Complex Analysis and Dynamics Seminar
Spring 2008 Schedule
Feb. 1: Fred Gardiner (Brooklyn College and GC of CUNY)
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.
The Minimum Norm Principle, Extremal Length and Lines of Minima
Feb. 8: Martin Bridgeman (Boston College)
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.
Hausdorff Dimension under Bending Deformations and the Weil-Petersson
Feb 22: Linda Keen (Lehman College and GC of CUNY)
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.
Enumerating Palindromes in Free Groups of Rank Two
Feb. 29: Peter Storm (University of Pennsylvania)
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
Deformation a Hyperbolic 4-Orbifold
Mar. 7: Matt Feiszli (Brown University)
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.
The Universal Teichmuller Space, Computer Vision, and Data Compression
Mar. 14: The seminar will feature two talks:
This will be a continuation of my February 1st talk.
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
4:00-5:00 pm: Shawn Rafalski (Williams College)
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.
Immersions of Hyperbolic Turnovers
Mar. 21: No meeting
Mar. 28: John Parker (University of Durham, UK)
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.
Unfaithful Triangle Groups and the Hunt for Complex Hyperbolic Lattices
Apr. 4: Igor Rivin (Temple University)
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.
Asymptotic Geometry of Convex Sets in Hyperbolic Space
Apr. 11: Saeed Zakeri (Queens College and GC of CUNY)
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.
Siegel Disks of a Class of Entire Maps
Apr. 18: Zheng Huang (CUNY Staten Island)
I will discuss the Weil-Petersson metric on Teichmuller space, its geometric properties
and some techniques involved in the study.
Introduction to the Weil-Petersson Metric on Teichmuller Space
Apr. 25: No meeting
May 2: Sudeb Mitra (Queens College of CUNY)
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.
Some Extensions of Holomorphic Motions
May 16: Fred Gardiner (Brooklyn College and GC of CUNY)
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
A Dirichlet Principle for Partial Measured Foliations in the Unit Disk