I am studying infinite translation surfaces, i.e. surfaces that are equipped with a translation structure. These can have infinite area or infinite topological type but they can also have finite area and finite topological type.
A striking difference to the theory of finite translation surfaces is that there can exist singularities that are not cone angle singularities. For instance, singularities coming from infinite covering constructions will usually be infinite angle singularities. However, there are also singularities that can not be simply described by an angle. I am particularly interested in these so-called wild singularities because they can have very complicated neighborhoods that carry a lot of information.
For many statements from the theory of finite translation surfaces, the analog for infinite translation surfaces is not true in full generality. To recover these statements, it is important to study different subclasses of infinite translation surfaces. For example, the study of the geodesic flow leads to the restriction on translation surfaces with a discrete set of singularities. I am interested in finding out which restrictions are necessary to recover which types of theorems.
Until August 2017, I was working on the DFG funded project which is called "Classification and deformation theory of infinite translation surfaces".
During the Sage Days 73, I wrote a small tool in Sage that visualizes the translation structure that is given by a differential on a subset of the complex plane. As soon as I have straightened the code, I will publish it here.
- Geometry and topology of wild translation surfaces. PhD thesis, Karlsruhe Institute of Technology, 2015.
- When wild singularities of translation surfaces imply infinite genus. Preprint.
- Rotational component spaces for infinite-type translation surfaces. With Lucien Clavier and Chenxi Wu, preprint.
- Notes on the Veech group of the Chamanara surface. With Frank Herrlich, notes.
- Dreiecksgruppen, die als Veechgruppen auftreten. Diploma thesis (in german), Karlsruhe Institute of Technology, 2011.
- Unendliche Translationsflächen. Lecture notes (in german), Karlsruhe Institute of Technology, 2013.
- On Loch Ness monsters and wild singularities. Geometry & Topology Seminar, Hamilton (Canada), November 2017
- The Veech group of the Chamanara surface. Oberseminar Algebra und Zahlentheorie, Saarbrücken (Germany), June 2017
- A class of infinite translation surfaces where almost every direction is uniquely ergodic. Complex Analysis and Dynamics Seminar, Graduate Center of CUNY (USA), December 2016
- A class of infinite translation surfaces where almost every direction is uniquely ergodic. Dynamics Seminar, University of Maryland (USA), September 2016
- Genus of wild translation surfaces. Simons Semester Workshop: Translation Surfaces and Dynamics, Będlewo (Poland), November 2015
- Topological types of infinite translation surfaces. Dynamics Seminar, Chicago (USA), September 2015
- What is infinite about infinite translation surfaces? Seminar of the Differential Geometry Research Group, Heidelberg (Germany), February 2015
- Singularities on infinite translation surfaces. International conference and workshop on surfaces of infinite type, Morelia (Mexico), August 2013