Dynamics Around Translation Surfaces

平移表面周围的动力学

• 批准号: 2350393
• 负责人: Jonathan Chaika
• 金额: $ 34.55万
• 依托单位: University of Utah
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2024
• 资助国家: 美国
• 起止时间: 2024-07-01 至 2027-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2350393&HistoricalAwards=false
• 关键词: Dynamics; Around; Translation; Surfaces

This award will support a project in dynamical systems. The mathematical field of dynamical systems seeks to understand how a system behaves as time evolves; it is an important subfield of mathematical analysis, which enjoys connections and applications to many other areas of the mathematical sciences. The systems at the heart of this project are connected to physics and geometry. A concrete example is that of a point mass traveling inside a polygon, which has elastic collision when it hits the sides. One focus of the project is to understand how prevalent randomness is in these systems. The PI will also investigate the structure of paths in related dynamical systems and aims to deepen our understanding of the connection between geometric and dynamical properties. This project will also stimulate the growth of the next generation of mathematicians by providing graduate student research opportunities. This project is concerned with two closely related dynamical systems: flows on translation surfaces and the SL(2,R) action on the space of translation surfaces. It seeks to better understand when the spectrum of (the one-parameter unitary group coming from) a flow on a translation surface is continuous (aside from a simple eigenvalue of 0). It seeks to understand the dynamics of the strictly upper triangular subgroup of SL(2,R) on spaces of translation surfaces. In particular, whether there are situations where the orbit closures are always constrained and how wild averages along orbits can behave.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

该奖项将支持动态系统中的项目。动态系统的数学领域试图了解系统的行为随时间的发展。它是数学分析的重要子领域，它享有与数学科学的许多其他领域的连接和应用。该项目核心的系统与物理和几何形状有关。一个具体的示例是在多边形内传播的点质量，该点在撞击侧面时具有弹性碰撞。 该项目的重点是了解这些系统中的普遍性。 PI还将研究相关动力学系统中路径的结构，并旨在加深我们对几何和动力学特性之间的联系的理解。 该项目还将通过提供研究生研究机会来刺激下一代数学家的成长。该项目与两个密切相关的动态系统有关：翻译表面上的流动和SL（2，R）在翻译表面空间上的动作。它试图更好地理解何时（一个参数统一组来自）在翻译表面上的流程是连续的（除了简单的特征值0）。它试图了解翻译表面空间上SL（2，r）严格上部三角形亚组的动力学。特别是，是否在某些情况下，轨道封闭始终受到限制，以及沿轨道沿途的狂野平均值。该奖项反映了NSF的法定任务，并且使用基金会的智力优点和更广泛的影响评估标准，被认为值得通过评估来获得支持。