Complex Analysis and Dynamics Seminar

Fall 2010 Schedule

Aug 27: Jason Behrstock (Lehman College and Graduate Center of CUNY)
Curve Complex Projections and Teichmuller Space

We will explain a certain natural way to project elements of the mapping class to simple closed curves on subsurfaces. Generalizing a coordinate system on hyperbolic space, we will use these projections to describe a way to parametrize the mapping class group in terms of these projections; we will also explain a similar parametrization for Teichmuller space. This point of view is useful in several applications; time permitting we shall discuss how we have used this to prove the Rapid Decay property for the mapping class group. This talk will include joint work with Kleiner, Minksy, and Mosher.

Sep. 3: Qian Yin (University of Michigan)
Lattes Maps and Combinatorial Expansion

A Lattes map is a rational map that is obtained from a finite quotient of a conformal torus endomorphism. We characterize Lattes maps by their combinatorial expansion behavior, and deduce new necessary and sufficient conditions for a Thurston map to be topologically conjugate to a Lattes map. In terms of Sullivan's dictionary, this characterization corresponds to Hamenstadt's entropy rigidity theorem.

Sep. 10: No meeting

Sep. 17: Dragomir Saric (Quuens College of CUNY)
Infinitesimal Liouville Currents, Cross-Ratios and Intersection Numbers

Many classical objects on a surface S can be interpreted as cross-ratio functions on the circle at infinity of the universal covering of S. This includes closed curves considered up to homotopy, metrics of negative curvature considered up to isotopy and, in the case of interest here, tangent vectors to the Teichmuller space of complex structures on S. When two cross-ratio functions are sufficiently regular, they have a geometric intersection number, which generalizes the intersection number of two closed curves. In the case of the cross-ratio functions associated to tangent vectors to the Teichmuller space, we show that two such cross-ratio functions have a well-defined geometric intersection number, and that this intersection number is equal to the Weil-Petersson scalar product of the corresponding vectors. This is joint work with F. Bonahon.

Sep. 24: Edson de Faria (Universidade de Sao Paulo, Brazil)
David Homeomorphims and Carleson Boxes

David homeomorphisms are very useful generalizations of quasi-conformal mappings introduced by G. David in 1988. They were used for the first time in Complex Dynamics by P. Haissinski to perform parabolic surgery on rational maps, and later by C. Petersen and S. Zakeri in their study of quadratic polynomials with Siegel disks. In this talk we answer a question posed by S. Zakeri some time ago, concerning the boundary values of such homemorphisms. We construct a family of examples of increasing homeomorphisms of the real line whose local quasi-symmetric distortion blows up almost everywhere, which nevertheless can be realized as the boundary values of David homeomorphisms of the upper half-plane. The construction of such David extensions uses Carleson boxes and a Borel-Cantelli argument. The talk is based on a paper with the same title, to appear in the Annales Academiae Scientiarum Fennicae.

Oct. 1: No meeting

Oct. 8: Igor Rivin (Temple University)
Finding Conformal Maps Between Molecules and Other Stories

We will talk about (and mostly around) the question of figuring out how close metrics on two surfaces (and often curves) are, and show that almost every problem (that can be answered, anyhow) reduces to convex optimization.

Oct. 15: Patrick Hooper (City College of CUNY)
Symmetry Groups of Translation Surface Covers

A translation surface is a Riemann surface equipped with a holomorphic 1-form. The group SL(2,R) acts on the moduli space of translation surfaces, and the subgroup fixing a surface is called the Veech group of the surface. I will discuss Veech groups of Z-covers of compact translation surfaces. In particular, I will use a theorem of Thurston (on stretch maps) to show that some of these Veech groups are of the first kind. I will explain some of the dynamical motivations for studying this question. This is joint work with Barak Weiss.

Oct. 22: No meeting (Linda Keen's conference at CUNY)

Oct. 29: Ege Fujikawa (Chiba University)
The fixed Point Theorem and the Nielsen Realization Problem for Asymptotic Teichmuller Modular Groups

We prove that every finite subgroup of the asymptotic Teichmuller modular group has a common fixed point in the asymptotic Teichmuller space under a certain geometric condition of a Riemann surface, and give an asymptotic version of the Nielsen realization problem.

Nov. 5: Moon Duchin (University of Michigan)
Lengths of Curves in Flat Metrics

It is a classical fact that the length spectrum is quite rigid in hyperbolic geometry: if one specifies the lengths of finitely many (well-chosen) curves on a surface S, then that information determines the hyperbolic metric on S. Suppose one is interested instead in singular flat metrics on S: the metrics induced by (semi-)translation structures or by quadratic differentials, which are of crucial importance in Teichmuller theory and billiards. How rigid is the length spectrum for these metrics? We answer this question and develop some new tools for studying the geometry of flat metrics. This is joint work with Christopher Leininger and Kasra Rafi.

Nov. 12: Linda Keen (Lehman College and Graduate Center of CUNY)
Discrete Groups Outside Teichmuller Spaces

In this talk we will show that there are rigid discrete groups outside certain Teichmuller spaces and that they converge to the boundary in a well-defined manner.

Nov. 19: Ara Basmajian (Hunter College and Graduate Center of CUNY)
The Orthogonal Spectrum of a Hyperbolic Manifold

Dec. 3: The seminar will feature two talks:

1:30-2:30: Yunping Jiang (Queens College and Graduate Center of CUNY)
On Bounded Geometry and Characterization of Rational Maps

2:40-3:40: Youngju Kim (Korea Institute for Advanced Study)
Quasiconformal Mappings and Cross Ratio on the Heisenberg Group

The Heisenberg group is identified with the natural boundary of infinity of the Siegel domain model for the complex hyperbolic plane. It is known that conformal mappings on the Heisenberg group preserve a cross ratio which is an analogue of the classical cross ratio of complex numbers. We will discuss how quasiconformal mappings distort the cross ratio on the Heisenberg group. This is a work in progress.

Dec. 10: Christian Wolf (City College of CUNY)
Regularity of Topological Pressure: From One to Two Dimensional Complex Dynamics

In this talk we discuss the thermodynamics of complex Henon maps which are small perturbations of one-dimensional polynomials. We derive regularity results of the generalized pressure function in a neighborhood of the degenerate map (i.e. the polynomial). We then apply these results to show uniqueness of the measure of maximal dimension as well as discontinuity of Hausdorff dimension at the boundary of the hyperbolicity locus.

Dec. 17: Sergiy Merenkov (University of Illinois at Urbana-Champaign)
Rigidity of Relative Schottky Sets

Let D be a domain contained in the standard n-sphere or the n-dimensional Euclidean space. A relative Schottky set S in D is a subset of D whose complement consists of disjoint open (geometric) balls. Relative Schottky sets are closely related to relative circle domains introduced by Z.-X. He and O. Schramm in mid 90s. In this talk I will discuss rigidity results for relative Schottky sets in relation to quasisymmetric (or quasiconformal) deformations. Some of the main tools used to establish these rigidity results are adapted from the theory of circle packings.

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