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1. Speaker: Masahiko Kanai (Japan, Tokyo)
Title: Cross ratio and its folks in geometry and dynamics
Abstract: This is a survey on cross ratio and its folks such as Schwarzian derivative, geodesic currents and paraKaehler structures emphasizing applications to geometry and dynamics.
2. Speaker: Fanny Kassel (France, Lille)
Title: Complete Lorentzian 3-manifolds of constant nonpositive curvature
Abstract: The 3-dimensional Minkowski space R^{2,1} and anti-de Sitter space AdS^3 are Lorentzian analogues of the 3-dimensional Euclidean space and real hyperbolic space. I will consider quotients of these spaces by discrete groups of isometries acting properly discontinuously, and will describe some of their geometric and spectral properties.
This includes:
- a double properness criterion for finitely generated actions on AdS^3,
- a double properness criterion for finitely generated free actions on R^{2,1}, which shows that the corresponding quotients (called Margulis spacetimes) are "infinitesimal analogues" of quotients of AdS^3,
- the topological tameness of finitely generated quotients of AdS^3 and of Margulis spacetimes,
- the Crooked Plane conjecture for Margulis spacetimes,
- a combinatorial parameterization of Margulis spacetimes by the arc complex,
- a construction of discrete spectrum of the Laplacian on quotients of AdS^3.
This is joint work with François Guéritaud, with Jeffrey Danciger and François Guéritaud, and with Toshiyuki Kobayashi.
3. Speaker: Takayuki Okuda(Japan, Tohoku)
Title: Extensions of proper actions on semisimple symmetric spaces
Abstract: Let $G$ be a linear semisimple Lie group and $(G,H)$ a symmetric pair. We denote by $G_\mathbb{C}$ the complexification of $G$ and we take a real form $G^c$ of $G_\mathbb{C}$ such that $H = G \cap G^c$ and $\mathrm{Lie}~G^c$ is the $c$-dual of $(\mathrm{Lie}~G, \mathrm{Lie}~H)$. Then the semisimple symmetric space $G/H$ is a submanifold of $G_\mathbb{C}/G^c$.
In this talk, we show that the following two conditions on a closed subgroup $L$ of $G$ are equivalent:
(1) The $L$-action on $G/H$ is proper.
(2) The $L$-action on $G_\mathbb{C}/G^c$ is proper.
As applications of this result, we also give classifications of the following two kinds of proper actions on $G/H$:
(a) Proper $SL(2,\mathbb{R})$-actions on $G/H$ via a Lie group homomorphism $SL(2,\mathbb{R}) \rightarrow G$.
(b) Symmetric pairs $(G,L)$ such that the $L$-action on $G/H$ is proper.
4. Speaker: Inkang Kim (Korea, KIAS)
Title: Entropy and deformation of real projective structure
Abstract: We discuss the deformation of real projective structures and the entropy of various probability measures invariant by the geodesic flow.
Title: Cross ratio and its folks in geometry and dynamics
Abstract: This is a survey on cross ratio and its folks such as Schwarzian derivative, geodesic currents and paraKaehler structures emphasizing applications to geometry and dynamics.
2. Speaker: Fanny Kassel (France, Lille)
Title: Complete Lorentzian 3-manifolds of constant nonpositive curvature
Abstract: The 3-dimensional Minkowski space R^{2,1} and anti-de Sitter space AdS^3 are Lorentzian analogues of the 3-dimensional Euclidean space and real hyperbolic space. I will consider quotients of these spaces by discrete groups of isometries acting properly discontinuously, and will describe some of their geometric and spectral properties.
This includes:
- a double properness criterion for finitely generated actions on AdS^3,
- a double properness criterion for finitely generated free actions on R^{2,1}, which shows that the corresponding quotients (called Margulis spacetimes) are "infinitesimal analogues" of quotients of AdS^3,
- the topological tameness of finitely generated quotients of AdS^3 and of Margulis spacetimes,
- the Crooked Plane conjecture for Margulis spacetimes,
- a combinatorial parameterization of Margulis spacetimes by the arc complex,
- a construction of discrete spectrum of the Laplacian on quotients of AdS^3.
This is joint work with François Guéritaud, with Jeffrey Danciger and François Guéritaud, and with Toshiyuki Kobayashi.
3. Speaker: Takayuki Okuda(Japan, Tohoku)
Title: Extensions of proper actions on semisimple symmetric spaces
Abstract: Let $G$ be a linear semisimple Lie group and $(G,H)$ a symmetric pair. We denote by $G_\mathbb{C}$ the complexification of $G$ and we take a real form $G^c$ of $G_\mathbb{C}$ such that $H = G \cap G^c$ and $\mathrm{Lie}~G^c$ is the $c$-dual of $(\mathrm{Lie}~G, \mathrm{Lie}~H)$. Then the semisimple symmetric space $G/H$ is a submanifold of $G_\mathbb{C}/G^c$.
In this talk, we show that the following two conditions on a closed subgroup $L$ of $G$ are equivalent:
(1) The $L$-action on $G/H$ is proper.
(2) The $L$-action on $G_\mathbb{C}/G^c$ is proper.
As applications of this result, we also give classifications of the following two kinds of proper actions on $G/H$:
(a) Proper $SL(2,\mathbb{R})$-actions on $G/H$ via a Lie group homomorphism $SL(2,\mathbb{R}) \rightarrow G$.
(b) Symmetric pairs $(G,L)$ such that the $L$-action on $G/H$ is proper.
4. Speaker: Inkang Kim (Korea, KIAS)
Title: Entropy and deformation of real projective structure
Abstract: We discuss the deformation of real projective structures and the entropy of various probability measures invariant by the geodesic flow.