The SL(2, R) action on spaces of differentials
The moduli space of holomorphic differentials on surfaces comes with a natural SL(2, R) action which has been the object of a huge amount of recent mathematical study. Starting from describing these structures on surfaces in terms of flat geometry and polygons, we will describe this action, its key properties, and recent important results on the geometry of the underlying spaces and the dynamics of the action.
Holonomy representations of complex projective structures
We will discuss the characterization of holonomy representations of complex projective structures on closed surfaces based. This lecture is based on the work of Gallo, Kapovich, and Marden.
Gallo Kapvoich Marden 2000, the monodromy groups of Schwarzian equations on closed Riemann surfaces.
Schwarzian derivatives and Epstein surfaces
An important analytic tool for studying complex projective structures on surfaces is the Schwarzian derivative. The Schwarzian measures how much a locally univalent map differs from a Mobius transformation and can be viewed as a measure of the “curvature” of the projective structure. Also associated to locally univalent map and a conformal metric on the domain is an Epstein surface (defined by C. Epstein) in hyperbolic 3-space. For the hyperbolic metric the Schwarzian explicitly determines the curvature of the Epstein surface and we will see how classical facts about the Schwarzian can be proved using the geometry of the Epstein surface.
Lyapunov exponents of linear surface group representations
I will discuss some dynamical invariants, the Lyapunov exponents, that one can associate to the data of a closed Riemann surface, together with a linear representation of its fundamental group. We will see that one can relate the Lyapunov spectrum to the stability properties of the associated flat bundle, in the sense of algebraic geometry, as was conjectured by Fei Yu and proved recently by Eskin, Kontsevich, Möller et Zorich. I will give an alternative proof that generalizes to the case where the base is a Kähler manifold and if time permits I will give some applications of this result. All this is based on joint works with Jeremy Daniel, Romain Dujardin and Victor Kleptsyn.
Branched complex projective structures on surfaces
We will discuss the nature of (X,G) stractures and branched (X,G) structures on surfaces, then focusing on the case where X=CP1 and G is PSL(2,C). We will define the pair developing map and holonomy representation and discuss the role of the holonomy. We will discuss sources of branched projective structures, and study moduli spaces of branched projective structure with prescribed holonomy. In particular we will discuss the cases of trivial, parabolic and fuchsian holonomy explaining some recent result and open problems.
Anosov representations: the basics and maybe more
This series of lectures aims at giving an overview on Anosov representations or Anosov subgroups. It will cover convex-cocompact subgroups of isometries of the hyperbolic spaces, and different characterizations of Anosov subgroups (bounday maps, growth of eigenvalues, etc.), and projective geometric structure associated to Anosov subgroups, and the structural stability of Anosov representations.
Suggested bibliography: M. Kapovich, B. Leeb, Discrete isometry groups of symmetric spaces, Spring 2015 MSRI Lecture Notes. Volume IV of Handbook of Group Actions. The ALM series, International Press, Eds. L.Ji, A.Papadopoulos, S-T.Yau. (2018) Chapter 5, p. 191-290.
and my survey Olivier GUICHARD — Groupes convexes--cocompacts en rang supérieur [d'après Labourie, Kapovich, Leeb, Porti, ...] Link
Right-angled Coxeter groups and affine actions
This series of lectures will be concerned with properly discontinuous actions by discrete groups on (real) affine space. We will give an overview of the history and of known results in the area, and focus on the construction of examples. These will mostly rely on infinitesimally deforming a representation of a discrete group in a Lie group, as such data can be re-interpreted as an affine action on the Lie algebra. We will show various criteria for properness and discuss a recent result obtained with F.Kassel and J.Danciger: any right-angled Coxeter group on N generators acts properly discontinuously on an affine N(N-1)/2-dimensional space.
Univ. Côte d’Azur
Fock-Goncharov coordinates, positivity and character varieties
I will first explain in these series of lectures the shearing coordinates of Bonahon—Thurston and how they are related to configurations of points in the projective line. Then, I will move to higher rank and explain the Fock—Goncharov coordinates, explaining how they are related to the notion of positivity of configurations of points in the Grassmannian of full flags. I will only explain the case of SL(n,R) giving only hints of the situation for all real split group. A familiarity with character varieties would help but is not necessary. No knowledge of Lie theory will be required.
Chern Institute of Mathematics, Nankai University
Harmonic maps for surface group representations
The equivariant harmonic maps play an important role in the non-abelian Hodge correspondence. In this mini-course, we will explain the explicit relationship between harmonic maps and Higgs bundles. Then we will discuss some selected topics on how harmonic maps help to understand surface group representations.
ETH Zurich, Switzerland
The bounded Borel class and complex representations of 3-manifold groups
We define the volume of a representation of a three-manifold group into SL(n,C), prove that it satisfies a Milnor--Wood type inequality and study the properties implied by its maximality. We indicate possible extensions to all complex simple Lie groups and we relate our results to recent results of Farre, Francaviglia and Savini.
Convex real projective structures on closed surfaces
Let S be a closed orientable surface of genus at least 2. A convex real projective structures on a surface S is a geometric structure locally modelled on the real projective plane, and whose developing map is homeomorphism onto a properly convex domain in the real projective plane. The deformation space of convex real projective structures is a key motivating example that is central to the study of higher Teichmuller theory. Choi-Goldman showed that this deformation space is canonically identified with the SL(3,R)-Hitchin component. Goldman also parameterized this deformation space, which later inspired similar parameterizations of the Hitchin component. Labourie and Loftin later independently showed that this deformation space is naturally identified with the bundle of holomorphic cubic differentials over Teichmuller space. In this minicourse, we will describe Goldman’s description of this deformation space, as well as the Labourie-Loftin description. We will also study the symplectic geometry on this deformation space, as well as mention several metrics.