Limits of invariants of translation surfaces

这个项目的主题是通过研究平移曲面的不变量的极限来探索无穷远处的几何。这一主题背后的指导问题是有限平移曲面序列如何收敛到无限平移曲面。平移曲面在许多不同的上下文中自然产生，如数学台球理论、Teichmüler空间或阿贝尔微分理论。有限平移曲面可以用沿平行且具有相同长度的边粘合的有限多个多边形来描述。近年来，当我们无限地粘合而不是有限多个多边形时，这个理论会发生怎样的变化的问题出现了。从这个问题出发，无限平移曲面领域得到了发展，并提供了更广泛的应用，例如在物理模型中。本研究项目的目的是研究平移曲面的四种不变量的收敛问题。这些是几何不变量(如直径或切格常数)、Veech群(测量平移曲面的对称性)、鞍接复合体(反映鞍形连接的组合性)和Siegel-Veech常数(计数问题)。理解这些限制还将揭示如何定义无限平移曲面的合适空间的问题。由于这四种方法使用了广泛的几何工具，因此具有大型数学家网络的优先计划《无限远的几何》是开展这一研究项目的理想框架。

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.