The Ahlfors-Bers Colloquium 2014 at Yale

Wendelin Werner (ETH Zürich): Some aspects of conformal invariance can survive within fractal carpets in the plane. In the present talk, I will survey how it is possible to make sense in a rather precise way of certain of these ideas in the special case of certain random -- yet very natural -- carpets, often referred too as conformal loop ensembles. In particular, after briefly introducing these random fractals, we shall describe the analog of a local Schramm-Loewner Evolution curve within this carpets, that can be viewed as a continuous critical percolation interface within these fractal domains. Most of this talk is based on joint work with Jason Miller and Scott Sheffield.

David Dumas (UIC): Thurston's skinning map is a holomorphic map between Teichmüllerspaces that arises in the construction of hyperbolic structures oncompact 3-manifolds. I will describe the theory and implementation ofa computer program that computes the images of skinning maps in somecases where the Teichmueller space has complex dimension one. The keyto the method is that each point in the image of the skinning maprepresents an intersection between two submanifolds of the SL(2,C)character variety of a surface group. The skinning image is computedby tracking the movement of these intersections as one of thevarieties (the Bers slice) is changed. This is joint work with Richard Kent.

Jon Chaika (University of Utah): A basic question in dynamical systems is when are two systemsisomorphic. Starting from rotations of the circle and flows on tori wewill talk about the fact that typical interval exchanges and flows on flatsurfaces are not isomorphic. In fact, they satisfy a stronger propertycalled being disjoint. We will mention some consequences of these resultsand open questions. This includes joint work with Vaibhav Gadre and PascalHubert.

Stergios Antonakoudis (Cambridge University): From a complex analytic perspective, Teichmüller spaces can be realized as contractible bounded domains in complex vector spaces by theBers embeddings. Bounded Symmetric domains constitute another class ofbounded domains that has been extensively studied in the past. In thistalk we will study isometric maps between these two important classes ofbounded domains equipped with their intrinsic Kobayashi metric.

Nick Makarov (Caltech): The talk will be about conformal dynamics of Schwarz reflections. Typical simple example:consider a cardioid sitting inside a closed disc; the dynamical system is generated by the reflectionin the cardioid and the reflection in the circle, the boundary of the disc.In a certain (but rather precise) sense, the Schwarz reflection is a hybrid of a rational map and a reflection group. Joint work with Seung-Yeop Lee.

Peter Haïssinsky (Toulouse): The talk will be devoted to discussing background andingredients for the proof of the following theorem: a finitely generated groupquasi-isometric to a convex-cocompact Kleinian group contains a finite index subgroupisomorphic to a convex-cocompact Kleinian group.

Kasra Rafi (University of Toronto): We study the large scale geometry of Teichmüller space equipped with the Teichmüller metric.We show that, except for low complexity cases, any self quasi-isometry of Teichmüller spaceis a bounded distance away from an isometry of Teichmüller space. Our approach is differentfrom the proof of rigidity of the mapping class group in that the arguments are local: we examinelarge balls in Teichmüller space instead of the asymptotic cone. This is jointwork with Alex Eskin and Howard Masur.

Sarah Koch (University of Michigan): In his last paper, "Entropy in Dimension One," W. Thurston completely characterized which algebraic integers arise as exp(entropy(f)), where f is a postcritically finite real map of a closed interval. On page 1 of this paper, there is a spectacular image of a subset in C comprised of roots of polynomials which come from entropy values associated to the dynamics of quadratic polynomials. This set displays some amazing fractal structure which can be (somewhat) understood when viewed as a distinguished subset of parameter space for a particular family of iterated function systems (IFS). In this talk, we investigate other distinguished subsets in parameter space for this IFS and prove some surprising results about their topology. We compare/contrast the discussion of this parameter space to the study of the parameter space for quadratic polynomials pc:z → z2+c. This is joint work with D. Calegari and A. Walker.

Jeremy Kahn (CUNY Graduate Center): In a largely expository talk, I will summarize the results leading up to the Virtual Haken and Virtual Fibered Theorem for three manifolds, including 1. The Geometrization Theorem of Thurston and Perelman 2. The Surface Subgroup Theorem of the speaker and Vladimir Markovic 3. The cube complex theory of Daniel Wise and others 4. Ian Agol's criterion for virtual fibering and his proof of the Virtual Haken Theorem The emphasis will be on the flow of ideas rather than on detailed statements and proofs.

Mladen Bestvina (University of Utah): I will survey recent progress on the geometry of Outerspace, and compare similarities and differences with Teichmüllerspace.