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 Self-Matings 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 self-matings of the above quadratics. Some general results on (quasi)-self-matings of starlike quadratics will also be discussed.

Feb. 27: Karan Puri (Rutgers University, Newark)
Factorization of Isometries of Hyperbolic 4-Space Through a Discreteness Condition

We discuss Gilman's discreteness condition for subgroups of isometries of hyperbolic 3-space and the question of its extension to dimension 4. This extension raises the question of whether an orientation-preserving isometry of hyperbolic 4-space can be factored into the product of two "half-turns" (orientation-preserving 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 4-manifolds 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 non-holomorphic Lefschetz fibrations. This is joint work with Hideki Miyachi.

Mar. 20: No seminar due to Fred Gardiner's 70th Birthday Conference on Mar. 20 and 21 at CUNY's Graduate Center

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 Weil-Petersson 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 2-dimensional homology classes. After considering some further algebraic properties and characterizations of the homology theory, we make connections between certain 2-dimensional 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 spin-off, 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 4-Space

In this talk, we will present a geometric classification of isometries acting on the hyperbolic 4-space.

May 15: Jane Gilman (Rutgers University, Newark)
The Non-Euclidean Euclidean Algorithm

There is an algorithm for determining when a non-elementary two-generator 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 re-interpret the algorithm as a geometric algorithm in the hyperbolic plane using the non-Euclidean distance. The algorithm then becomes a type of Euclidean algorithm using hyperbolic distance, that is, a non-Euclidean Euclidean algorithm. This formulation of the algorithm simplifies the proof of polynomial time complexity.

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