# 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.