Problems in Geometry and Dynamics

几何和动力学问题

• 批准号: 1204471
• 负责人: Richard Schwartz
• 金额: $ 31.23万
• 依托单位: Brown University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-07-01 至 2016-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1204471&HistoricalAwards=false
• 关键词: Problems; Geometry; Dynamics

Richard Schwartz plans to continue his work in geometry and dynamical systems. The first area of his focus will be the dynamics of particles in the complement of planar polygonal regions. Building on his solution of the 1960 Moser-Neumann question, Schwartz plans to continue developing the theory, finding connections to such subjects as Diophantine approximtion, self-similar tilings, and polytope exchange transformations. The second ares of his focus will be on optimization problems, such as the question of how a finite number of electrons optimally distribute themselves on the sphere. Specifically, Schwartz would like to extend his solution of Thomson's 5-electron problem to the case of other power laws. Finally, Schwartz plans to study a variety of geometric iterations, such as high dimensional iterated barycentric subdivision. For the most part, Schwartz's research involves studying what happens when one makes a simple geometric construction over and over again. For instance, one could follow a ball as it bounces endlessly inside a triangular table, or follow the motion of traffic as it moves along an infinite grid of one-way streets according to certain rules, or study the shapes one gets when one cracks a tetrahedron into smaller and smaller tetrahedra. Often these simple constructions, when done repeatedly, produce very beautiful and unexpected patterns. Schwartz likes to study such problems, at first, by doing computer experiments and then, later, by trying to construct rigorous proofs that the phenomena uncovered by the experiments actually exist.

理查德施瓦茨计划继续他的工作在几何和动力系统。 他关注的第一个领域将是平面多边形区域中粒子的动力学。基于他对1960年莫泽-诺依曼问题的解决方案，施瓦茨计划继续发展该理论，寻找与丢番图近似、自相似镶嵌和多面体交换变换等主题的联系。他的第二个重点战神将是优化问题，例如有限数量的电子如何最佳地分布在球体上的问题。 具体来说，施瓦茨想把他的解决方案汤姆逊的5电子问题的情况下，其他的权力法律。 最后，Schwartz计划研究各种几何迭代，如高维迭代重心细分。在很大程度上，Schwartz的研究涉及研究当一个人一遍又一遍地做一个简单的几何构造时会发生什么。例如，我们可以跟踪一个球在三角形桌子内无限反弹，或者跟踪交通的运动，因为它根据某些规则沿着单向街道的无限网格移动，或者研究当我们将一个四面体分裂成越来越小的四面体时得到的形状。这些简单的构造，如果反复进行，通常会产生非常美丽和意想不到的图案。 施瓦茨喜欢研究这类问题，首先是通过计算机实验，然后是试图构建严格的证据，证明实验发现的现象确实存在。