Topics in Geometry and Dynamics

这项研究项目涉及分析当一个简单的几何结构无限重复时出现的模式。一个与空间中形状的排列有关的几何问题的例子是：如果一个人(无休止地)将等边三角形粘合在一起，使得每个顶点周围有7个三角形，那么产生的曲面是否可能嵌入到3维空间中？一个与平面几何形状有关的问题的例子是确定无摩擦球在多边形桌内侧(无休止地)滚动并从侧面反弹时可能扫出的形状。这样的过程既发生在自然界中，也发生在纯粹抽象的数学背景中，并经常产生神秘、复杂和美丽的图案。这些过程与数论、几何学和物理学有关。这项研究项目通过数值模拟来研究这些过程，以产生假说，这些假说随后通过严格的数学论证得到证实或反驳。从更专业的角度讲，首席研究员将继续研究几何动力学，集中在多面体交换变换和外部台球上。这项工作的一个共同主题是构造非紧致低维动力系统的高维紧致。另一个共同的主题是重整化现象的出现。在感兴趣的情况下，有一个纤维丛，其中基空间是参数空间，每个纤维是基于它所在的参数的多面体交换变换。然后，目标是找到基本空间的辅助变换，其将一点上的光纤的动力学与变换下的点的图像上的光纤的动力学联系起来。例如，当一个人研究一个过程的符号动力学时，这似乎就会发生，这个项目的目的是找到一个普遍的理论来解释这一点。主要研究人员还将研究一些几何迭代，例如五角形映射，一种与可积系统和簇代数相关的多边形空间上的射影自然映射。最后，首席研究人员计划解决一些长期悬而未决的几何学问题，例如方形钉子猜想。

This research project concerns the analysis of the patterns that emerge when a simple geometric construction is repeated indefinitely. An example of a geometric problem having to do with the arrangement of shapes in space is: If one (endlessly) glues together equilateral triangles so that 7 triangles go around each vertex, is it possible to embed the resulting surface in 3-dimensional space? An example of a problem having to do with geometric shapes in the plane is to determine the possible shapes swept out by the path of a frictionless ball as it rolls (endlessly) around the inside of a polygonal table and bounces off the sides. Such processes occur both in nature and in a purely abstract mathematical context, and often produce mysterious, intricate, and beautiful patterns. These processes have connections to number theory, geometry, and physics. This research project studies these processes via numerical simulation to generate hypotheses that are subsequently confirmed or disproved via rigorous mathematical justification.Speaking more technically, the principal investigator will continue investigations into geometric dynamics, concentrating on polytope exchange transformations and outer billiards. One common theme in this work is the construction of higher dimensional compactifications of non-compact low-dimensional dynamical systems. Another common theme is the appearance of renormalization phenomena. In the situation of interest, one has a fiber bundle where the base space is the parameter space and each fiber is a polytope exchange transformation based on the parameter it lies over. The goal is then to find an auxiliary transformation of the base space which relates the dynamics at the fiber over one point to the dynamics of the fiber over the image of the point under the transformation. For instance, this seems to happen when one studies the symbolic dynamics of a process, and this project aims to find a general theory for this. The principal investigator will also study a number of geometric iterations, such as the pentagram map, a projectively natural map on the space of polygons that is related to integrable systems and cluster algebras. Finally, the principal investigator plans to attack a number of longstanding unsolved problems in geometry, such as the square peg conjecture.