Topics in geometry and dynamics

理查德施瓦茨计划继续他的研究几何动力学。他研究的一个中心焦点是外台球的研究。2006年，施瓦茨解决了这个有50年历史的课题的核心问题，即关于台球外轨道稳定性的莫泽-诺依曼问题。 施瓦茨将继续发展的理论，探索连接到自相似平铺，限制集离散群体，重整化。 Schwartz还计划继续研究无理多边形台球，重点是三角形台球问题，该问题询问是否每个三角形台球桌都有周期性台球路径。Schwartz计划研究基本的几何迭代，例如迭代重心细分定义的几何迭代。Schwartz提出的研究中的共同主题是分析当一个简单的几何构造重复时会发生什么。一遍又一遍。例如，在行星运动的玩具模型外台球中，一个点（卫星）根据简单的几何规则围绕凸平面形状（行星）的外部移动。在解决这个系统的一个50年前的问题时，Schwartz发现了凸形的例子，以及点的起始位置，使得点在遵循游戏规则的同时无限地远离形状。有人可能会说，这些形状是行星的例子，其卫星可以漫游到太空中。像外台球这样的迭代几何结构在智力上很有吸引力，在科学上也很重要，因为它们的简单定义有时会导致极其丰富和神秘的行为，就像自然科学中的类似系统一样。

Richard Schwartz plans to continue his research in geometric dynamics. A central focus of his research is the study of outer billiards. In 2006, Schwartz resolved the central problem in this 50-year-old subject, the Moser-Neumann problem concerning the stability of outer billiards orbits. Schwartz will continue to develop the theory, exploring connections to self-similar tilings, limit sets of discrete groups, and renormalization. Schwartz also plans to continue studying irrational polygonal billiards, with a focus on the Triangular Billiards Problem, which asks if every triangular shaped billiard table has a periodic billiard path. Finally, Schwartz plans to study basic geometric iterations such as those defined by iterated barycentric subdivision.The common theme in Schwartz's proposed research is the analysis of what happens when a simple geometric construction is repeated over and over again. For example, in outer billiards, a toy model for planetary motion, a point (the satellite) moves around the outside of a convex planar shape (the planet) according to a simple geometric rule. Resolving a 50 year old question about this system, Schwartz found examples of convex shapes, and starting positions for the point, such that the point wanders unboundedly far away from the shape while following the rules of the game. One might say that these shapes are examples of planets whose satellites can wander off into space. Iterated geometric constructions such as outer billiards are intellectually appealing and scientifically important because their simple definitions sometimes lead to extremely rich and mysterious behavior, as do analogous systems in the natural sciences.