Complex Analysis and Dynamics Seminar

Spring 2008 Schedule

Feb. 1: Fred Gardiner (Brooklyn College and GC of CUNY)
The Minimum Norm Principle, Extremal Length and Lines of Minima

To a surface, a conformal structure on it, and a measured foliation there corresponds an extremal length, which is realized by minimizing a Dirichlet integral. The extremal metic realizing the extremal length is the square root of the absolute value of a holomorphic quadratic differential q whose horizontal trajectories and vertical measure realize the measure class of the measured foliation. As the conformal structure varies the extremal length varies differentiably and the derivative is realized by q. Lines of minima are Teichmueller lines obtained by minimizing over conformal structures the product of extremal lengths of transversely realizable measured foliations. Along the line the product is constantly equal to the same minimum value and at every point not on the line the product is larger.

Feb. 8: Martin Bridgeman (Boston College)
Hausdorff Dimension under Bending Deformations and the Weil-Petersson Metric

We analyse how the Hausdorff dimension of the limit set of a Kleinian group changes near the fuchsian locus in quasifuchsian space of a surface. We describe a metric on Teichmuller space obtained by taking the second derivative of Hausdorff dimension and show that this new metric is bounded below by the classical Weil-Petersson metric. We use this to relate the change in Hausdorff dimension under bending along a measured lamination to the length in the Weil-Petersson metric of the associated earthquake vector of the lamination. This is joint work with Ed Taylor.

Feb 22: Linda Keen (Lehman College and GC of CUNY)
Enumerating Palindromes in Free Groups of Rank Two

In this talk, based on joint work with Jane Gilman, I will talk about a new enumeration scheme for generators for free groups of rank two. These generators are palindromes, or products of palindromes. I will indicate how this can be used to study Kleinian groups that are free groups on two generators.

Feb. 29: Peter Storm (University of Pennsylvania)
Deformation a Hyperbolic 4-Orbifold

It is well known that Thurston's beautiful deformation theory of hyperbolic structures is mostly useless in dimensions > 3. Steve Kerckhoff and I have been studying a new example of a hyperbolic deformation in 4 dimensions which produces an infinite number of new hyperbolic 4-orbifolds with interesting properties. The talk will attempt to motivate this work. It will be aimed at a general geometry/topology audience.

Mar. 7: Matt Feiszli (Brown University)
The Universal Teichmuller Space, Computer Vision, and Data Compression

The two classical models of the universal Teichmuller space provide an isomorphism between quasisymmetric self-maps of the circle and quasicircles. Motivated by computer vision we consider the case when the quasicircles are smooth; we take domains with smooth boundaries and study paths through shape space which join them to the unit disk. We first develop a differential equation for the conformal map from the convex hull boundary to the disk; the conformal structure of the hyperbolic convex hull can be understood in terms of the medial axis of the domain, and this differential equation acts as a retraction along the medial axis. This in turn leads to explicit estimates for geometric quantities like geodesic lengths, extremal distances, and boundary derivatives; these estimates have natural interpretations in terms of the medial axis. Finally, we will review Kolmogorov's theory of epsilon-entropy and apply our geometric results to produce a family of compressors for 2-dimensional shapes: given a shape, the compressor produces an epsilon-approximating shape with short expected description length. We will discuss the relevant constructions and the differential of the isomorphism between self-maps of the circle and shapes in order to get some understanding of the performance of the method.

Mar. 14: The seminar will feature two talks:

2:00-3:00 pm: Fred Gardiner (Brooklyn College and GC of CUNY)
The Minimum Norm Principle for Measured Foliations and Extremal Length Geometry of Teichmuller Space

This will be a continuation of my February 1st talk.

4:00-5:00 pm: Shawn Rafalski (Williams College)
Immersions of Hyperbolic Turnovers

Take two copies of a hyperbolic triangle with interior angles Pi/p, Pi/q and Pi/r, where p, q and r are integers, and identify these two triangles together in the natural way along their boundaries. The result is called a hyperbolic turnover, and is a specific example of a two-dimensional hyperbolic orbifold. In this talk, we will see that mapping a turnover by an immersion (which is not an embedding) into a hyperbolic three-orbifold places strong restrictions on, among other things, the volume of the three-orbifold.

Mar. 21: No meeting

Mar. 28: John Parker (University of Durham, UK)
Unfaithful Triangle Groups and the Hunt for Complex Hyperbolic Lattices

A lattice is a group of isometries of a metric space that acts properly discontinuously and for which the quotient space has finite volume. A triangle group is a group generated by reflections in the sides of a triangle. We know relatively few examples of complex hyperbolic lattices. Deligne and Mostow, using ideas that go back to Picard, gave a family of lattices which are triangle groups with extra relations. These include the first examples of non-arithmetic complex hyperbolic lattices due to Mostow. Recently Deraux constructed a new example of a complex hyperbolic lattice that is also a triangle group with extra relations. In this talk I will give an elementary account of the above constructions and then outline a programme (which is joint work with Julien Paupert) for finding other triangle groups that may be candidates for lattices.

Apr. 4: Igor Rivin (Temple University)
Asymptotic Geometry of Convex Sets in Hyperbolic Space

We study convex sets C of finite but non-zero volume in Hn and En. We show that the intersection of any such set with the ideal boundary of Hn has Minkowski (and thus Hausdorff) dimension of at most (n-1)/2, and this bound is sharp. In the hyperbolic case, we show that for any k <= (n-1)/2 there is a bounded section S of C through any prescribed point p, and we show an upper bound on the radius of the ball centered at p containing such a section. We show similar bounds for sections through the origin of convex body in En, and give asymptotic estimates as 1 << k << n.

Apr. 11: Saeed Zakeri (Queens College and GC of CUNY)
Siegel Disks of a Class of Entire Maps

Let f be an entire map of the form P(z) exp(Q(z)), where P and Q are polynomials. Following a method pioneered by Shishikura, we show that if f has a Siegel disk of bounded type rotation number centered at the origin, then the boundary of this Siegel disk is a quasicircle passing through a critical point of f. This unifies and generalizes several previously known results.

Apr. 18: Zheng Huang (CUNY Staten Island)
Introduction to the Weil-Petersson Metric on Teichmuller Space

I will discuss the Weil-Petersson metric on Teichmuller space, its geometric properties and some techniques involved in the study.

Apr. 25: No meeting

May 2: Sudeb Mitra (Queens College of CUNY)
Some Extensions of Holomorphic Motions

In this talk we will discuss holomorphic motions over some infinite dimensional parameter spaces. We will give a simple example of a holomorphic motion of a finite set over a simply connected parameter space that cannot be extended to a holomorphic motion of the sphere. We will then show that a holomorphic motion of a closed set E in the Riemann sphere can be extended to a quasiconformal motion of the sphere, in the sense of Sullivan and Thurston, if and only if the given holomorphic motion of E is induced by a holomorphic map into the Teichmueller space of E. We will also discuss some properties of continuous motions of the sphere. If time permits, we will give an example of a holomorphic motion of a four-point set over the punctured unit disk that cannot be extended to a quasiconformal motion of the sphere.

May 9: No meeting due to The Fourth Ahlfors-Bers Colloquium at Rutgers-Newark

May 16: Fred Gardiner (Brooklyn College and GC of CUNY)
A Dirichlet Principle for Partial Measured Foliations in the Unit Disk

The classical Dirichlet principle says that to find a harmonic function in a plane domain with given boundary values while holding the boundary values, one can look among all functions with the given boundary values and minimize the Dirichlet integral. A similar principle holds for measured foliations. The proof involves the notion of extremal length of a measured foliation.

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