Complex Analysis and Dynamics Seminar

Spring 2009 Schedule

Feb. 6: Laura DeMarco (University of Illinois at Chicago)
Escape Combinatorics for Polynomial Dynamics
In this talk, I will introduce a combinatorial method for studying the
dynamics of complex polynomials. It can be used to distinguish topological
conjugacy classes of polynomials, to study global structure of the moduli
space of polynomials, or to (re)prove that the Mandelbrot set is connected.
This is joint work with Kevin Pilgrim.

Feb. 13: No meeting

Feb. 20: Ross Flek (Graduate Center of CUNY)
Boundaries of Bounded Fatou Components of Quadratic Maps and the Structure
of Julia Sets of SelfMatings of Starlike Quadratics
I will discuss a recent paper (joint with Linda Keen) which
addresses some of the aspects of my thesis. We characterize
those external rays that land on the boundary of bounded Fatou components of hyperbolic
and parabolic quadratic polynomials. For maps outside the main cardioid of the
Mandelbrot set, we prove that these rays form a Cantor subset of the circle at
infinity. Our techniques involve constructions based on the theory of orbit
portraits and unit disk laminations for quadratic maps. This classification
is important since it provides a way to characterize buried Julia sets of a
class of degree two rational maps which are conjugate to selfmatings of the
above quadratics. Some general results on (quasi)selfmatings of
starlike quadratics will also be discussed.

Feb. 27: Karan Puri (Rutgers University, Newark)
Factorization of Isometries of Hyperbolic 4Space Through a Discreteness Condition
We discuss Gilman's discreteness condition for subgroups of isometries
of hyperbolic 3space and the question of its extension to dimension 4.
This extension raises the question of whether an orientationpreserving
isometry of hyperbolic 4space can be factored into the product of two
"halfturns" (orientationpreserving isometries of order 2). We follow
techniques developed by Wilker to address this question and use these to make
a construction analogous to Gilman's.

Mar. 6: William Goldman (University of Maryland at College Park)
Deformation Spaces of Surface Group Representations
In this talk I will survey two closely
related dynamical systems on the space of flat
bundles over orientable surfaces. One is discrete
and the other is continuous. The discrete dynamical
system arises from the action of the mapping class
group. The continuous system is generated by
Hamiltonian flows of character functions on the
surface.

Mar. 13: Hiroshige Shiga (Tokyo Institute of Technology)
Monodromy and Complex Structures of Surface Fibrations
We consider real 4manifolds which
are smooth surface fibrations over a smooth oriented surface. It is known
that the Lefschetz fibration, a special kind of surface
fibration, plays an important role by the remarkable works of
Donaldson and Gompf and there are many works on the Lefschetz
fibrations. If the fibration admits a certain complex structure, it can be
viewed as a holomorphic family of Riemann surfaces and
therefore the Teichmuller theory comes in. In this talk, we shall
consider the monodromies of surface fibrations and give an example of
nonholomorphic Lefschetz fibrations. This is joint work with Hideki
Miyachi.


Mar. 27: Yunping Jiang (Queens College and Graduate Center of CUNY)
Function Model for the Teichmuller Space of a Closed Hypberbolic Riemann Surface
We introduce a function model for the Teichmuller space of a
closed hyperbolic Riemann surface. From this function model, we define a
new metric by using the maximum norm on the function space. We prove that the identity
map from the Teichmuller space equipped with the usual Teichmuller metric
to the Teichmuller space equipped with this new metric is uniformly
continuous with continuous inverse. In particular, the new metric induces the same
topology as the Teichmuller metric. We also show a relation between the pressure metric and the WeilPetersson
metric viewed by this function model.

Apr. 3: Ara Basmajian (Hunter College and Graduate Center of CUNY)
Hyperbolic Motions as Commutators II
This will be a continuation of my talk in the Fall.

Apr. 10 and 17: No seminar (Spring Recess)

Apr. 24: Joel Zablow (Long Island University, Brooklyn)
The Dehn Quandle, its Homology, and an Application to Lefschetz Fibrations
The Dehn quandle on an orientable surface is defined by the action of Dehn
twists about circles, upon the collection of (isotopy classes of) such circles.
We consider some relations in this quandle and a general quandle homology
theory, applied to the Dehn quandle. The relations turn out to be
2dimensional homology classes. After considering some further algebraic
properties and characterizations of the homology theory, we make connections
between certain 2dimensional homology classes and Lefschetz fibrations over a
disk. We then raise some conjectures and questions regarding generalizing the
Dehn quandle to laminations on surfaces.

May 1: Sudeb Mitra (Queens College of CUNY)
Holomorphic Families of Mobius Groups
A normalized holomorphic motion of a closed set in the Riemann sphere, defined
over a simply connected complex Banach manifold, can be extended to a
normalized quasiconformal motion of the sphere, in the sense of Sullivan and
Thurston. In this talk, we will show that if the given holomorphic motion has
a group equivariance property, then the extended quasiconformal motion will
also have the same property. As a spinoff, we obtain a generalization of a
theorem of Bers on holomorphic families of isomorphisms of Mobius groups. If
time permits, we may discuss an application. This is part of a joint work
with Hiroshige Shiga.

May 8: Youngju Kim (Lehman College of CUNY)
Classification of Isometries Acting on Hyperbolic 4Space
In this talk, we will present a geometric classification of isometries
acting on the hyperbolic 4space.

May 15: Jane Gilman (Rutgers University, Newark)
The NonEuclidean Euclidean Algorithm
There is an algorithm for determining when a nonelementary twogenerator
subgroup of PSL(2,R) is or is not discrete. There are several different ways
in which to interpret the algorithm: as a geometric algorithm, as a type of
BSS machine or as an algorithm where the entries in the two matrices are
algebraic numbers lying in a finite extension of the rationals. The different
forms of the algorithm were found by Gilman and/or Jiang to be of polynomial
time complexity. In this talk we reinterpret the algorithm as a geometric
algorithm in the hyperbolic plane using the nonEuclidean distance. The
algorithm then becomes a type of Euclidean algorithm using hyperbolic
distance, that is, a nonEuclidean Euclidean algorithm. This formulation of
the algorithm simplifies the proof of polynomial time complexity.
