The study of Teichmüller curves has become an active field today advanced by major impetus of Fields medalists as Yoccoz, Kontsevich, McMullen, Okounkov , Avila and Mirzakhani within the last thirty years. A tantalizing problem is to relate the arithmetic properties of Teichmüller curves defined by origamis with combinatorial data as their associated dessins d’enfants at infinity. It is e.g. widely open which fields of definition Teichmüller curves in fixed strata can have. Using computer tools, we will progress towards this and other geometric problems by a combination of experimental and theoretical methods.
