Complex Analysis and Dynamics Seminar
Department of Mathematics

Past seminars:
Fall 2006,
Spring 2007

Spring 2015: 
Feb 6: Igor Rivin (Temple University)
I will discuss various models of 3dimensional manifolds, and their statistical properties.

Feb 13: No seminar

Feb 20: Tanya Firsova (Kansas State University)
Henon maps are maps of the form $(x,y) \mapsto (p(x)+ay, x)$.
Dynamically nontrivial automorphisms of ${\mathbb C}^2$ are conjugate to
compositions of Henon maps. For onedimensional holomorphic maps the
dynamics of the map is to a large extent determined by the orbits of
the critical points. Since Henon maps are biholomorphisms of ${\mathbb C}^2$, they
do not have critical points in the classical sense. However, there is
a way to define an appropriate analog, called critical locus. Critical
locus is a Riemann surface. I will give a topological description of the
critical locus for Henon maps that are small perturbations of
quadratic polynomials with disconnected Julia set. This proves the
conjecture of J. Hubbard. I will also show that the critical loci are
quasiconformally equivalent. The last part of the talk is a joint work
with Misha Lyubich.

Feb 27: The seminar will feature two talks:
Labourie and I independently proved that on a closed oriented surface $S$ of genus $g$ at least $2$, a convex
real projective structure is equivalent to a pair $(\Sigma,U)$, where $\Sigma$ is a conformal structure and $U$
is a holomorphic cubic differential. It is then natural to allow $\Sigma$ to go to the boundary of the moduli
space of Riemann surfaces. The bundle of cubic differentials then extends over the boundary to form the bundle of
regular cubic differentials, which is an orbifold vector bundle over the DeligneMumford compactification
$\bar{\mathcal M}_g$ of moduli space.
We define regular convex real projective structures corresponding to the regular cubic differentials over nodal Riemann surfaces and define a topology on these structures. Our topology is an extension of Harvey's use of the Chabauty topology to analyze $\bar {\mathcal M}_g$ via limits of Fuchsian groups. The main theorem is that the total space of the bundle of regular cubic differentials over $\bar {\mathcal M}_g$ is homeomorphic to the space of regular real projective structures. The proof involves several analytic inputs: a recent result of BenoistHulin on the convergence of some invariant tensors on families of convex domains converging in the GromovHausdorff sense, a recent uniqueness theorem of DumasWolf for certain complete conformal metrics, and some old techniques of the author to specify the real projective end of a surface in terms of the residue of a regular cubic differential.
2:553:55 Barak Weiss (Tel Aviv University)
Suppose a light source is placed in a polygonal hall of mirrors (so light can bounce off the walls).
Does every point in the room get illuminated? This elementary geometrical question was open from the 1950s
until Tokarsky (1995) found an example of a polygonal room in which there are two points which do not
illuminate each other. Resolving a conjecture of HubertSchmollTroubetzkoy, in joint work with Lelievre
and Monteil we prove that if the angles between walls is rational, every point illuminates all but at most finitely
many other points. The proof is based on recent work by Eskin, Mirzakhani and Mohammadi in the ergodic theory
of the $SL(2,R)$ action on the moduli space of translation surfaces. The talk will serve as a gentle introduction
to the amazing results of Eskin, Mirzakhani and Mohammadi.

Mar 6: Hugo Parlier (University of Fribourg and Hunter College of CUNY)
Given a metric space $X$ and a positive real number $d$, the chromatic number of
$(X,d)$ is the minimum number of colors needed to color points of the metric space
such that any two points at distance $d$ are colored differently. When $X$ is a metric
graph (and $d$ is $1$) this is the usual chromatic number of a graph. When $X$ is the
euclidean plane (the $d$ is irrelevant) the chromatic number is known to be between
$4$ and $7$ (finding the exact value is known as the HadwigerNelson problem).
For the hyperbolic plane even less is known and it is not even known whether or
not it is bounded by a quantity independent of $d$.

