Topics in Geometry and Dynamics

这个项目涉及几何学和动力学系统的研究，其中一些利用计算机实验来研究人们知之甚少的简单问题。他们的想法是，新的强大的计算工具将提供前几代数学家无法获得的洞察力。一个这样的问题被称为汤姆森问题，它问的是一些电子将如何在球体上排列，以使它们的总势能最小化。这样的问题有广泛的应用，因为在许多科学领域，知道如何在空间中均匀分布点、传感器或能源是非常有用的。首席研究员最近解决了球面上五点案件的很大一部分问题，并计划继续发展这一理论。另一个问题是关于太阳系的稳定性，这是一种高度简化的天体力学模型，称为外部台球。众所周知，存在相应轨道无界的初始条件；本项目旨在进一步发展一种理论，在特殊情况下提供这些轨道的详细理论图景。该项目正在研究的另一个问题是著名的正方形钉子问题，该问题询问平面上的每个环上是否有四个点构成正方形的角点。在技术方面，该项目侧重于以下几个方面。首先，计划推广与5电子问题的相变猜想有关的结果。主要研究人员证明了存在一个常数c使得三角双金字塔是Riesz S势的能量极小化当且仅当S不大于c。我们猜想某个正方形底的金字塔是Riesz S势的能量极小化当且仅当S不小于c。第二，该项目的目标是继续发展格子模式，这是一种组合结构，它为每个有理数分配一个包含在立方体中的多面体曲面的集合。当模型在一个坐标方向上切片时，它给出的循环可以精确地模拟具有相同参数的外部台球相对于风筝的某些轨道。当模型在其他两个坐标方向上切片时，它给出的循环在组合上与Truchet平铺和角落渗流中的循环同构。第三，首席研究员计划继续对约旦环中内接矩形的空间进行研究。他证明了每个Jordan环都与一个连通的矩形集合相关联，使得该环的每个点(至多有4个例外点)都是该集合中一个矩形的顶点。这一结果导致了一个问题，即一个矩形如何沿着一对圆弧连续滑动，这个问题类似于研究微分方程式的解。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.