Complex Analysis and Dynamics Seminar

Fall 2009 Schedule

Sep. 4: No meeting

Sep. 11: Jeremy Kahn (Stony Brook University)
A Priori Bounds for Bounded-Primitive Renormalization

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.

Sep. 18: No meeting

Sep. 25: Jun Hu (Brooklyn College and Graduate Center of CUNY)
Conformally Natural Barycentric Extensions of Continuous Circle Maps

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.

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 Area-Length 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 non-compact 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):

1:30-2:30 pm: Dragomir Saric (Queens College of CUNY)
Circle Homeomorphisms and Shears

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).

3:00-4:00 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 Arnoux--Yoccoz 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 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.

Nov. 20: The seminar will feature two talks (note the different times):

1:30-2:30: Zeno Huang (CUNY Staten Island)
Quasi-Fuchsian Manifolds and Teichmuller Geometry: An Introduction

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.

3:00-4:00: Biao Wang (University of Toledo)
Minimal Surfaces in Quasi-Fuchsian 3-Manifolds

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.

Nov. 27: No meeting

Dec. 4: The seminar will feature two talks (note the different times):

1:30-2:30: Zhou Zhang (University of Michigan)
Topological and Analytic Aspects of the Kahler-Ricci Flows

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.

3:00-4:00: Julio Rebelo (University of Toulouse, France)
Stationary Measures for Non-Discrete Groups of Diff(S1)

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.

Dec. 11: Justin Malestein (Temple University)
Self-intersections 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 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.

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