Spring 2013 Schedule

Feb 1: No meeting

Feb 8: Joseph Maher (College of Staten Island and Graduate Center of CUNY)
Statistics for Teichmuller Geodesics
We describe two ways of picking a geodesic "at random" in a
space, one coming from the standard Lebesgue measure on the visual sphere,
and the other coming from random walks. The spaces we are interested in
are hyperbolic space and Teichmuller space, together with some discrete
group action on the space. We investigate the growth rate of word length
as one moves along the geodesic, and we show these growth rates are
different depending on how one chooses the geodesic. This is joint work
with Vaibhav Gadre and Giulio Tiozzo.

Feb 15: Araceli Bonifant (University of Rhode Island)
Antipode Preserving Cubic Rational Maps and Herman Rings
In this talk, I will discuss a family of cubic rational maps which
carry antipodal points of the Riemann sphere to antipodal
points, with emphasis on the abundance of Herman rings.

Feb 22: Frederick Gardiner (Brooklyn College and Graduate Center of CUNY)
Compactifying Teichmuller Space; Horocycles and Wedges
There is a horoball compactification of Teichmuller space based
on extremal length that corresponds to Mumford's compactification of
moduli space. Elaborating this idea and Grotzsch's argument leads to the following topics:
1. The ReichStrebel inequalities
2. The minimum Dirichlet principle for measured foliations
3. Horoballs and wedges
4. Tangential and radial limits
5. Julia's lemma
6. Cylindrical (JenkinsStrebel) differentials
7. Dehn multitwists
8. Masur's theorem that Teichmuller's space is not CAT(0)
9. Analogies to the DenjoyWolf theorem on iterating holomorphic
self maps of the disc
10. Measured foliations and Thurston's compactification

Mar 1: Alex Kheyfits (Bronx Community College of CUNY)
ValironTitchmarsh Theorem for Subharmonic Functions in ${\mathbb R}^n$ with Masses on a HalfLine
Let $f(z)$ be an entire function of order $\rho,\; 0<\rho <1$, with negative zeros and let $n(r)$ be the counting function of its zeros. If the finite limit $(*) \lim_{r \to \infty} r^{\rho}n(r)=\Delta $ exists, then it is straightforward to prove the existence of the limit
$$
\lim_{r \to \infty} r^{\rho} \log f(r e^{i \theta}) = \frac{\pi \mathbf{\Delta}}{\sin \pi \rho}e^{i \rho \theta},\;\; \theta \in (\pi, \pi).
$$
The ValironTitchmarsh theorem gives the tauberian converse of this statement, namely, if the limit over the positive halfline
$$
\lim_{r \to \infty} r^{\rho} \log f(r) = \frac{\pi \mathbf{\Delta}}{\sin \pi \rho}
$$
exists, then the limit $(*)$ exists. In this talk, the theorem is extended to subharmonic functions in $\mathbb{R}^n,\; n\geq 3$, with the Riesz associated masses on a ray.

Mar 8: Clifford Earle (Cornell University)
Teichmueller Space of a Quadratic Differential
The Teichmuller space of a Fuchsian group $G$ is defined by Bers
as the set of quadratic differentials in universal Teichmuller space that
are $G$invariant. We reverse the process: Starting with a quadratic
differential $\phi$ in universal Teichmuller space, we study the group of all
$g$ in $\text{PSL}(2,{\mathbb R})$ such that $\phi$ is $g$invariant.
There is an unexpected (by me) connection with an old paper of Charles Kalme.

Mar 15: Jun Hu (Brooklyn College and Graduate Center of CUNY)
Thurston's Earthquake Measure Characterization of the Asymptotic Teichmuller Space on the Hyperbolic Plane
Let $MLb(D)$ be the collection of Thurston bounded measured geodesic laminations on
the hyperbolic plane $D$. We introduce an equivalence relation on $MLb(D)$ such
that the earthquake measure map induces a bijection between the asymptotic
Teichmuller space $AT(D)$ and the quotient space $AMLb(D)$ of $MLb(D)$ under
the equivalent relation. Furthermore, we introduce a topology on $AMLb(D)$
under which the bijection is a homeomorphism between $AT(D)$ and $AMLb(D)$
with respect to the Teichmuller metric on $AT(D)$. Corresponding results are
also developed for a bijection and then a homeomorphism between the tangent
space $AZ(S^1)$ of $AT(D)$ at a base point and $AMLb(D)$ with respect to the
asymptotic crossratio norm on $AZ(S^1)$ and the topology on $AMLb(D)$.
This is joint work with Jinhua Fan.

Mar 22: Joel Zablow (Long Island University)
Families of Relations in Mapping Class Groups and Beyond, Parametrized by Knots
For an oriented surface $F$ with mapping class group $MCG(F)$, it is known that if a mapping class $f$ fixes a simple closed curve $c$, then $f$ commutes with Dehn twists about $c$. For a given nonseparating curve $c$ in $F$, we give a construction for creating families of such commutation relations, where the mapping class $f$ can involve arbitrarily long products of elements in $MCG(F)$. These families are parametrized by knots (here, mainly pretzel knots) admitting particular "colorings" by element of Dehn quandles. In these quandles, there is a notion of two elements having "intersection number" 0 or 1. We show the above construction generalizes to produce such relations in other groups associated to quandles with such intersection number 1 elements, e.g. $SL(2,{\mathbb Z})$,
$SL(2,{\mathbb R})$, braid groups, symmetric groups, symplectic groups, and certain Artin groups.

