Complex Analysis and Dynamics Seminar

Fall 2008 Schedule

Sep. 12: Ara Basmajian (Hunter College and GC of CUNY)
Hyperbolic Motions as Commutators

We call a (not necessarily orientation-preserving) isometry of order two a "half-turn." In dimensions two and three it is well known that any non-trivial isometry of hyperbolic space is the product of two half-turns, and is the commutator of two orientation-preserving isometries. In this talk, we will discuss the situation in higher dimensions. This is joint work with Bernard Maskit.

Sep. 19: No meeting

Sep. 26: Jun Hu (Brooklyn College and GC of CUNY)
Kobayashi Metric on the Teichmuller Space of Symmetric Circle Homeomorphisms

We show that Kobayashi and Teichmuller metrics coincide on the Teichmuller space of symmetric circle homeomorphisms. This is joint work with Yunping Jiang and Zhe Wang.

Oct. 3: Katsuhiko Matsuzaki (Okayama University and Wesleyan University)
Asymptotically Elliptic Modular Transformations of Teichmuller Space

We survey several properties of the action of a Teichmuller modular transformation that has a fixed point on the asymptotic Teichmuller space.

Oct. 10: No meeting

Oct. 17: Xiaobo Liu (Columbia University)
Quantum Teichmuller Spaces

We will briefly give the motivation to quantize the Teichmuller space, then introduce the algebraic formalism, representation theory, and the construction of invariants. The case of the 1-puncture torus will be discussed in detail.

Oct. 24: Dragomir Saric (Queens College of CUNY)
The Mapping Class Group Cannot be Realized by Homeomorphisms

Let S be a closed surface of genus at least two. The mapping class group MC(S) is the quotient of the group Homeo(S) of homeomorphisms of S by the normal subgroup Homeo0(S) of those which are homotopic to the identity. The natural projection Homeo(S) → MC(S)=Homeo(S)/Homeo0(S) is a group homomorphism. Nielsen asked whether this projection has a homomorphic section MC(S) → Homeo(S), i.e., whether MC(S) can be realized as a subgroup of Homeo(S). We show the answer is negative for all surfaces of genus at least two. This is joint work with V. Markovic.

Oct. 31: No meeting

Nov. 7: Hugo Parlier (IGAT Institute, Switzerland)
Bers' Constant for Punctured Spheres and Hyperelliptic Surfaces

Lipman Bers showed that one can cut a finite area hyperbolic surface along disjoint "short" curves so that the result is a set of three holed spheres. Here the term "short" means that the length of each curve is bounded by a constant (Bers' constant) which only depends on the topology of the surface and not on the metric. The best upper and lower bounds on Bers' constant are due to Peter Buser who also conjectured the existence of a universal constant C such that Bers' constant is bounded above by C times the square root of the area (which is linear in the Euler characteristic). The goal of this talk is to present a solution to this conjecture for punctured spheres and hyperelliptic surfaces. This is joint work with Florent Balacheff.

Nov. 14: Jane Hawkins (University of North Carolina at Chapel Hill)
Poles in Parameter Space of Iterated Elliptic Functions

We discuss the iteration of the Weierstrass ℘-function (and other elliptic functions) with a view to understanding the dependence of the dynamics on the underlying period lattice. We look at one example in detail that illustrates many of the general phenomena; namely, square lattices. We show that the parameter space shows both the expected Mandelbrot structure and the poles that are evident in the corresponding Julia sets as well.

Nov. 21: Jeremy Kahn (Stony Brook University)
Random Ideal Triangulations and the Weil-Peterson Ehrenpreis Conjecture

The Ehrenpreis conjecture states that given any two compact hyperbolic Riemann surfaces, or any two non-compact finite area hyperbolic Riemann surfaces, there are finite covers of the two surfaces that are arbitrarily close in the Teichmuller metric. We prove the same statement for the normalized Weil-Petersson metric, in the case where the two surfaces are non-compact. In the course of doing so we construct "Random ideal triangulations" of the covers, where fairly accurate estimations of the proportion of each immersed triangle can be made.

Dec. 5: Melkana Brakalova (Fordham University)
On Conformality at a Point Using a Geometric Approach

This talk will focus on the local behavior of homeomorphisms in the plane that are absolutely continuous on lines with an a.e. positive Jacobian. Such homeomorphisms are generalizations of quasiconformal mappings; in particular, their real dilatation need not be bounded. We present a geometric necessary and sufficient condition for conformality of such maps at a point using extremal length techniques. (Recall that a homeomorphism f, normalized by f(0)=0, is conformal at z=0 if f(z)/z tends to a non-zero limit as z tends to 0.) To obtain such a condition, we study the behavior of the images of annuli, families of radial segments, and arcs of logarithmic spirals in those annuli, as z approaches 0, in terms of their extremal lengths. We provide representation formulas and estimates for such extremal lengths using directional dilatations and apply them to obtain analytic sufficient (but not necessary) conditions for conformality that extend previously known results, new and old, including the well-known Teichmuller-Wittich-Belinski theorem.

Dec. 12: The seminar will feature two talks:

2:00-3:00 pm: Yair Minsky (Yale University)
Weil-Petersson Geodesics, Bounded Geometry and Bounded Combinatorics

We develop aspects of a boundary theory for geodesic rays in the moduli space of surfaces with its Weil-Petersson metric. To each ray we associate an "ending lamination", in analogy with the vertical foliations associated to geodesics in the Teichmuller metric. However the relationship is considerably less explicit and robust in our setting. The correspondence between rays and laminations is a bijection for recurrent rays and has dynamical consequences. We are also able to give a combinatorial criterion on the laminations which is equivalent to precompactness for the rays. Many other questions remain open. This is Joint work with J. Brock and H. Masur.

3:30-4:30 pm: Ege Fujikawa (Chiba University, Japan)
The Intermediate Teichmuller Spaces

To consider the structure of the moduli space of a Riemann surface of infinite type, we introduce a new space which we call the intermediate Teichmuller space, between the Teichmuller space and the moduli space. The stable quasiconformal mapping class group is a group of quasiconformal mapping classes of a Riemann surface that are homotopic to the identity outside some topologically finite subsurface. Its analytic counterpart is a group of mapping classes that act on the asymptotic Teichmuller space trivially. The intermediate Teichmuller space is the quotient space of the Teichmuller space by the asymptotically trivial mapping class group. Its automorphism group is canonically isomorphic to the asymptotic Teichmuller modular group.

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