Mathematical Sciences: The Geometry and Topology of Manifolds and Groups

这个项目涉及几个公开的问题，以及几何、拓扑学和群论之间的界面问题。具体地说，Bestvina打算继续与R.D.爱德华兹合作，研究希尔伯特-史密斯猜想，一个关于流形拓扑的经典猜想，以及一个与R-树上的群作用动力学有关的研究计划。R-树的概念，即以实数为模型的树状空间，近年来已经发展成为拓扑学和群论中的一个强有力的研究工具。了解R-树将使我们更深入地理解负曲线空间的万能覆盖的退化。根据这一新理论，首先在三维流形中发现的某些现象，特别是特征子流形、表面同态的分类和测量的层片的作用，具有更广泛的意义，正如Gromov、Rips和许多其他人的工作所清楚的那样。Mess打算研究有关离散和几何结构的各种问题，特别是与3-流形有关的问题。过去十年在这一领域的工作清楚地表明了几何分析在发展新见解方面的力量。考虑到这一点，梅斯试图更多地了解几何结构通过层合的变形，并了解一致准对称群。一个对象的对称性群在试图理解该对象时起着重要的作用。相反，抽象群通常作为空间的对称群出现。在这个项目所涉及的情况下，空间具有分形性，这使得空间的分形性几何和群的性质之间产生了迷人的相互作用。(分形几何的特征是在无限的衰退中，在越来越小的尺度上重复类似的结构。它是近年来作为计算机建模的自然理论构造而形成的一个概念，并且作为实际问题，它使计算机图形学能够从非常简单的程序中描绘出异常逼真的山脉、云等。)

This project concerns several open problems and questions at the interface between geometry, topology, and group theory. Specifically, Bestvina intends to continue working with R. D. Edwards on the Hilbert-Smith Conjecture, a classical conjecture on the topology of manifolds, and on a program of research related to the dynamics of group actions on R-trees. The notion of an R-tree, i.e. a tree-like space modelled on the real numbers, has developed into a powerful research tool in topology and group theory in recent years. Understanding R-trees will lead to a deeper understanding of the degeneration of the universal cover of a negatively curved space. Certain phenomena first discovered in 3-dimensional manifolds, in particular, the characteristic submanifold, the classification of surface homeomorphisms, and the role of measured laminations, have a wider significance in light of this new theory, as has become clear through the work of Gromov, Rips, and many others. Mess intends to work on various problems about discrete and geometric structures, especially in connection with 3-manifolds. Work in this area over the past decade has made clear the power of geometric analysis in developing new insights. With this in mind, Mess seeks to understand more about the deformations of geometric structures by means of laminations, and to understand uniformly quasisymmetric groups. The group of symmetries of an object plays an important role in an attempt to understand the object. Conversely, abstract groups often arise as groups of symmetries of a space. In the cases this project deals with, the spaces have fractal nature, which makes for a fascinating interplay between the fractal geometry of the spaces and the properties of the groups. (Fractal geometry is characterized by the repetition of similar structures on ever smaller scales in an infinite recession. It is a notion that has come into its own in recent years as a natural theoretical construct for a computer to model, and, as a practical matter, it has enabled computer graphics to depict unusually realistic mountains, clouds, etc. from remarkably simple programs.)