Anosov representations of word hyperbolic groups into reductive Lie groups provide a generalization of convex cocompact representations to higher real rank. I will explain how these representations can be used to construct properly discontinuous actions on homogeneous spaces. For a rank-one simple group G, this construction covers all proper actions on G, by left and right multiplication, of quasi-isometrically embedded discrete subgroups of G×G; in particular, such actions remain proper after small deformations, and we can describe them explicitly. This is joint work with F. Guéritaud, O. Guichard, and A. Wienhard. 

We discuss coordinates for representation varieties of 3-manifold groups. These coordinates are 3-dimensional analogues of the Fock-Goncharov coordinates on higher Teichmuller spaces.

I will describe joint work with Katzarkov, Noll, and Simpson, which introduces the notion of a versal harmonic map to a building associated with a given spectral cover of a Riemann surface, generalizing to higher rank the leaf space of the foliation defined by a quadratic differential. A motivating goal is to develop a geometric framework for studying spectral networks that affords a new perspective on their role in the theory of Bridgeland stability structures and the WKB theory of differential equations depending on a small parameter. This talk will focus on the WKB aspect: I will discuss the sense in which the asymptotic behavior of the Riemann-Hilbert correspondence is governed by versal harmonic maps to buildings. 

In this talk we study Fock-Goncharov generalized shearing parameters for representations of a punctured surface group in PGL(3,K) for an ultrametric field K, acting on the associated (real) Euclidean building.  The main motivation is to understand degenerations of real convex projective structures on surfaces.  We will explain how to interpret these parameters in the Euclidean building, using a geometric classification of triples of ideal chambers.  Under simple open conditions on the parameters, we construct a nice invariant subspace and an associated finite  A2-complex encoding the marked length spectrum. This allows to describe degenerations of convex projective strucures in an open cone of parameters.

This talk will be about the theory of compactification of the character variety.
Methods coming from algebraic geometry and tropical geometry can be
used to construct and to study some of these compactifications. With
these methods it is often possible to get some results by direct
computations. We will mostly discuss the special case of deformation
spaces of convex real projective structures on open surfaces, a subset
of the character variety of the free group in SL(3,R).
This is joint work with S. Choi.
 

Hitchin component for $SL(n,R)$ is the component in the space of surface group representations into $SL(n,R)$ which can deform to Fuchsian locus. The Hitchin component is in correspondence with the moduli space of $SL(n,R)$-Higgs bundles. I will introduce recent work with Brian Collier on asymptotic behaviors of families in Hitchin component in terms of certain families of Higgs bundles. Namely, given a family of Higgs bundles by scaling Higgs field by $t$, we analyze the asymptotic behavior of the corresponding representations as $t$ goes to $\infty$ in two special cases.

Schiffer variations produce isoperiodic deformations on the moduli space of abelian differentials on algebraic curves of genus g, and define there an interesting algebraic foliations. I'll explain that the dynamics of this latter can be understood through the dynamics of the lattice Sp(2g,Z) on the homogeneous space Sp(2g,R)/Sp(2g-2,R). This transfer principle is based on a topological property of the period map (its fibers are connected). The fact that this property holds answers a question of McMullen, and generalizes a previous work of Simpson. This is a joined work with Calsamiglia and Francaviglia