Schedule

Title Local rigidity problem of smooth group actions (Part I)

Abstract Some dynamical systems, like Anosov systems, are rigid under perturbations, but the rigidity is just topological and smooth structure of the systems varies under perturbation. However, some smooth group actions exhibit smooth local rigidity, i.e., any perturbations do not change smooth structure of the actions. For example, in 1993, Ghys showed smooth local rigidity of the standard action of cocompact subgroup of PSL(2,R) on the circle. In 1997, Katok and Spatzier also proved smooth local rigidity of many homogeneous group actions related to semi-simple Lie groups of real rank greater than one. Actions related to simple Lie groups of real rank one, e.g., SO(n,1) or SU(n,1) are still mysterious, but, the speaker and his collaborator proved local rigidity of some actions related to such Lie groups. 

In this series of talks, we explore local rigidity problem of smooth group actions, mainly focusing on actions related to simple Lie group of real rank one. In the first talk, we give a panorama of locally rigid actions from dynamical point of view. In the second and last talks, we focus on rigidity/non-rigidity of some specific actions related to simple Lie groups of real rank one. In the second talk, we discuss the case PSL(2,R). In this case, the action have perturbations originated from perturbations of lattices of PSL(2,R) and existence of other perturbations has been an open problem until the speaker constructed such perturbations. We see how to construct non-classical perturbation using dynamical ideas. In the last talk, we show local rigidity of some homogeneous actions related to higher dimensional simple Lie groups of real rank one. Again, dynamical ideas play central roles in the proof.

Title Relative simplicity and Tsuboi metric

Abstract Many transformation groups on manifolds are simple, but their universal coverings are not. In this talk, we introduce the concept of relative simplicity of groups instead of simplicity of groups. We show that the universal coverings of many transformation groups are relatively simple and we define a new metric, which is called Tsuboi metric. This is a joint work with Mitsuaki Kimura (Osaka Dental University), Yoshifumi Matsuda (Aoyama Gakuin University), Takahiro Matsushita (Shinshu University), Ryuma Orita (Niigata University).

Title An exotic embedding of the hyperbolic plane via convex bodies

Abstract Similarly to Euclidean spaces, there is an infinite-dimensional analog for the algebraic hyperbolic spaces. It can be defined for example as (the completion of) the direct limit of all the finite-dimensional hyperbolic spaces. With this definition, the hyperbolic plane naturally sits inside as a totally geodesic plane. In this talk, I will discuss another way to embed the hyperbolic plane which is still equivariant with respect to PSL(2,R), but not totally geodesic. This construction is based on a previous work of Debin and Fillastre on the hyperbolic geometry of convex bodies. This is a joint work with F. Fillastre and Y. Long.

Speaker list
- Wonyong Jang (KAIST):  Profinite rigidity and orderabilities of groups
- Seungyeol Park (KAIST):  Cusp openings on hyperbolic Coxeter orbifolds
- Leonardo Dinamarca (KIAS): Growth for diffeomorphism of the interval
- Hongjun Lee (KAIST): Matsumoto dicotomy on foliated S^1 bundle
- Seunghoon Hwang (SNU): Projective Coxeter Groups of Finite Covolumes
- Kangrae Park (SNU): Diophantine Approximation on the Kleinian Circle Packing
- Seong Yoon Kim (SNU): Algebraic Mapping Class Group Rigidity
- Sunghwan Ko (SNU): Finiteness of integral representation arising from truncation d-polytopes (d>=3)
- Juseop Lee (KAIST): Orbifold Lickorish-Wallace Theorem
- Yongho Seo (KAIST): Hausdorff Quotients on the Full Reflective Locus

Title Local rigidity problem of smooth group actions (Part II)

Abstract Some dynamical systems, like Anosov systems, are rigid under perturbations, but the rigidity is just topological and smooth structure of the systems varies under perturbation. However, some smooth group actions exhibit smooth local rigidity, i.e., any perturbations do not change smooth structure of the actions. For example, in 1993, Ghys showed smooth local rigidity of the standard action of cocompact subgroup of PSL(2,R) on the circle. In 1997, Katok and Spatzier also proved smooth local rigidity of many homogeneous group actions related to semi-simple Lie groups of real rank greater than one. Actions related to simple Lie groups of real rank one, e.g., SO(n,1) or SU(n,1) are still mysterious, but, the speaker and his collaborator proved local rigidity of some actions related to such Lie groups. 

In this series of talks, we explore local rigidity problem of smooth group actions, mainly focusing on actions related to simple Lie group of real rank one. In the first talk, we give a panorama of locally rigid actions from dynamical point of view. In the second and last talks, we focus on rigidity/non-rigidity of some specific actions related to simple Lie groups of real rank one. In the second talk, we discuss the case PSL(2,R). In this case, the action have perturbations originated from perturbations of lattices of PSL(2,R) and existence of other perturbations has been an open problem until the speaker constructed such perturbations. We see how to construct non-classical perturbation using dynamical ideas. In the last talk, we show local rigidity of some homogeneous actions related to higher dimensional simple Lie groups of real rank one. Again, dynamical ideas play central roles in the proof.

