Complex Analysis and Dynamics Seminar
Fall 2009 Schedule
Sep. 4: No meeting
Sep. 11: Jeremy Kahn (Stony Brook University)
We say that an infinitely renormalizable quadratic polynomial has
bounded-primitive 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.
A Priori Bounds for Bounded-Primitive Renormalization
Sep. 18: No meeting
Sep. 25: Jun Hu (Brooklyn College and Graduate Center of CUNY)
We re-develop 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.
Conformally Natural Barycentric Extensions of Continuous Circle Maps
Oct. 2: Frederick Gardiner (Brooklyn College and Graduate Center of CUNY)
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.
Tracking Moving Points on a Riemann Surface
Oct. 9: Guangyuan Zhang (Tsinghua University of Beijing and Graduate Center of CUNY)
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.
Best Bound for the Area-Length Ratio in Ahlfors' Covering Surface Theory
Oct. 16: Joseph Maher (CUNY Staten Island)
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.
Generic Elements in the Mapping Class Group
Oct. 23: Eran Makover (Central Connecticut State University)
In a joint work with Hugo Parlier, we investigate the growth rate of maximal systole length of compact
and non-compact hyperbolic surfaces. This is work in progress, and most of the problems are still open.
Systole Growth on Riemann Surface
Oct. 30: The seminar will feature two talks (note the different times):
The space of homeomorphisms Homeo(S1) of the unit circle
S1 is a classical topological group which acts on S1.
It contains many important subgroups such as the group
Diff(S1) of diffeomorphisms of S1, the group
QS(S1) of quasisymmetric maps S1, the group
Symm(S1) of symmetric maps of S1, the group
Mob(S1) of Mobius maps which preserve S1, and
many more. We use shear coordinates on the Farey tessellation to parametrize
the co-adjoint orbit spaces Mob(S1) \ Homeo(S1),
Mob(S1) \ QS(S1) and
Mob(S1) \ Symm(S1). To our best knowledge, this
gives the only known explicit parametrization of the universal Teichmuller
space T(H) = Mob(S1) \ QS(S1).
1:30-2:30 pm: Dragomir Saric (Queens College of CUNY)
Circle Homeomorphisms and Shears
3:00-4:00 pm: Laura DeMarco (University of Illinois at Chicago)
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.
Polynomial Dynamics and Critical Heights
Nov. 6: Joshua Bowman (Stony Brook University)
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 Arnoux--Yoccoz family of surfaces.
Flat Surfaces: Triangulations, Tessellations, and Limits
Nov. 13: Michael Shub (University of Toronto)
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 x-axis,
i.e. 1/y2. 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, Jean-Pierre 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.
Geodesics in the Condition (Number) Metric
Nov. 20: The seminar will feature two talks (note the different times):
This talk is intended to introduce various background material for the study of hyperbolic
3-manifolds and its interaction with Teichmuller theory. I will talk on the foliations
and incompressible surfaces in quasi-Fuchsian 3-manifolds, induced quasiconformal mapping
between the conformal structures on these surfaces, as well as distances in Teichmuller space.
1:30-2:30: Zeno Huang (CUNY Staten Island)
Quasi-Fuchsian Manifolds and Teichmuller Geometry: An Introduction
3:00-4:00: Biao Wang (University of Toledo)
We prove that if a quasi-Fuchsian 3-manifold 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.
Minimal Surfaces in Quasi-Fuchsian 3-Manifolds
Nov. 27: No meeting
Dec. 4: The seminar will feature two talks (note the different times):
The goal of this talk is to provide a beginner's guide to the
study of the Kahler-Ricci flows. Towards the end, we can also get a
taste of some recent progress in this field and the motivation behind.
1:30-2:30: Zhou Zhang (University of Michigan)
Topological and Analytic Aspects of the Kahler-Ricci Flows
3:00-4:00: Julio Rebelo (University of Toulouse, France)
Consider a (countable) subgroup G of Diff(S1) that is not
discrete in the sense that it contains a non-trivial sequence of
elements converging to the identity. The basic example of these groups is provided by a
non-solvable finitely-generated 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 non-degenerate. As an application, we shall derive a rigidity result for measurable
conjugacies between such groups.
Stationary Measures for Non-Discrete Groups of Diff(S1)
Dec. 11: Justin Malestein (Temple University)
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 self-intersections of a nontrivial element
of the k-th term of the lower central series and derived series of the fundamental
group of a surface. This is joint work with Andrew Putman.
Self-intersections of Curves Deep in the Lower Central Series of a Surface Group