Mathematical Sciences: Geometry and Topology

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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. ***