Mathematical Sciences: The Geometry and Topology of Manifolds and Groups

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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.)