David Fisher - Quasi-isometric embeddings of non-uniform lattices.
We study quasi-isometric embeddings of one non-uniform lattice in another. For example, we show that any quasi-isometric embedding of SL(n,Z) into SL(n, Z[i]) is a bounded distance from a homomorphism on a subgroup of finite index. The rigidity phenomenon holds more generally but is surprisingly much more fragile than the rigidity theorems for quasi-isometries of non-uniform lattices. This is joint work with Thang Nguyen. I will also discuss some related prior work with K. Whyte.
Dave Futer - Abundant quasfuchsian surfaces in cusped hyperbolic 3-manifolds
I will discuss a proof that a cusped hyperbolic 3-manifold M contains an abundant collection of immersed, quasifuchsian surfaces. These surfaces are abundant in the sense that their boundaries separate any pair of points on the sphere at infinity. As a corollary, we recover Wise's theorem that the fundamental group of M is cubulated. This is joint work with Daryl Cooper.
Camille Horbez - ThePacman compactification of Outer space
Sarah Koch - Deforming rational maps
For a rational map on the Riemann sphere, there is an associated deformation space (originally defined by A. Epstein). Epstein proved that this deformation space is a complex submanifold of a certain Teichmueller space. The arguments in his proof are local; not much is known about the global topology of the deformation space in general. We present a concrete example of this construction in which the topology can be relatively understood. These deformation spaces are related to dynamically defined subvarieties in the moduli space of rational maps. I will mention some fundamental open questions in complex dynamics concerning the topology of these objects. This is joint work with E. Hironaka.
Balázs Strenner - Construction of pseudo-Anosov maps and a conjecture of Penner
There are many constructions of pseudo-Anosov elements of mapping class groups of surfaces. Some of them are known to generate all pseudo-Anosov mapping classes, others are known not to. In 1988, Penner gave a very general construction of pseudo-Anosov mapping classes, and he conjectured that all pseudo-Anosov mapping classes arise this way up to finite power. This conjecture was known to be true on some simple surfaces, including the torus, but has otherwise remained open. I will discuss the proof (joint work with Hyunshik Shin) that the conjecture is false for most surfaces.
Tengren Zhang - Degeneration of Hitchin representations
I will describe an analog of the Fenchel-Nielsen coordinates on the Hitchin component, and then use these coordinates to define a large family of deformations in the Hitchin component called "internal sequences". Then, I will explain some geometric properties of these internal sequences, which allows us to conclude some structural similarities and differences between the higher Hitchin components and Teichmuller space.