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 orientationpreserving) isometry of order two a
"halfturn." In dimensions two and three it is well known that any nontrivial
isometry of hyperbolic space is the product of two halfturns, and is the
commutator of two orientationpreserving 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 1puncture 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
Homeo_{0}(S) of those which are homotopic to the identity. The natural
projection Homeo(S) → MC(S)=Homeo(S)/Homeo_{0}(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 WeilPeterson Ehrenpreis Conjecture
The Ehrenpreis conjecture states that given any two compact hyperbolic Riemann surfaces,
or any two noncompact 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 WeilPetersson metric,
in the case where the two surfaces are noncompact. 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 nonzero 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 wellknown
TeichmullerWittichBelinski theorem.

Dec. 12: The seminar will feature two talks:
2:003:00 pm: Yair Minsky (Yale University)
WeilPetersson Geodesics, Bounded Geometry and Bounded Combinatorics
We develop aspects of a
boundary theory for geodesic rays in the moduli space of surfaces
with its WeilPetersson 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.
3:304:30 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.
