In this project we will study geometric structures and the structure of representation varieties. These topics are a natural outgrowth of previous research programs that focused on low-dimensional hyperbolic manifolds. The areas of Kleinian groups and the geometry of 3-manifolds have seen an amazing amount of progress over the last decade. As a result these areas are in a position to refocus their efforts. The situation has become more like that of surfaces where one is interested in families and spaces of structures. It is important to try to understand the genealogy of 3-manifolds, how they are related by various topological and geometric operations. This has led to the study of many other types of geometric structures, such as projective and Lorentzian structures, that don't have an invariant metric. These structures are interesting in their own right. They also provide a context in which to view the relation between the different eight 3-dimensional metric geometries, leading to the concept of transitional geometry. Furthermore, there are interesting questions about geometric structures in other dimensions, involving an array of different Lie groups, such as hyperbolic structures in high dimensions, complex and real projective structures on surfaces, and representations of surface groups into higher rank Lie groups.
The idea of studying various types of metrics on spaces dates back at least to the late 19th century and the work of Poincare, Klein, and others. Much of their motivation came from the desire to understand physical phenomena. Modern physics, beginning with Einstein, has led to an even greater need for sophisticated mathematics to understand the physical universe, particularly that coming from metric geometry. Although the physical world is not a completely homogeneous one like the type of structures studied in this project, hyperbolic and Lorentzian geometry are believed to represent useful models for understanding physical phenomena. Geometry in dimension 3 is particularly appealing because it is visually accessible to many people, including beginning mathematics students and those with less technical backgrounds. It also has led to the creation of a number of graphical interfaces that have been widely utilized.
