Translation surfaces and their orbit closures: An introduction for a broad audience
Translation surfaces and their orbit closures: An introduction for a broad audience
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平移表面及其轨道闭合：面向广大受众的介绍
DOI：
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发表时间：
2014
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通讯作者：
A. Wright
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文献类型：
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作者：
A. Wright
平移曲面可以通过多边形以基本的方式定义，并且在各种基本动力系统的研究中自然出现。它们也可以被定义为黎曼曲面上的阿贝尔微分，并且具有与黎曼曲面的模空间相关的称为层的模空间。在每一层上都有GL（2，R）作用，要解决平移曲面的大多数问题，首先必须知道在此作用下轨道的闭合性。此外，这些轨道闭包本身就具有根本的意义，现在已知它们是代数变种，可以将具有非凡代数几何和平坦性质的平移曲面参数化。近年来，随着新工具和新思想的涌入，轨道闭合的研究大大加速了数学的各个领域。
这项调查是邀请来自不同背景的数学家熟悉这个问题。很少的背景知识，超出了黎曼曲面及其余切束的定义，是假定的，并给予最优先考虑的是提出一个观点的问题，是在一次访问和连接到许多领域的数学。
Translation surfaces can be defined in an elementary way via polygons, and arise naturally in in the study of various basic dynamical systems. They can also be defined as Abelian differentials on Riemann surfaces, and have moduli spaces called strata that are related to the moduli space of Riemann surfaces. There is a GL(2,R) action on each stratum, and to solve most problems about a translation surface one must first know the closure of its orbit under this action. Furthermore, these orbit closures are of fundamental interest in their own right, and are now known to be algebraic varieties that parameterize translation surfaces with extraordinary algebro-geometric and flat properties. The study of orbit closures has greatly accelerated in recent years, with an influx of new tools and ideas coming diverse areas of mathematics.
This survey is an invitation for mathematicians from different backgrounds to become familiar with the subject. Little background knowledge, beyond the definition of a Riemann surface and its cotangent bundle, is assumed, and top priority is given to presenting a view of the subject that is at once accessible and connected to many areas of mathematics.