Davis and Orr investigate several issues concerning the classification of manifolds, as well as the classification of embeddings of manifolds in manifolds, i.e. knot theory. A variety of tools are used: surgery theory, localization and completion of groups, rings and spaces, index theory, von Neumann algebras, Whitney disks techniques, algebraic K- and L-theory, and controlled topology. Some problems of interest are the study of connected sums of manifolds and corresponding Nil and UNil phenomena, the concordance classification of knots, actions on a product of spheres, and the ribbon/slice problem. The goal, as usual in geometric topology, is to use a variety of algebraic, geometric, and analytic techniques to find and compute invariants for classification.
Geometric topology is the study of manifolds. An n-dimensional manifold is a set of points locally modeled on n-dimensional Euclidean space. For instance, a 2-manifold is a surface and looks like a plane near each point. Many physical phenomenon are represented by manifolds, and as such, understanding the global structure of a manifold, and what possible manifolds exist, is fundamental to the sciences, as well as to mathematics. Additionally, one asks how manifolds can sit within manifolds, a subject known as knot theory. Knot theory has been used to model genetic structures and chemical bonds.
