The character variety of maximal Sp(2n,R)-representations of the fundamental group G of a surface S is homeomorphic to a real semi algebraic set X and as such admits a compactification by the set of closed points in the real spectrum of the coordinate ring of X. This compactification has very good topological properties and the mapping class group acts on it with virtually abelian stabilizers. Its ideal points are represented by certain Sp(2n,F) representations of G over various real closed fields F. Characterizing these representations and understanding their geometric properties is an ongoing theme of our research. In this talk we focus on the R-building B(n,F) associated to Sp(2n,F) and will explain why the corresponding G-action is proper in the sense of Korevaar and Schoen, implying the existence of an equivariant harmonic map for every complex structure h on S. We also explain that if the systole of the G-action on B(n,F) is strictly positive, the energy functional, as a function of h, is proper on Teichmueller space. Such actions with strictly positive systole form an open set in the real spectrum boundary, that is non-empty as soon as n is at least 2. joint work with A.Iozzi, A.Parreau and B. Pozzetti.

Symmetric spaces are special Riemannian manifolds where at each point the geodesic symmetry is an isometry.  Such a space has a Lie group of isometries acting transitively and we will study the structure of these spaces by using the interplay between the Riemannian and the Lie group theory aspect. For instance we will see that a symmetric space is a product of irreducible ones, and that the latter come in three types: compact type (positive curvature), Euclidean type (zero curvature) and non-compact type (negative curvature), with a remarkable duality between the compact and non-compact type. Then we will turn to symmetric spaces of non-compact type and establish the connection between regular elements and maximal flat totally geodesic subspaces; with this at hand, we will obtain the root space decomposition. In the last part we will study more closely the symmetric spaces that admit an invariant complex  structure (hermitian symmetric). These form the natural framework for the study of that part of higher Teichmueller theory concerned with maximal representations of surface groups.

Prerequisite: some basic knowledge of differential and Riemannian geometry, ideally corresponding to the eight first chapters of M. do Carmo, "Riemannian Geometry", Birkhauser, 1992; the book is freely available in pdf format on google.

Symmetric spaces are special Riemannian manifolds where at each point the geodesic symmetry is an isometry.  Such a space has a Lie group of isometries acting transitively and we will study the structure of these spaces by using the interplay between the Riemannian and the Lie group theory aspect. For instance we will see that a symmetric space is a product of irreducible ones, and that the latter come in three types: compact type (positive curvature), Euclidean type (zero curvature) and non-compact type (negative curvature), with a remarkable duality between the compact and non-compact type. Then we will turn to symmetric spaces of non-compact type and establish the connection between regular elements and maximal flat totally geodesic subspaces; with this at hand, we will obtain the root space decomposition. In the last part we will study more closely the symmetric spaces that admit an invariant complex  structure (hermitian symmetric). These form the natural framework for the study of that part of higher Teichmueller theory concerned with maximal representations of surface groups.

Prerequisite: some basic knowledge of differential and Riemannian geometry, ideally corresponding to the eight first chapters of M. do Carmo, "Riemannian Geometry", Birkhauser, 1992; the book is freely available in pdf format on google.