Mathematical Sciences: Geometry and Topology

这个项目的目标是理解非正曲率空间的大尺度几何，有限生成群的边界和有限秩自由群的自同构群。其他研究方向是研究齐次度量空间中线排列的位形空间和机械连杆与某些有限生成群（Artin和Coxeter群）的表示理论的关系，以及从拓扑学的角度研究“Riemann-Hilbert问题”。该项目部分关注几何、代数和微分方程中自然产生的物体的对称性结构。古希腊人理解柏拉图立体的对称性，例如立方体。柏拉图立体是正弯曲的，它们有有限的对称性。在光谱的另一端是负弯曲的物体，如罗巴切夫斯基双曲平面（在埃舍尔的绘画中经常出现），这些物体通常具有无限多的对称性。因此，这些对称性的结构既有趣又难以理解。该项目的另一部分是研究联系。这些是由连接在接头上的刚性杆构成的机械装置。用钉子固定连杆的另一端后，如何描述连杆的一端所勾勒出的可能形状是一个经典问题。例如，指南针可以看作是一个（非常简单的）连杆，它画出一个圆。该项目探索了由连杆绘制的形状与代数方程解的空间的某些对称性之间的惊人关系。项目的最后一部分是关于微分方程组产生的对称性。＊＊＊

9626633 Bestvina The goal of this project is to understand large-scale geometry of spaces of nonpositive curvature, boundaries of finitely generated groups and groups of automorphisms of free groups of finite rank. Other directions of research are study of configuration spaces of line arrangements and mechanical linkages in homogeneous metric spaces in relation with the representation theory of certain classes of finitely generated groups (Artin and Coxeter groups) and investigation of the ``Riemann-Hilbert problem'' from the topological point of view. The project focuses in part on the structure of symmetries of objects naturally arising in geometry, algebra, and differential equations. The ancient Greeks understood symmetries of Platonic solids, e.g., a cube. Platonic solids are positively curved, and they have a finite number of symmetries. At the opposite end of the spectrum are negatively curved objects, such as the Lobachevsky hyperbolic plane (featured frequently in Escher paintings), and these typically have infinitely many symmetries. Consequently, the structure of these symmetries is both more interesting and harder to understand. Another part of the project is the study of linkages. These are mechanical devices constructed from rigid rods connected at joints. It is a classical problem to describe the possible shapes traced out by an end of a linkage after affixing another end with a nail. For example, a compass can be regarded as a (very simple) linkage, and it traces out a circle. This project explores the surprising relationship between the shapes traced out by a linkage and certain symmetries of spaces of solutions of algebraic equations. The final part of the project is concerned with symmetries arising from a system of differential equations. ***