# Complex Analysis and Dynamics Seminar

## Department of MathematicsGraduate Center of CUNY

### Spring 2015:

#### Feb 6: Igor Rivin (Temple University)Random 3-Manifolds

I will discuss various models of 3-dimensional manifolds, and their statistical properties.

#### Feb 20: Tanya Firsova (Kansas State University)Critical Locus for Complex Henon Maps

Henon maps are maps of the form $(x,y) \mapsto (p(x)+ay, x)$. Dynamically non-trivial automorphisms of ${\mathbb C}^2$ are conjugate to compositions of Henon maps. For one-dimensional 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: 1:50-2:50 John Loftin (Rutgers University, Newark)Cubic Differentials and Limits of Convex $RP^2$ Strucures under Neck Pinches

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 Deligne-Mumford 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 Benoist-Hulin on the convergence of some invariant tensors on families of convex domains converging in the Gromov-Hausdorff sense, a recent uniqueness theorem of Dumas-Wolf 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.

#### 2:55-3:55 Barak Weiss (Tel Aviv University)Everything is Illuminated (Except at Most Finitely Many Points)

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 Hubert-Schmoll-Troubetzkoy, 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)Chromatic Numbers for Hyperbolic Surfaces

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 Hadwiger-Nelson 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 $d$.
This talk is about finding different bounds on the chromatic number of hyperbolic surfaces and is based on joint work with Camille Petit.

#### Mar 13: Anita Rojas (University of Chile)Decomposable Abelian Varieties, Studying the Case of Jacobians

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.
The action of a finite group $G$ on an abelian variety $A$ induces a decomposition of $A$ into $G-$invariant factors, called the isotypical decomposition of $A$. It comes from the decomposition of the group algebra $\mathbb{Q}[G]$ of $G$ over the rationals, into simple algebras. Hence each factor corresponds to a rational irreducible representation of $G$.
The approach of considering group actions, group algebra decompositions and subvarieties defined by idempotents has been fruitful. In this talk we will discuss some recent advances, as well as some work in progress, regarding these questions. All these results come from the interplay between algebra and geometry, and were developed in collaboration with A. Carocca, H. Lange, R.E. Rodriguez and J. Paulhus.

#### Mar 20: Fred Gardiner (Brooklyn College and Graduate Center of CUNY)The Caratheodory and Kobayashi Metrics on Teichmuller Space

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)Computing the Teichmüller Polynomial

The Teichmüller polynomial of a fibered 3-manifold 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 pseudo-Anosov 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 17: Jon Fickenscher (Princeton University)A Bound of Boshernitzan

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)Collar Lemma for Hitchin Representations

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 Gye-Seon Lee.

#### May 1: Matthieu Astorg (University of Toulouse)The Dynamical Teichmuller Space Immerses into the Moduli Space of Degree $d$ Rational Maps

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 15: Bram Petri (University of Fribourg and Brown University)The Systole of a Random Surface

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 non-contractible 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.