Mar 13: Anita Rojas (University of Chile)
A completely decomposable abelian variety is one that is isogenous to a product of elliptic curves.
In 1993, Ekedahl and Serre asked several questions about completely decomposable Jacobian varieties,
some of them are still open. In particular they asked if there are completely decomposable Jacobian varieties
in any dimension $g\geq 2$. In the same work, the authors presented a list of dimensions in which there are
completely decomposable Jacobian varieties. Nevertheless, besides stopping in dimension $1297$ leaving open
the question whether there are higher dimensional completely decomposable Jacobian varieties, their list has some gaps.
These questions have motivated several articles approaching their answers through different methods.
We use group actions as the main tool.

Mar 20: Fred Gardiner (Brooklyn College and Graduate Center of CUNY)
I show that for vector fields $V$ that are tangent to cylindrical differentials that are squares of abelian differentials, the Caratheodory and Kobayashi infinitesimal forms are equal, that is,
$$\text{Car}(\tau,V) = \text{Kob}(\tau,V),$$
and for vector fields $V$ that are tangent to cylindrical differentials that are separating, the Caratheodory and Kobayashi infinitesimal forms are unequal, that is,
$$\text{Car}(\tau,V)< \text{Kob}(\tau,V).$$

Mar 27: Ferran Valdez (UNAM, Mexico)
The Teichmüller polynomial of a fibered 3manifold plays a useful role in the construction of mapping classes
with small entropy (small stretch factor). In this talk, we explain what this polynomial is and we provide an
algorithm that computes the Teichmüller polynomial of the fibered face associated to a pseudoAnosov mapping
class of a disc homeomorphism. This algorithm is based on the results of Penner and Papadopoulos
on train tracks and elementary operations on them (folding and splitting). This is joint work with Erwan Lanneau.

Apr 3: No seminar

Apr 10: No seminar

Apr 17: Jon Fickenscher (Princeton University)
In 1985, Boshernitzan showed that a minimal symbolic dynamical system with a linear complexity bound must have a finite number of probability invariant ergodic measures. We will discuss methods to sharpen this bound in general and provide cases in which the bound may already be reduced. This is ongoing work with Michael Damron.

April 24: Tengren Zhang (University of Michigan)
The classical collar lemma is an important result and a useful tool used to study hyperbolic surfaces.
In particular, it implies that if two geodesics intersect on a surface, then there is an explicit lower bound of
the length of one in terms of the length of the other. We will explain an analog of this statement for Hitchin representations,
which are natural generalizations of the holonomy representations of hyperbolic surfaces. This is joint work with GyeSeon Lee.

May 1: Matthieu Astorg (University of Toulouse)
Teichmuller theory's goal is to study deformations of the
complex structure of a Riemann surface. In the 1980's,
McMullen and Sullivan introduced an analog of this theory in the
context of iterations of a rational map $f$.
In particular, they constructed a "dynamical Teichmuller space" which is
a simply connected complex manifold,
with a holomorphic map $F$ defined on Teich($f$) and taking values in the
space of rational maps of the same degree
as $f$, and whose image is exactly the quasiconformal conjugacy class of
$f$. A natural question, raised in their
article, is to know if this map $F$ is an immersion: it turns out the
answer is affirmative. A. Epstein has an
unpublished proof of this; we will expose a different approach.

May 8: No seminar

May 15: Bram Petri (University of Fribourg and Brown University)
There are multiple notions of random surfaces. In this talk a random surface will be a surface constructed
by randomly gluing together an even number of triangles that carry a fixed metric. If one chooses a specific
hyperbolic metric then the set of all possible surfaces obtained by performing this procedure will be dense
in every moduli space of compact surfaces. This means that using this construction, one can ask questions
about the geometry of a typical hyperbolic surface. The model lends itself particularly well to studying high
genus surfaces. For example it turns out that the expected value of the length of the shortest noncontractible
curve, the systole, of such a surface converges to a constant. In this talk I will explain what goes into
the proof of this fact and how this relates to the theory of random regular graphs.
