# Research keywords

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**:

_{1}(S),G) = Hom

^{*}(π

_{1}(S),G)/G,

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^{+}(H^{2}), 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(π_{1}(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**.