Monday, 10 December 2018
Time Speaker Title Resources
10:00 to 11:15 Ken Bromberg TBA
11:15 to 11:45 -- Tea Break
11:45 to 13:00 Francois Gueritaud Right-angled Coxeter groups and affine actions - Lecture 1

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.

13:00 to 15:00 -- Lunch
15:00 to 16:15 Qiongling Li Harmonic maps for surface group representations - Lecture 1

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.

16:15 to 16:30 -- Tea Break
Tuesday, 11 December 2018
Time Speaker Title Resources
10:00 to 11:15 Olivier Guichard Anosov representations: the basics and maybe more - Lecture 1

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

11:15 to 11:45 -- Tea Break
11:45 to 13:00 Francois Gueritaud Right-angled Coxeter groups and affine actions - Lecture 2

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.

13:00 to 15:00 -- Lunch
15:00 to 16:15 Qiongling Li Harmonic maps for surface group representations - Lecture 2

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.

16:15 to 16:30 -- Tea Break
Wednesday, 12 December 2018
Time Speaker Title Resources
10:00 to 11:15 Olivier Guichard Anosov representations: the basics and maybe more - Lecture 2

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

11:15 to 11:45 -- Tea Break
11:45 to 13:00 Bertrand Deroin Lyapunov exponents of linear surface group representations - Lecture 1

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.

13:00 to 15:00 -- Lunch
15:00 to 16:15 Ken Bromberg TBA
16:15 to 16:30 -- Tea Break
Thursday, 13 December 2018
Time Speaker Title Resources
10:00 to 11:15 Francois Gueritaud Right-angled Coxeter groups and affine actions - Lecture 3

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.

11:15 to 11:45 -- Tea Break
11:45 to 13:00 Bertrand Deroin Lyapunov exponents of linear surface group representations - Lecture 2

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.

13:00 to 15:00 -- Lunch
15:00 to 16:15 Olivier Guichard Anosov representations: the basics and maybe more - Lecture 3

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

16:15 to 16:30 -- Tea Break
Friday, 14 December 2018
Time Speaker Title Resources
10:00 to 11:15 Stefano Francavigli Branched complex projective structures on surfaces (Lecture 1)

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. 

11:15 to 11:45 -- Tea Break
11:45 to 13:00 Bertrand Deroin Lyapunov exponents of linear surface group representations - Lecture 3

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.

13:00 to 15:00 -- Lunch
15:00 to 16:15 Alessandra Iozzi The bounded Borel class and complex representations of 3-manifold groups (Lecture 1)

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.

16:15 to 16:30 -- Tea Break
Monday, 17 December 2018
Time Speaker Title Resources
10:00 to 11:15 Stefano Francaviglia Branched complex projective structures on surfaces (Lecture 2)

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. 

11:15 to 11:45 -- Tea Break
11:45 to 13:00 Tengren Zhang Convex real projective structures on closed surfaces - Lecture 1

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.

13:00 to 15:00 -- Lunch
15:00 to 16:15 Alessandra Iozzi The bounded Borel class and complex representations of 3-manifold groups (Lecture 2)

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.

16:15 to 16:30 -- Tea Break
Tuesday, 18 December 2018
Time Speaker Title Resources
10:00 to 11:15 Francois Labourie Fock-Goncharov coordinates, positivity and character varieties - Lecture 1

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.

11:15 to 11:45 -- Tea Break
11:45 to 13:00 Alessandra Iozzi The bounded Borel class and complex representations of 3-manifold groups (Lecture 3)

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.

13:00 to 15:00 -- Lunch
15:00 to 16:15 Tengren Zhang Convex real projective structures on closed surfaces - Lecture 2

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.

16:15 to 16:30 -- Tea Break
Wednesday, 19 December 2018
Time Speaker Title Resources
10:00 to 11:15 Francois Labourie Fock-Goncharov coordinates, positivity and character varieties - Lecture 2

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.

11:15 to 11:45 -- Tea Break
11:45 to 13:00 Tengren Zhang Convex real projective structures on closed surfaces - Lecture 3

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.

13:00 to 15:00 -- Lunch
15:00 to 16:15 Jayadev Athreya The SL(2, R) action on spaces of differentials - Lecture 1

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.

16:15 to 16:30 -- Tea Break
Thursday, 20 December 2018
Time Speaker Title Resources
10:00 to 11:15 Francois Labourie Fock-Goncharov coordinates, positivity and character varieties - Lecture 3

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.

11:15 to 11:45 -- Tea Break
11:45 to 13:00 Shinpei Baba Holonomy representations of complex projective structures - Lecture 1

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.

Reference:
Gallo Kapvoich Marden 2000, the monodromy groups of Schwarzian equations on closed Riemann surfaces.

13:00 to 15:00 -- Lunch
15:00 to 16:15 Jayadev Athreya The SL(2, R) action on spaces of differentials - Lecture 2

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.

16:15 to 16:30 -- Tea Break
Friday, 21 December 2018
Time Speaker Title Resources
10:00 to 11:15 Shinpei Baba Holonomy representations of complex projective structures - Lecture 2

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.

Reference:
Gallo Kapvoich Marden 2000, the monodromy groups of Schwarzian equations on closed Riemann surfaces.

11:15 to 11:45 -- Tea Break
13:00 to 15:00 -- Lunch
16:15 to 16:30 -- Tea Break