Title Rigidity of Riemannian foliations with locally symmetric leaves

Abstract We discuss rigidity properties of Riemannian foliations whose leaves are locally symmetric spaces. The classification of Riemannian foliations can be reduced to that of Lie foliations, which are obtained as quotients of fiber bundles over Lie groups, by the Molino theory. Zimmer characterized minimal Lie foliations whose leaves are symmetric spaces of non-compact type as pullbacks of standard homogeneous foliations on double coset spaces. We strengthen and generalize Zimmer's result by showing that any minimal Lie foliation whose leaves are locally isometric to a symmetric space of non-compact type without a Poincaré disk factor is smoothly conjugate to a standard homogeneous foliation on a double coset space. This talk is based on joint work with Gaël Meigniez.

Title Equidistribution of polynomially bounded o-minimal curves in homogenous spaces

Abstract In this talk, I will present a recent application of o-minimal structures, a branch of model theory, combined with techniques from homogeneous dynamics, to establish new uniform distribution results for a broad class of unbounded curves in general homogeneous spaces. Specifically, I will discuss my recent joint work with N. Shah and H. Xing in which we generalize Ratner’s theorem on the equidistribution of unipotent flows and Shah’s theorem on the equidistribution of polynomial trajectories. As an example, consider a curve in SL(n, R) whose entries are rational functions. Under an explicit and verifiable non-contraction condition, we show that the averages along this curve in SL(n, R)/SL(n, Z) converge to a homogeneous measure.

Title Local rigidity problem of smooth group actions (Part III)

Abstract Some dynamical systems, like Anosov systems, are rigid under perturbations, but the rigidity is just topological and smooth structure of the systems varies under perturbation. However, some smooth group actions exhibit smooth local rigidity, i.e., any perturbations do not change smooth structure of the actions. For example, in 1993, Ghys showed smooth local rigidity of the standard action of cocompact subgroup of PSL(2,R) on the circle. In 1997, Katok and Spatzier also proved smooth local rigidity of many homogeneous group actions related to semi-simple Lie groups of real rank greater than one. Actions related to simple Lie groups of real rank one, e.g., SO(n,1) or SU(n,1) are still mysterious, but, the speaker and his collaborator proved local rigidity of some actions related to such Lie groups. 

In this series of talks, we explore local rigidity problem of smooth group actions, mainly focusing on actions related to simple Lie group of real rank one. In the first talk, we give a panorama of locally rigid actions from dynamical point of view. In the second and last talks, we focus on rigidity/non-rigidity of some specific actions related to simple Lie groups of real rank one. In the second talk, we discuss the case PSL(2,R). In this case, the action have perturbations originated from perturbations of lattices of PSL(2,R) and existence of other perturbations has been an open problem until the speaker constructed such perturbations. We see how to construct non-classical perturbation using dynamical ideas. In the last talk, we show local rigidity of some homogeneous actions related to higher dimensional simple Lie groups of real rank one. Again, dynamical ideas play central roles in the proof.

Title Counting totally geodesic submanifolds in infinite volume rank one spaces

Abstract In this talk, we focus on totally geodesic submanifolds (TGS) in geometrically finite rank one locally symmetric spaces of infinite volume. We first show that there are at most finitely many maximal TGS of finite volume of the given manifold. We then provide explicit upper bounds for the number of TGS with volume less than T. These bounds are polynomial in T, and are obtained by quantitative result accounting for the isolation of TGS. Our results extend previous work of Mohammadi-Oh on real hyperbolic 3-manifolds to the general rank-one locally symmetric spaces. This is ongoing joint work with Hee Oh.

Title Partially hyperbolic flows on flat vector bundles with  an application to complete affine manifolds.

Abstract Let N be a manifold of dimension m with a flat vector bundle given by a representation $\rho:\pi_1(N) \ra GL(n,R)$ where $\pi_1(N)$ is finitely generated. $\rho$ is a k-partially hyperbolic holonomy representation if the flat bundle pulled back over the unit tangent bundle of a sufficiently large compact submanifold of N splits into expanding, neutral, and contracting subbundles along the geodesic flow, where the expanding and contracting subbundles are k-dimensional with k < n/2. Suppose that each element of $\rho(\pi_1(N))$ has an eigenvalue of norm 1, or, alternatively, each element of it has a singular value that is uniformly bounded above and below.  We show that $\rho$ is a P-Anosov representation for a parabolic subgroup P of GL(n,R) if and only if $\rho$ is a partially hyperbolic representation. We are going to primarily employ representation theory techniques. This had never been done over the full general linear group. As an application, we will show this holds when N is a complete affine n-manifold and $\rho$ is a linear part of the holonomy representation. Also, we will end with some questions involving the word hyperbolicity of closed complete affine manifolds.