# Complex Analysis and Dynamics Seminar

## Department of MathematicsGraduate Center of CUNY

### Spring 2018:

#### Feb 9: Daniele Alessandrini (University of Heidelberg) Geometric Structures with Quasi-Hitchin Holonomy

Higher Teichmuller Theory is a way to generalize Teichmüller Theory to higher rank Lie groups. I will describe some manifolds admitting real and complex projective structures whose holonomy is a Hitchin or a Quasi-Hitchin representation. This generalizes the Thurston’s theories of Fuchsian and Quasi-Fuchsian representations to higher rank Lie groups. The results come from a joint work with Qiongling Li and a joint work with Sara Maloni and Anna Wienhard.

#### Feb 16: Joseph Maher (College of Staten Island, CUNY) Random Mapping Classes Have Generic Foliations

A pseudo-Anosov element of the mapping class group determines a quadratic differential, which lies in the principal stratum if all zeroes are simple, equivalently, if the corresponding foliations have trivalent singularities. We show that this occurs with asymptotic probability one for random walks on the mapping class group, and furthermore, the hitting measure on the boundary gives weight zero to foliations with saddle connections. This is joint work with Vaibhav Gadre.

#### Feb 23: David Aulicino (Brooklyn College of CUNY) Trajectories on the Platonic Solids

Given any of the five Platonic solids, can we find a straight-line trajectory on the surface of the solid that starts and ends at the same vertex without passing through any other vertex? It was proven for the tetrahedron, octahedron, cube, and icosahedron that there is no trajectory from a vertex to itself that does not pass through another vertex. We will give a simple proof of this for the tetrahedron and outline the proof for the other solids. Finally, we will show that there does indeed exist such a trajectory on the dodecahedron, and using translation surfaces, we give a complete classification of such trajectories. All of the necessary theory of translation surfaces will be developed and the connection to $k$-differentials will be mentioned. This is joint with Jayadev S. Athreya and Pat Hooper.

#### March 2: Tao Chen (Laguardia Community College of CUNY) Shell Components of Extended Family of the Tangent Map

Each hyperbolic component of the Mandelbrot set consists of quadratic maps with an attracting periodic cycle. Similarly, we consider the family of maps $f_\lambda=\lambda \tan^p z^q$. Each component of the set of $\lambda$ such that $f_\lambda$ has an attracting cycle is called a shell component. In this talk, we mainly give a topological and combinatorial description of shell components. This is joint work with Linda Keen.

#### March 9: Enrique Pujals (Graduate Center of CUNY) Dynamics of Mild Dissipative Diffeomorphisms

We discuss a class of volume-contracting surface diffeomorphisms whose dynamics is intermediate between one-dimensional dynamics and general surface dynamics.

#### March 16: Babak Modami (Stony Brook University) Teichmuller Geodesics with Higher Dimensional Limit Sets

One of the features of Teichmuller geodesics is the interplay between the geometric and dynamical properties of the geodesics and foliations on surfaces. An example of this interplay is a result of H. Masur saying that the limit set of a Teichmuller geodesic with a uniquely ergodic (vertical) foliation is a point in the Thurston compactification of the Teichmuller space. However, this is not the case for Teichmuller geodesics with non uniquely ergodic foliations. I present my joint work with Kasra Rafi and Ania Lenzhen where we construct Teichmuller geodesics whose limit sets have dimensions greater than 1.

#### March 23: Tarik Aougab (Brown University) Weil-Petersson Metrics for Moduli of Graphs

Fix a finite graph $G$ whose fundamental group has rank $n$ at least $2$, and so that each vertex has valence at least $3$. The moduli space of metrics on $G$, denoted $M(G)$, is then naturally identified with a (normalization of) the positive orthant of $R^{|E|}$ where $|E|$ denotes the number of edges of $G$. Taking motivation from thermodynamics, Policott-Sharp and Kao defined a pair of Riemannian metrics on $G$ which resemble the Weil-Petersson metric on the moduli space of hyperbolic surfaces. We extend and generalize their work by proving:
(1) For any $G$, both metrics are incomplete on $M(G)$;
(2) The metric completion of the Kao metric for $M(R)$ is infinite diameter when $R$ is a rose;
(3) The completion of the Policott-Sharp metric for $M(R)$ (again for $R$ a rose) is compact, and agrees with the simplicial completion of the simplex in outer space associated to $R$.
This represents joint work with Matt Clay and Yo'av Rieck.

#### April 13: Christian Wolf (City College of CUNY) Regularity of the Localized Entropy Function

Let $f : X \to X$ be a continuous map on a compact metric space with finite topological entropy. Further, assume that the entropy map $\mu \to h_{\mu}(f)$ is upper semi-continuous. It is well known that this implies the continuity of the localized entropy function of a given continuous potential $\varphi : X \to {\mathbb R}$. In this talk we show that this result does not carry over to the case of higher-dimensional potentials $\Phi : X \to {\mathbb R}^m$. Namely, we construct for a shift map $f$ a 2-dimensional Lipschitz continuous potential $\Phi$ with a discontinuous localized entropy function.

#### April 20: Huiping Pan (Fudan University) Existence of Closed Geodesics Through a Regular Point on Translation Surfaces

In this talk, we will discuss the existence of closed geodesics through a regular point on translation surfaces. We show that on any translation surface, every regular point is contained in either zero or infinitely many simple closed geodesics. Moreover, the set of points that are not contained in any simple closed geodesic is finite. We also construct explicit examples showing that such points exist. For a surface in any hyperelliptic component, we show that this finite exceptional set is actually empty. This is a joint work with Duc-Manh Nguyen and Weixu Su.

#### May 4: Brice Loustau (Rutgers University) Bi-Lagrangian Structures and Teichmüller Theory

A Bi-Lagrangian structure in a manifold is the data of a symplectic form and a pair of transverse Lagrangian foliations. Equivalently, it can be defined as a para-Kähler structure, i.e. the para-complex equivalent of a Kähler structure. After discussing interesting features of bi-Lagrangian structures in the real and complex settings, I will show that the complexification of any Kähler manifold has a natural complex bi-Lagrangian structure. I will then specialize this discussion to moduli spaces of geometric structures on surfaces, which typically have a rich symplectic geometry. We will see that some of the recognized geometric features of these moduli spaces are formal consequences of the general theory while revealing new other features, and derive a few well-known results of Teichmüller theory. Time permits, I will present the construction of an almost hyper-Kähler structure in the complexification of any Kähler manifold. This is joint work with Andy Sanders.

#### May 11: Yunping Jiang (Queens College and Graduate Center of CUNY) Expendability of Holomorphic Motions

In this talk, we discuss necessary and sufficient conditions for extending holomorphic motions of subsets of the Riemann sphere over hyperbolic Riemann surfaces.