# Research

I study 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.

## Research activities

- Principal Investigator in DFG Research Training Group "Asymptotic Invariants and Limits of Groups and Spaces" (GRK 2229)
- Project Leader of the project “Limits of invariants of translation surfaces” (RA 2857/3-1) in the DFG Priority Programme "Geometry at Infinity" (SPP 2026)
- DFG Research Fellowship (2016-2017): "Classification and deformation theory of infinite translation surfaces" (RA 2857/1-1)

## Publications

### Preprints

- Realizing groups as symmetries of infinite translation surfaces. With Mauro Artigiani, Chandrika Sadanand, Ferrán Valdez, and Gabriela Weitze-Schmithüsen.

### Published

- Conjugacy classes of big mapping class groups. With Jesús Hernández Hernández, Michael Hrušák, Israel Morales, Manuel Sedano, and Ferrán Valdez, Journal of the London Mathematical Society. Second Series 106 (2022), no. 2, 1131-1169.
- Rigidity of the saddle connection complex. With Valentina Disarlo and Robert Tang, Journal of Topology 15 (2022), 1248-1310.
- Expected covering radius of a translation surface. With Howard Masur and Kasra Rafi, International Mathematics Research Notices 10 (2022), 7967-8002.
- Large-scale geometry of saddle connection graphs. With Valentina Disarlo, Huiping Pan, and Robert Tang, Transactions of the American Mathematical Society 374 (2021), 8101-8129.
- Cusp excursion in hyperbolic manifolds and singularity of harmonic measure. With Giulio Tiozzo, Journal of Modern Dynamics 17 (2021), 183-211.
- Renormalizing an infinite rational IET. With W. Patrick Hooper and Kasra Rafi, Discrete & Continuous Dynamical Systems - Series A 40 (2020), no. 9, 5105-5116.
- Rotational component spaces for infinite-type translation surfaces. With Lucien Clavier and Chenxi Wu, Geometriae Dedicata 201 (2019), 57-80.
- Wild translation surfaces and infinite genus. Algebraic & Geometric Topology 18 (2018), no. 5, 2661-2699.
- Geometry and topology of wild translation surfaces. PhD thesis, Karlsruhe Institute of Technology, 2015.

### Other

- The Veech group of the golden ladder.
- The work of Stäckel and Rodenberg. Appendix for Platonic solids and high genus covers of lattice surfaces by Jayadev S. Athreya, David Aulicino, and W. Patrick Hooper. To appear in Experimental Mathematics.
- Notes on the Veech group of the Chamanara surface. With Frank Herrlich.
- Fields and Groups. Lecture notes, University of Toronto, 2018/2019 (please email if you are interested in a copy).
- Introduction to Analysis. Lecture notes, University of Toronto, 2019 (please email if you are interested in a copy).
- Unendliche Translationsflächen. Lecture notes (in German), Karlsruhe Institute of Technology, 2013.
- Dreiecksgruppen, die als Veechgruppen auftreten. Diploma thesis (in German), Karlsruhe Institute of Technology, 2011.

