RTG Study Seminar -- Fall 2008

Thursdays 4:10-6:00 -- East Hall 3866

PAST SEMINARS: Winter 08 Fall 07  Winter 07 Fall 06  Winter 06 

To schedule a talk, or for more information, please contact Pekka Pankka (pankka 'at' umich.edu).

Thursday, December 11.

Speaker: Moon Duchin, UM.
Title: Currents II.

Abstract: Details on Bonahon's geodesic currents and the properties of intersection number.

Thursday, December 4.

Speaker: Juan Souto, UM.
Title: Dynamics of the geodesic and horocyclic flows of hyperbolic surfaces.

Abstract: (TBA)

Thursday, November 27.

No seminar. Thanksgiving.

Thursday, November 20.

No seminar.

Thursday, November 13.

Speaker: Moon Duchin, UM.
Title: Bonahon's theory of geodesic currents.

Abstract: (TBA)

Thursday, November 6.

Speaker: Aaron Magid, UM.
Title: One-parameter families of hyperbolic manifolds (cont.)

Abstract: Given a 1-parameter family of hyperbolic metrics on a manifold, one can associate a cohomology class that represents the infinitesmal change in the metric at each time throughout the deformation. This cohomology class will be a 1-form in H^1(M;E) where M is the manifold and E is the bundle of Killing fields on M. Once we obtain a cohomology class, we can use the Hodge theorem to find a harmonic representative. Calabi-Weil used this to show that any infinitesmal deformation of a closed hyperbolic manifold is trivial. This discussion is motivated by Hodgson, Kerckhoff, and Bromberg's more recent work on noncompact hyperbolic manifolds. They prove a generalized Hodge theorem and use this to show that hyperbolic cone-manifolds are locally rigid relative to the cone angle. (cont.)

Thursday, October 30. 5-6pm. Note the exceptional time!

Speaker: Aaron Magid, UM.
Title: One-parameter families of hyperbolic manifolds

Abstract: Given a 1-parameter family of hyperbolic metrics on a manifold, one can associate a cohomology class that represents the infinitesmal change in the metric at each time throughout the deformation. This cohomology class will be a 1-form in H^1(M;E) where M is the manifold and E is the bundle of Killing fields on M. Once we obtain a cohomology class, we can use the Hodge theorem to find a harmonic representative. Calabi-Weil used this to show that any infinitesmal deformation of a closed hyperbolic manifold is trivial. This discussion is motivated by Hodgson, Kerckhoff, and Bromberg's more recent work on noncompact hyperbolic manifolds. They prove a generalized Hodge theorem and use this to show that hyperbolic cone-manifolds are locally rigid relative to the cone angle.

Thursday, October 23.

Speaker: Lars Louder, UM.
Title: A topological characterization of hyperbolic groups (After Bowditch) (cont.)

Abstract: A nonelementary word hyperbolic group G acts by homeomorphisms on its boundary, a perfect, metrisable compactum. The associated action on the space of distinct triples in the boundary is is properly discontinuous and cocompact. I will give a proof, due to Bowditch, of the converse: If G acts properly discontinuously and cocompactly on the space of distinct triples of a perfect metrisable compactum, then G is actually hyperbolic.

Thursday, October 16.

Speaker: Lars Louder, UM.
Title: A topological characterization of hyperbolic groups (After Bowditch)

Abstract: A nonelementary word hyperbolic group G acts by homeomorphisms on its boundary, a perfect, metrisable compactum. The associated action on the space of distinct triples in the boundary is is properly discontinuous and cocompact. I will give a proof, due to Bowditch, of the converse: If G acts properly discontinuously and cocompactly on the space of distinct triples of a perfect metrisable compactum, then G is actually hyperbolic.

Thursday, October 9.

No seminar.

Thursday, October 2.

Speaker: Pekka Pankka, UM.
Title: Geometric rigidity of conformal matrices (after Faraco and Zhong).

Abstract: A classical theorem of Liouville(-Gehring-Reshetnyak) states that a conformal mapping from a domain in R^n to R^n is a restriction of a Möbius mapping if n > 2. An interesting related question is to try to estimate the distance of a general mapping to the set of Möbius mappings in terms of the distance of the differential to the set of conformal matrices. I will discuss recent results of Faraco and Zhong into this direction.

Speaker: Qian Yin, UM.
Title: The existence of closed hyperbolic manifolds

Abstract: I will talk about a proof of the existence of closed hyperbolic manifolds with dimension is less or equal to three. Then show the generalization of this approach to any dimension. I will give a concrete example at the end if time permits.

Thursday, September 18.

Speaker: Mario Bonk, UM.
Title: Horned spheres and wild reflections

Abstract: In the 1950's R.H. Bing showed that there are wild reflections on R^3. By definition these are involutions that are not conjugate to linear involutions. In my talk I will discuss his construction which uses Alexander horned spheres and decomposition spaces.

Thursday, September 11.

Speaker: Juan Souto, UM.
Title: Small eigenvalues of the Laplacian

Abstract: I will discuss some older results due to Buser about eigenvalues of the Laplacian of hyperbolic surfaces and then explain the proof of a theorem of Otal asserting that $\lambda_{2g-2}>1/4$ for a closed hyperbolic surface of genus $g$. This result is optimal.