Limits of invariants of translation surfaces

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The theme of this project is to explore geometry at infinity by studying the limits of invariants of translation surfaces. The guiding question behind the theme is that of how a sequence of finite translation surfaces converges to an infinite translation surface.Translation surfaces arise naturally in many different contexts such as the theory of mathematical billiards, of Teichmüller spaces, or of Abelian differentials. Finite translation surfaces can be described by finitely many polygons that are glued along edges which are parallel and have the same length. In recent years, the question has arisen how the theory changes when we glue infinitely instead of finitely many polygons. From that question the field of infinite translation surfaces has evolved and offers more broad applications, for example to physical models.The goal of this research project is to study the convergence of four types of invariants of translation surfaces. These are geometric invariants (such as the diameter or the Cheeger constant), Veech groups (measuring the symmetry of a translation surface), saddle connection complexes (reflecting the combinatorics of saddle connections), and Siegel–Veech constants (counting problems).Understanding these limits will also shed light on the question of how to define a suitable space of infinite translation surfaces.As these four approaches use a wide range of geometrical tools, the Priority Programme "Geometry at infinity" with the large network of mathematicians is an ideal frame to carry out this research project.