## Selected Talks

- Topological behaviour of conjugacy classes of big mapping class groups. Geometry Seminar, Leipzig (Germany), April 2024
- The realization problem for Veech groups. Long-distance seminar on Geometric Group Theory in Mexico, online, February 2024
- Topological behaviour of conjugacy classes of big mapping class groups. Oberseminar Algebra und Zahlentheorie, Saarbrücken (Germany), February 2024
- The realization problem for Veech groups. Renormalization and Visualization in Geometry, Dynamics and number theory, Marseille (France), December 2023
- Introduction to big mapping class groups. Group actions on Cantor sets, Oaxaca (Mexico), September 2023
- Symmetries of surfaces of infinite topological type (a.k.a. big mapping class groups). Colloquium, Morelia (Mexico), September 2023
- The saddle connection complex. Prospectos in Topología, Morelia (Mexico), August 2023.
- Big mapping class groups as Polish groups. Séminaire Géométrie et groupes discrets, Paris (France), February 2023
- Realization of Veech groups of infinite translation surfaces. Workshop on Surfaces in the frontier, Manizales (Colombia) / online, February 2023
- Periodic directions on translation surfaces in genus 2. DMV Annual Meeting, Berlin (Germany), September 2022
- Big mapping class groups as Polish groups. DMV Annual Meeting, Berlin (Germany), September 2022
- Topological behaviour of conjugacy classes of big mapping class groups. Hyperbolic Lunch, Toronto (Canada), August 2022
- Topological behaviour of conjugacy classes of big mapping class groups. BiSTRO mini-conference, online, May 2022
- Periodic directions on translation surfaces in genus 2. Number Theory Seminar, Oregon State University (USA) / online, May 2022
- Billiards and translation surfaces. Celebrating Women in Mathematics, online, May 2022
- Saddle connection complex: coarse and fine. Geometry and Topology Seminar, Bristol (United Kingdom) / online, March 2022
- Examples of infinite rational IETs. Mostly Teichmüller Spaces Seminar, New York (USA) / online, February 2022
- Interval exchange transformations and translation surfaces in genus 2. Bremen Online Dynamics Seminar, Bremen (Germany) / online, January 2022
- Trees everywhere (The saddle connection complex). 19. Karlsruher Weihnachtsworkshop on Geometry and Number Theory, Karlsruhe (Germany), December 2021
- Topological behaviour of conjugacy classes of big mapping class groups. Geometry and Topology Seminar, Haifa (Israel), November 2021
- Saddle connection complex: coarse and fine. Hyperbolic Geometry Seminar, New York (USA) / online, November 2021
- Topological behaviour of conjugacy classes of big mapping class groups. Swiss Knots, Fribourg (Switzerland), July 2021
- Saddle connection complex: coarse and fine. Quasiworld Seminar, online, June 2021
- The Veech group of the golden ladder. Big Surf(aces) Seminar, online, January 2021
- Large-scale geometry of the saddle connection complex. Séminaire Teich, Marseille (France) / online, December 2020
- What does your average translation surface look like? Nearly Carbon Neutral Geometric Topology Conference, online, June 2020
- Random walks on hyperbolic manifolds. Oberseminar Algebra und Zahlentheorie, Saarbrücken (Germany), February 2020
- Coarse geometry of the saddle connection complex. Lost in Translation Surfaces, University of Luxembourg (Luxembourg), January 2020
- The saddle connection complex. Holomorphic Differentials in Mathematics and Physics, MSRI, Berkeley (USA), November 2019
- Degenerating directions on infinite translation surfaces. Dynamics of Parabolic Flows, Zürich (Switzerland), July 2019
- Cusp excursion for random walks on hyperbolic manifolds. Boundaries of Random Walks and Applications, Brunswick (USA), June 2019
- The saddle connection complex. Flat Surfaces and Dynamics on Moduli Space, II, Oaxaca (Mexico), May 2019
- Cusp excursion for random walks on hyperbolic manifolds. Bloomington Geometry Workshop, Bloomington (USA), April 2019
- Asymptotics of the expected diameter of translation surfaces. Geometry, Groups, and Dynamics seminar, University of Illinois at Urbana-Champaign (USA), February 2019
- An infinite rational IET with nice behaviour. Horowitz Seminar, Tel Aviv University (Israel), December 2018
- The saddle connection complex. Workshop on Dynamics and Moduli Spaces of Translation Surfaces, Fields Institute, Toronto (Canada), October 2018
- Estimating the volume of the thin part. Working seminar of the Thematic Program on Teichmüller Theory and its Connections to Geometry, Topology and Dynamics, Fields Institute, Toronto (Canada), October 2018
- Average diameter of translation surfaces. Conference Teichmüller dynamics, mapping class groups and applications, Institut Fourier (France), June 2018
- 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

## Conferences organized

- AIM workshop "Surfaces of infinite type", San Jose, California, April/May 2019
- Hengstberger Symposium "Geometry and dynamics of surfaces beyond finite-type", Heidelberg, Germany, July 2024.

## Programming

During the Sage Days 73, I wrote a tool in Sage that visualizes the translation structure that is given by a differential on a subset of the complex plane. If I'll ever straighten and document the code, I will publish it here. Until then, it is available upon email request.