# Complex Analysis and Dynamics Seminar

## Spring 2012 Schedule

#### Feb 3: Jun Hu (Brooklyn College and Graduate Center of CUNY) Conformally Natural Extensions of Circle Maps

In this talk, I will give a brief overview of the works on conformally natural extensions of circle maps.

#### Feb 10: Frederick Gardiner (Emeritus, Brooklyn College and Graduate Center of CUNY) Double Poles Extremal Problems

We discuss the following topics: 1. Alternative definitions of extremal length; 2. Extremal problems for measured foliations; 3. The minimum norm principle; 4. The heights mapping; 5. Double pole extremal and a solution to Slodkowski's extension theorem for holomorphic motions; 6. Uniformization; 7. A direct construction of the Weierstrass $\wp$-function.

#### Feb 17: Igor Rivin (Temple University) Critical Points of Random Polynomials

I will discuss the distribution of critical points of a polynomial with the prescribed distribution of zeros.

#### Feb 24: The seminar will feature two talks: 2:00-3:00 Xavier Buff (University of Toulouse) Dynamics on Stable Curves and Rescaling Limits (after M. Arfeux)

Consider a sequence of conjugacy classes of rational maps degenerating in the moduli space. What happens in the limit? Rescaling limits appear in the work of Epstein and have recently been studied by Kiwi, DeMarco and others. The approach we will use is inspired by the work of Epstein on the deformation space $\text{Def}_A^B(f)$ and by the work of Selinger and Koch on Thurston's theorem. Our contribution is to use admissible covers between marked stable curves.

#### 3:30-4:30 Mike Wolf (Rice University) Note: This talk will be in Room 8404The Weil-Petersson Hessian of Length

The length of a closed curve defines a function on Teichmuller space, and we give a short derivation of a concise geometric formula for the Hessian of that function with respect to the Weil-Petersson metric on that space. The convexity of length is apparent and we are also led to an easy proof of Wolpert's result on the coincidence of the Thurston and Weil-Petersson metrics along with some other conclusions.

#### Mar 2: Jane Gilman (Rutgers University at Newark) Conformal Automorphism Groups, Adapted Bases, and Rewriting Systems

Recently there has been renewed interest in the action of a conformal automorphism group on a compact Riemann surface as new more sophisticated group theoretic tools have been applied to the conformal, geometric and topological problem. I will summarize older results obtained when I originated the concept of an adapted homology bases for prime order automorphisms and extend these results to new results for arbitrary finite groups. I will use Broughton's more recent concept of a generating vector combined with the older methods of curve lifting-cutting-pasting and the less ad hoc method of Schreier-Reidemeister rewriting process and elimination of generators and relations. I will survey more recent results of Wootton, Broughton, Weaver, Anderson, Rodriguez and others.

#### Mar 9: Ilesanmi Adeboye (Wesleyan University) Volume in Complex Hyperbolic Geometry

Complex hyperbolic space is the complex analogue of (real) hyperbolic space. The half-plane model of the hyperbolic plane is also a model for complex hyperbolic 1-space. In higher dimensions, complex hyperbolic manifolds are the simplest examples of Riemannian manifolds of variable negative curvature.
In the first half of this talk, I will define complex hyperbolic space and describe elementary aspects of its geometry. In the second half, I will prove an explicit lower bound for the volume of a complex hyperbolic orbifold that depends only on dimension.

#### Mar 16: Dierk Schleicher (Jacobs University, Bremen) News about Newton: Global Dynamics and Efficient Root Finding

Newton's method as a root finder is locally very fast, but the global dynamics seemed difficult to understand, even for complex polynomials in one variable. In particular, there may be open sets of starting points that converge to no roots (but to attracting cycles of higher periods). We describe how to turn Newton's method into an efficient root finder that, in the expected case, needs no more than $O(d^2 \log^5d + d\log|\log \varepsilon |)$ iterations to find all roots of a degree $d$ polynomial with precision $\varepsilon$. We also answer a question by Smale to classify all polynomials for which there are attracting cycles of higher periods (joint with Mikulich).

#### March 23: Joshua Bowman (Stony Brook University) Basins of infinity for polynomial maps of $\mathbb{C}^2$

Basins of attraction in holomorphic dynamics are well understood in dimension 1, much less so in higher dimensions. We will consider regular polynomial maps of $\mathbb{C}^n$ (maps which extend to endomorphisms of $\mathbb{P}^n$) and describe some tools for studying their basins of infinity. In dimension $n = 2$, the analysis becomes somewhat simpler, and we show that there exist maps of $\mathbb{C}^2$ whose basins of infinity have infinitely generated second homology.

#### March 30: Babak Modami (Yale University) Weil-Petersson Geodesics, Stability and Divergence

The Weil-Petersson (WP) metric is an incomplete metric on the Teichmüller space of a surface, with negative sectional curvatures. The WP completion locus is a union of strata which intersect each other in a pattern encoded by the curve complex of the surface. In this talk we provide examples of diverging WP geodesic rays which travel closer and closer to a chain of completion strata, as well as closed WP geodesics in the thin part of the moduli space. These constructions are based on stability of a class of hierarchy paths in the pants graph, a quasi-isometric model for the WP metric, as well as some synthetic properties of the WP metric and its geodesics.

#### Apr 20: Dick Canary (University of Michigan) Dynamics on Character Varieties

Given a closed surface $F$, the mapping class group $\text{Mod}(F)$ of $F$ acts on the $\text{PSL}(2,{\mathbb R})$-character variety. This action is properly discontinuous on the components associated to the Teichmuller spaces of $F$ and of $F$ with the opposite orientation. It is conjectured to act ergodically on all other components.
In this talk, we consider a compact 3-manifold $M$ with boundary. The outer automorphism group of the fundamental group of $M$ acts on the $\text{PSL}(2,{\mathbb C})$-character variety of $M$. It acts properly discontinuously on the interior of the space of (characters of) discrete faithful representations. We will discuss the dynamics of the action on the entire character variety. In particular, we will discuss work of Canary, Lee, Magid, Minsky and Storm which exhibits domains of discontinuity for this action which are larger than the interior of the space of discrete faithful representations.

#### May 4: Clifford Earle (Emeritus, Cornell University) Simple closed Geodesics on Riemann Surfaces

Let $C$ be a simple closed geodesic on a hyperbolic Riemann surface of finite area. There are several geometric quantities associated to $C$. One is its arc length. Others are the module and area of a collar about $C$. Another is the minimal dilatation of a Dehn twist about $C$. We state some inequalities among these numbers. Some are classical, and some are obtained by plumbing with Kleinian groups. This is part of a joint project with Albert Marden.

#### May 11: Sudeb Mitra (Queens College and Graduate Center of CUNY) From Teichmüller Contraction to Schwarz's Lemma

The principle of Teichmüller contraction was first introduced by Gardiner. Later, Earle proved a sharp form of Teichmüller contraction and used it to obtain a form of Schwarz's lemma for the Teichmüller space of a hyperbolic Riemann surface.
Let $E$ be a closed set in the Riemann sphere containing $0$, $1$, $\infty$, and let $T(E)$ denote its Teichmüller space. In this talk, we will first extend Earle's sharp form of Teichmüller contraction to the space $T(E)$. Using a lifting theorem, we will then discuss isometries of holomorphic maps from the open unit disk into $T(E)$. Finally, we will obtain a Schwarz's lemma for $T(E)$, generalizing Earle's theorem.
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