Mar 29: No meeting

Apr 5: Saeed Zakeri (Queens College and Graduate Center of CUNY)
On Margulis Cusps of Hyperbolic $4$Manifolds
We study the geometry of the Margulis region associated with an irrational screwtranslation
$g$ acting on ${\mathbb H}^4$. This is an invariant domain with the parabolic fixed point of $g$
on its boundary which generalizes the notion of an invariant horoball for a translation in dimensions $\leq 3$.
The boundary of the Margulis region is described by a function $B_\alpha : [0,\infty) \to {\mathbb R}$ which solely
depends on the irrational rotation angle $\alpha \in {\mathbb R}/{\mathbb Z}$ of $g$. We obtain an asymptotically universal
upper bound for $B_\alpha(r)$ as $r \to \infty$ for arbitrary $\alpha$, as well as lower bounds when $\alpha$
is Diophatine and optimal bounds when $\alpha$ is of bounded type. We investigate the implications of these
results for the geometry of Margulis cusps of hyperbolic $4$manifolds. Among other things, we prove that such cusps
are biLipschitz rigid. This is joint work with Viveka Erlandsson.

Apr 12: Rubi Rodriguez (Catholic University of Chile)
Decomposition of Jacobian and Abelian Varieties with Group Actions
We will describe the decomposition of Abelian varieties with
automorphism group $G$ as a product of $G$simple subabelian varieties.
We will also give geometric interpretations for the factors in the
case of Jacobian varieties of Riemann surfaces with nontrivial automorphism group.

Apr 19: Eric Bedford (Indiana University)
Dynamics of Automorphisms and PseudoAutomorphisms of Complex Manifolds
We will survey the compact complex surfaces with automorphisms of positive entropy. Then we look at the higher dimensional situation and discuss the existence of pseudoautomorphisms with positive entropy.

Apr 26: Sergiy Merenkov (University of Illinois at UrbanaChampaign)
Uniqueness Properties of Schottky Maps
If $D$ is a domain in the complex plane, a relative Schottky set in $D$ is the residual set obtained by removing from $D$ open geometric discs whose closures are pairwise disjoint. A Schottky map is a local homeomorphism between relative Schottky sets that is conformal (in an appropriate sense) and whose derivative is continuous. In this talk, I will discuss uniqueness properties of such maps. The main result states that under mild geometric assumptions a Schottky map from an open subset of a relative Schottky set $S$ into $S$ is the identity provided that it fixes a point and its derivative is $1$ at this point.

May 3: Kentaro Ito (Nagoya University)
Convergence and Divergence of Kleinian Punctured Torus Groups
In this talk, we give a necessary and sufficient condition for a sequence of Kleinian punctured torus groups to converge.
This result tells us that every exotically convergent sequence of Kleinian punctured torus groups is obtained by the method
due to Anderson and Canary (Invent. Math., 1996). Thus we obtain a complete description of the set of points at which the
space of Kleinian punctured torus groups selfbumps.

May 10: Robert Suzzi Valli (Graduate Center of CUNY)
Nonsimple Closed Geodesics on Orbifold Surfaces
Let $\mathbb{H}$ be the hyperbolic plane and suppose $\Gamma$ is a Fuchsian group. In general the quotient $X=\mathbb{H}/\Gamma$ is an orbifold surface. In the case that $\Gamma$ contains elliptic elements, their fixed points in $\mathbb{H}$ project to cone points on $X$. The presence of cone points requires a finer notion of paths and homotopy on $X$ with the goal of defining the orbifold fundamental group, $\pi_1(X,b)$, and obtaining an isomorphism between $\pi_1(X,b)$ and $\Gamma$ (analagous to the surface case). With this at our disposal we will study once selfintersecting closed geodesics on $X$ which are disjoint from the cone points, called figure eight geodesics. We identify the shortest figure eight geodesic on a triangle group orbifold ($\mathbb{H}/\Gamma$, where $\Gamma$ is a triangle group), and then use this to find the shortest figure eight geodesic among all orbifold surfaces without cone points of order two.

May 17: Sandra Hayes (City College of CUNY)
The Real Dynamics of Bieberbach's Example
Bieberbach constructed in 1933 domains in ${\mathbb C}^2$ which were
biholomorphic to ${\mathbb C}^2$ but omitted an open set. The existence
of such domains was unexpected because the analogous statement for the onedimensional complex plane is false. The
special domains Bieberbach considered are given as basins of
attraction of a cubic Henon map. This classical method of construction is one of the first applications of dynamical
systems to complex analysis.
In this talk, the boundaries of the real sections of Bieberbach's
domains will be shown to be smooth. The real Julia sets of
Bieberbach's map will be calculated explicitly and illustrated
with computer generated graphics.
