Dynamics Around Translation Surfaces

该奖项将支持动力系统方面的一个项目。动力系统的数学领域试图理解系统如何随着时间的演变而表现；它是数学分析的一个重要分支，与数学科学的许多其他领域有着联系和应用。这个项目的核心系统与物理和几何相关。一个具体的例子是一个质点在多边形内部运动，当它与多边形的侧面碰撞时，会发生弹性碰撞。该项目的一个重点是了解这些系统中普遍存在的随机性。PI还将研究相关动力系统中的路径结构，旨在加深我们对几何和动力特性之间联系的理解。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。本项目关注两个密切相关的动力系统：平移表面上的流动和平移表面空间上的SL（2，R）作用。它试图更好地理解平移表面上的流（来自单参数酉群）的谱何时是连续的（除了简单的特征值为0）。它试图理解SL（2，R）的严格上三角子群在平移曲面空间上的动力学。特别是，是否存在轨道闭包总是受限的情况以及沿轨道的野平均如何表现。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.