I mainly study the Veech groups of flat surfaces, which represent the `self-affine similarity of surfaces’. The action of the matrices of self-affine deformations in the Veech group is regarded as a Möbius transformation, where the Fuchsian model parameterizes the family of affine deformations of a given surface.
In cases of surfaces with high self-symmetry, such as origami (square-tiled surfaces, not traditional origami), it is difficult to compute Veech groups in a practical process. I have obtained calculation results using a supercomputer. In the background, there are interesting facts indicating a relation to number theory and the theory of triangulated category, from which I am seeking feedback and applications.