Complex Analysis and Dynamics Seminar

Fall 2009 Schedule

Sep. 4: No meeting

Sep. 11: Jeremy Kahn (Stony Brook University)
A Priori Bounds for BoundedPrimitive Renormalization
We say that an infinitely renormalizable quadratic polynomial has
boundedprimitive type if we can find an infinite sequence of primitive
renormalization times such that the ratio between consecutive terms of the sequence
is bounded. We prove that any such polynomial has the a priori bounds:
there is a lower bound on the modulus of all renormalizations.
This implies that the Mandelbrot set is locally connected at the associated parameter values.

Sep. 18: No meeting

Sep. 25: Jun Hu (Brooklyn College and Graduate Center of CUNY)
Conformally Natural Barycentric Extensions of Continuous Circle Maps
We redevelop some results of Douady and Earle on conformally natural barycentric extensions of circle
homeomorphisms through a geometric approach. This approach enables us to generalize some aspects of
the theory to continuous circle maps of degree greater than 1. This is joint work with Oleg Muzician.

Oct. 2: Frederick Gardiner (Brooklyn College and Graduate Center of CUNY)
Tracking Moving Points on a Riemann Surface
We consider the natural Teichmuller and Kobayashi metrics on the complex manifold
of motions of n distinct points on a fixed Riemann surface and the properties of the
holomorphic evaluation map. When there is just one point this is the Bers' fiber space
and when there is more than one point the manifold has an interesting boundary.

Oct. 9: Guangyuan Zhang (Tsinghua University of Beijing and Graduate Center of CUNY)
Best Bound for the AreaLength Ratio in Ahlfors' Covering Surface Theory
Let f be a holomorphic mapping from a disk D into the Riemann sphere S
omitting 0, 1 and infinity. A basic consequence of Ahlfors' covering surface theory is
the inequality A < h L , where h is a constant independent of f,
A is the area of the image of the disk D, and
L is the length of the image of boundary of D, both counting multiplicity.
In this talk, we will give the best lower bound for the constant h. The starting point
is a classical isoperimetric inequality published in 1905.

Oct. 16: Joseph Maher (CUNY Staten Island)
Generic Elements in the Mapping Class Group
This is an introductory talk, intended to be accessible to
graduate students. The mapping class group of a surface is the group of
homeomorphisms of the surface to itself, up to isotopy, and is of interest
in a variety of different areas of mathematics. One way to investigate
this group is to study the geometry of the spaces on which it acts, such
as Teichmuller space, or the complex of curves. We will describe some
notions of what it might mean for a mapping class group element to be
generic, and discuss some collections of generic elements.

Oct. 23: Eran Makover (Central Connecticut State University)
Systole Growth on Riemann Surface
In a joint work with Hugo Parlier, we investigate the growth rate of maximal systole length of compact
and noncompact hyperbolic surfaces. This is work in progress, and most of the problems are still open.

Oct. 30: The seminar will feature two talks (note the different times):
1:302:30 pm: Dragomir Saric (Queens College of CUNY)
Circle Homeomorphisms and Shears
The space of homeomorphisms Homeo(S^{1}) of the unit circle
S^{1} is a classical topological group which acts on S^{1}.
It contains many important subgroups such as the group
Diff(S^{1}) of diffeomorphisms of S^{1}, the group
QS(S^{1}) of quasisymmetric maps S^{1}, the group
Symm(S^{1}) of symmetric maps of S^{1}, the group
Mob(S^{1}) of Mobius maps which preserve S^{1}, and
many more. We use shear coordinates on the Farey tessellation to parametrize
the coadjoint orbit spaces Mob(S^{1}) \ Homeo(S^{1}),
Mob(S^{1}) \ QS(S^{1}) and
Mob(S^{1}) \ Symm(S^{1}). To our best knowledge, this
gives the only known explicit parametrization of the universal Teichmuller
space T(H) = Mob(S^{1}) \ QS(S^{1}).
3:004:00 pm: Laura DeMarco (University of Illinois at Chicago)
Polynomial Dynamics and Critical Heights
The moduli space of polynomials has a natural decomposition based on escape rates of critical
points. I will describe this structure and explain how it relates to the organization (and classification)
of topological conjugacy classes. This is joint work with Kevin Pilgrim.

