation belongs to the quasi-Fuchsian space QF(S). We show that when c tends to the zero vector, this construction limits to Kra’s plumbing construction. When S is the once punctured torus, the holonomy representations of these structures belong to the "linear" slice of quasi-Fuchsian space QF(S), as defined by Komori and Yamashita. We will end the talk discussing some questions and conjectures for these slices suggested by computer pictures created jointly with Yamashita.

Giuseppe Martone: Tree-type limits of Hitchin representations

Hitchin singled out a preferred component of the character variety of representations from the fundamental group of a surface to PSL(d,R) containing a preferred copy of the Teichmüller space of the surface. 

In joint work, T. Zhang and I associated to each Hitchin representation a geometrically meaningful measure, called a geodesic current. Recently, Burger, Iozzi, Parreau, and Pozzetti used these geodesic currents to analyze the asymptotic behavior of sequences of Hitchin representations.

On the other hand, in previous work we used the Fock-Goncharov/Bonahon-Dreyer coordinates for the Hitchin component to describe some of the asymptotic geometry of sequences of Hitchin representations.

In this talk, we use these points of view to describe sequences of Hitchin representations whose limiting behavior is analogous to the one of a sequence of hyperbolic metrics.

Lorenzo Ruffoni: Tame CP1-structures on the thrice-punctured sphere with triangular holonomy.

Gallo, Kapovich and Marden showed that any non-elementary representation of the fundamental group of a closed surface into PSL(2,C) can be realized as the holonomy of a complex projective structure on it. Such a structure is far from being unique, and they asked for a characterization of non-uniqueness phenomena in terms of explicit geometric surgeries, such as grafting. Together with Sam Ballas, Phil Bowers and Alex Casella we are investigating the analogous questions in a non compact setting, namely in the case of structures on the thrice-punctured sphere, with holonomy representations generated by elliptics. Under some tameness assumptions on the behavior of the developing maps at the cusps, which are for instance satisfied by structures induced by constant curvature metrics with conical points (as in the work of Mondello and Panov), we show how grafting can be used to navigate the space of structures having the same holonomy.

Andy Sanders: Symmetric spaces, regular immersions, and a priori inequalities

A continuously differentiable mapping between two manifolds is an immersion if the differential is injective at each point.  When the target manifold is a symmetric space, a refinement of this condition is possible, by asking that the differential takes values in the interior of a Weyl chamber. We call such an immersion regular, and in this talk we will outline some basic properties of regular immersions.

After these preliminary discussions, we will turn to the topic of Anosov representations, and indicate some differential geometric methods which complement the mostly dynamical/synthetic understanding of such representations.  In particular, we will give some examples which relate a priori inequalities derived from the study of partial differential equations to the property of being a regular immersion in the setting of Anosov representations.  All inquiries and results in this talk are joint work with Johannes Horn.  

Peter Smillie: Hyperbolic surfaces in Minkowski 3-space

To essentially every lower semicontinuous function on the circle, there corresponds a unique properly embedded spacelike surface in Minkowski 3-space that is locally isometric to the hyperbolic plane. Sometimes, the corresponding surface is globally isometric to the hyperbolic plane, but other times it fails to be complete. So there is some condition on lower semicontinuous functions on the circle that corresponds to completeness of the surface. I will give a partial description of this condition, and discuss applications to actions of discrete groups on Minkowski 3-space. Everything is joint with Francesco Bonsante and Andrea Seppi.

Nicolas Tholozan: Minimal representations of punctured spheres

Let Γ be the fundamental group of the Riemann sphere with n ≥ 3 punctures. The non-Abelian Hodge correspondance draws a bijection between representations of Γ into the Lie group SU(p, q) and so-called parabolic SU(p, q) Higgs bundles on the Riemann sphere.

With Jérémy Toulisse, we prove that a purely topological condition on such a representation ρ (namely, a bound on its Toledo invariant) forces the corresponding Higgs bundle to be a variation of Hodge structure of weight 1. This implies that a non-empty open subset of the character variety of Γ into SU(p, q) consists of representations with surprising properties. (They map every simple closed curve to an elliptic element, their mapping class group

orbits are bounded...)

Jérémy Toulisse: Maximal surfaces in pseudo-hyperbolic space

The notion of maximal representations of a surface group into an Hermitian Lie group gives a natural generalization of Fuschian representations into PSL(2,R). In a joint work with Brian Collier and Nicolas Tholozan, we study these representations in rank 2 via their action on the pseudo-hyperbolic space H^2,n. We prove that any maximal representation preserves a unique maximal surface into H^2,n. As a consequence, we prove a conjecture of Labourie. If time permits, I will discuss an extension to the « universal » case (i.e. without group action) that is a joint work with François Labourie and Mike Wolf.

A recent work by Mazzeo-Swoboda-Weiss-Witt describes a stratum of the frontier of the space of SL(2,C) surface group representations in terms of 'limiting configurations' which solve a degenerated version of Hitchin's equations on a Riemann surface.  We interpret these objects in (a mapping class group invariant way in) terms of the hyperbolic geometric objects of shearings of pleated surfaces. We study limits of holonomies of complex projective structures from this perspective.  (Joint with Andreas Ott, Jan Swoboda, and Richard Wentworth). 

Maxime Wolff: Rigidity and geometricity for actions of surface groups on the circle.

We consider representations of surface groups into the group of homeomorphisms of the circle. In her phd thesis, K. Mann proved that the geometric representations (ie, those which factor through a faithful and discrete morphism into PSL(2,R), as well as the lifts of these representations) are rigid (ie, all their deformations are semi-conjugate: they have the same rotational dynamics). This generalizes a celebrated theorem of Matsumoto. In a collaboration with her, we proved that all rigid representations are obtained in this way. In this mini-course I will first give some introductory background, and then try to

present the theorem of K. Mann following an approach due to S. Matsumoto, and finally I will try to present its converse.