My work belongs to the field of Differential Geometry and Geometric Topology, with a focus on the geometry and topology of surfaces, e.g. hyperbolic surfaces.
My main research interest is in the areas of Teichmüller Theory and Higher Teichmüller Theory. The main objects of study are the character varieties of surface groups:
the set of conjugacy classes of reductive representations of the surface group into a reductive Lie group G.
Teichmüller Theory corresponds to the case when G = PSL(2,R) = Isom+(H2), the oriented isometries of the hyperbolic plane; this is a broad research area that can be approached from different points of view. (1) The complex side of the theory studies the parameter spaces of Riemann surfaces and sees the Teichmüller spaces T(S) as complex manifolds. (2) The point of view of geometric structures sees T(S) as parameter space of hyperbolic structures on S. (3) The same objects can be seen as discrete subgroups of Lie groups or as special representations. (4) The study of the asymptotic behavior of a sequence of hyperbolic structures and how they degenerate to measured foliations and measured laminations leads to Thurston's compactification of T(S).
Higher Teichmüller Theory deals with character varieties X(π1(S),G) of higher rank Lie groups. In my current research I explore this general case from the same points of view mentioned above. (1) The complex point of view is here closely related with Labourie's conjecture on uniqueness of equivariant minimal surfaces in symmetric spaces. (2) The point of view of geometric structures relates representations with parabolic geometries, like real and complex projective geometries or the geometry of flag manifolds. (3) The point of view of special representations leads to the study of Anosov representations and their dynamical properties. Anosov representations are one of the most promising way to understand discrete subgroups of Lie groups beyond the theory of lattices. (4) In the study of the asymptotic behavior and degenerations of a sequence of representations I introduced techniques coming form tropical geometry, a polyhedral version of algebraic geometry.
To solve problems in this field, I often use the analytical tools provided by Higgs bundles and the non-abelian Hodge correspondence obtained by solving Hitchin's equations. I am also applying the tool of spectral networks, directly coming from theoretical physics.
In Teichmüller Theory I was among the first who studied Teichmüller Spaces for surfaces of infinite topological type showing how the equivalence between the different points of view above breaks down in this more general case, and can be restored only under suitable conditions. I also worked at the theory of Thurston's asymmetric metric (a.k.a. Lipschitz metric) for surfaces with boundary.