Nov. 6: Joshua Bowman (Stony Brook University)
Flat Surfaces: Triangulations, Tessellations, and Limits
After reviewing some of the basic definitions and results regarding flat surfaces,
we will discuss their Delaunay triangulations, which are canonical for generic
surfaces, and some applications of these, including tessellations of Teichmuller
disks and an extension of the ArnouxYoccoz family of surfaces.

Nov. 13: Michael Shub (University of Toronto)
Geodesics in the Condition (Number) Metric
The hyperbolic metric is constructed by multiplying the usual Riemannian inner product
structure on the upper half plane by the the square of the inverse of the distance to the xaxis,
i.e. 1/y^{2}. In the new metric the function  ln y is convex. We investigate
similar constructions in other spaces. We call the resulting Riemann structures condition Riemann structures
because our main interest is the condition number of linear algebra and polynomial system solving.
The fact that the condition number is only Lipschitz, not smooth, complicates the situation.
Connections are drawn to the complexity of polynomial system solving by homotopy methods.
This is joint work with Carlos Beltran, JeanPierre Dedieu and Gregorio Malajovich. The lecture will preview
some of the material that will be included in the series of lectures I will be giving as a Topics in Complexity
course at GC this spring.

Nov. 20: The seminar will feature two talks (note the different times):
1:302:30: Zeno Huang (CUNY Staten Island)
QuasiFuchsian Manifolds and Teichmuller Geometry: An Introduction
This talk is intended to introduce various background material for the study of hyperbolic
3manifolds and its interaction with Teichmuller theory. I will talk on the foliations
and incompressible surfaces in quasiFuchsian 3manifolds, induced quasiconformal mapping
between the conformal structures on these surfaces, as well as distances in Teichmuller space.
3:004:00: Biao Wang (University of Toledo)
Minimal Surfaces in QuasiFuchsian 3Manifolds
We prove that if a quasiFuchsian 3manifold M contains a simple closed geodesic with complex length L=l+it such
that t/l >> 1, then it contains at least two minimal surfaces which are incompressible in M.

Nov. 27: No meeting

Dec. 4: The seminar will feature two talks (note the different times):
1:302:30: Zhou Zhang (University of Michigan)
Topological and Analytic Aspects of the KahlerRicci Flows
The goal of this talk is to provide a beginner's guide to the
study of the KahlerRicci flows. Towards the end, we can also get a
taste of some recent progress in this field and the motivation behind.
3:004:00: Julio Rebelo (University of Toulouse, France)
Stationary Measures for NonDiscrete Groups of Diff(S^{1})
Consider a (countable) subgroup G of Diff(S^{1}) that is not
discrete in the sense that it contains a nontrivial sequence of
elements converging to the identity. The basic example of these groups is provided by a
nonsolvable finitelygenerated group G admitting a generating set
sufficiently close to the identity. Suppose in addition
that G possesses no finite orbit. A consequence of a theorem of
Deroin, Kleptsyn and Navas is that G has a unique stationary measure μ on the circle
(for a fixed probability measure ν on G). In this talk we will show
that μ must be absolutely continuous with respect to Lebesgue measure provided
that ν is nondegenerate. As an application, we shall derive a rigidity result for measurable
conjugacies between such groups.

Dec. 11: Justin Malestein (Temple University)
Selfintersections of Curves Deep in the Lower Central Series of a Surface Group
In this talk, we relate topological aspects of a curve on
a surface to its algebraic propertries. In particular, we will give various
estimates of the minimal number of selfintersections of a nontrivial element
of the kth term of the lower central series and derived series of the fundamental
group of a surface. This is joint work with Andrew Putman.
