Topics in Geometry and Dynamics

几何和动力学主题

• 批准号: 1807320
• 负责人: Richard Schwartz
• 金额: $ 38.32万
• 依托单位: Brown University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2018
• 资助国家: 美国
• 起止时间: 2018-07-01 至 2022-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1807320&HistoricalAwards=false
• 关键词: Topics; Geometry; Dynamics

This project concerns research in geometry and dynamical systems, some of which uses computer experimentation to study simply-stated questions about which little is known. The idea is that new and powerful computing tools will provide insight that was not available to earlier generations of mathematicians. One such question, known as Thomson's problem, asks how some number of electrons will arrange themselves on the sphere so as to minimize their total potential energy. Such a problem has wide-ranging applications because in many scientific fields it is quite useful to know how to evenly distribute points or sensors or energy sources in a space. The principal investigator recently resolved a large part of the story for the case of five points on the sphere and plans to continue to develop this theory. Another problem asks about the stability of the solar system in a highly simplified model of celestial mechanics called outer billiards. It is known that there exist initial conditions for which the corresponding orbits are unbounded; this project aims to further develop a theory that provides detailed theoretical pictures of these orbits in a special case. Another problem under investigation in the project is the famous square peg problem, which asks if every loop in the plane has four points on it that make the corners of a square. In technical terms, the project focuses on the following areas. First, it is planned to extend results related to a phase-transition conjecture for the 5-electron problem. The principal investigator proved that there exists a constant c such that the triangular bi-pyramid is the energy minimizer for the Riesz s-potential if and only if s is not greater than c. It is conjectured that some pyramid with square base is the energy minimizer for the Riesz s-potential if and only if s is not less than c. Second, the project aims to continue developing the plaid mode, a combinatorial construction that assigns to each rational number a collection of polyhedral surfaces contained in a cube. When the model is sliced in one coordinate direction, it gives loops which accurately model certain orbits of outer billiards with respect to the kite with the same parameter. When the model is sliced in the other two coordinate directions it gives loops that are combinatorially isomorphic to those found in Truchet tilings and in corner percolation. Third, the principal investigator plans to continue an ongoing study of the space of inscribed rectangles in a Jordan loop. He has established that every Jordan loop has associated to it a connected set of rectangles such that every point of the loop, with at most 4 exceptional points, is the vertex of one of the rectangles in the set. This result leads to a question of how a rectangle can slide continuously along a pair of arcs, a problem akin to studying the solutions of a differential equation.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.

该项目涉及几何和动力系统的研究，其中一些使用计算机实验来研究知之甚少的简单问题。这个想法是，新的和强大的计算工具将提供洞察力，是不提供给前几代的数学家。 一个这样的问题，被称为汤姆逊问题，询问一些电子将如何在球体上排列，以使它们的总势能最小化。 这样的问题具有广泛的应用，因为在许多科学领域中，知道如何在空间中均匀地分布点或传感器或能量源是非常有用的。 首席研究员最近解决了很大一部分的故事的情况下，五个点的领域，并计划继续发展这一理论。 另一个问题是关于太阳系在一个高度简化的天体力学模型中的稳定性，这个模型被称为外台球。 已知存在相应轨道无界的初始条件;该项目旨在进一步发展一种理论，提供这些轨道在特殊情况下的详细理论图像。 该项目正在研究的另一个问题是著名的方形桩问题，该问题询问平面上的每个环是否有四个点构成正方形的四个角。 从技术上讲，该项目侧重于以下领域。 首先，计划扩展与5电子问题的相变猜想有关的结果。 主要研究者证明了存在一个常数c，使得三角双锥是Riesz s-势的能量极小当且仅当s不大于c。 证明了某个方底金字塔是Riesz s-势的能量极小的充要条件是s不小于c.其次，该项目旨在继续开发格子模式，这是一种组合结构，为每个有理数分配一个立方体中包含的多面体表面集合。 当模型在一个坐标方向上切片时，它给出了精确地模拟外部台球相对于风筝具有相同参数的某些轨道的回路。 当模型在其他两个坐标方向上切片时，它给出了与Truchet平铺和角渗透中发现的那些组合同构的循环。 第三，首席研究员计划继续进行正在进行的研究的空间内接矩形在约旦循环。 他已经建立，每一个约旦循环已与它相关联的一组连接的矩形，使每一个点的循环，最多有4个例外点，是顶点之一的矩形中的一组。 这一结果引出了一个问题，一个矩形如何能够沿着沿着一对弧连续滑动，这一问题类似于研究微分方程的解。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。