Problems in Geometry and Dynamics

理查德·施瓦茨计划继续他在几何和动力系统方面的工作。他关注的第一个领域将是平面多边形区域互补中的粒子的动力学。在他对1960年Moser-Neumann问题的解决方案的基础上，Schwartz计划继续发展这一理论，找到与丢番图近似、自相似平铺和多面体交换变换等主题的联系。他关注的第二个领域将是优化问题，例如有限数量的电子如何在球体上以最佳方式分布的问题。具体地说，施瓦茨希望将他对汤姆森5电子问题的解决方案推广到其他幂定律的情况。最后，施瓦茨计划研究各种几何迭代，例如高维迭代重心细分。在很大程度上，施瓦茨的研究涉及一遍又一遍地研究当一个简单的几何结构时会发生什么。例如，当一个球在三角形桌子内无休止地弹跳时，你可以跟随它，或者按照一定的规则，当它沿着无限的单向街道网格移动时，你可以跟踪它的运动，或者研究当一个四面体分裂成越来越小的四面体时，你会得到什么形状。通常，这些简单的构造，当重复进行时，会产生非常美丽和意想不到的图案。施瓦茨喜欢研究这样的问题，一开始，通过做计算机实验，然后，通过试图构建严格的证据，证明实验发现的现象确实存在。

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.