Manifolds are geometric objects that can be parametrized using the standard numerical variables of algebra and calculus. They are fundamental objects for much of modern mathematics and its applications to the other sciences, where they model physical space as well as many important data sets. The principal investigator is planning to study properties of higher dimensional manifolds. Here "higher dimensional" means that at least five variables are needed to parametrize the manifold (think of a data set depending on five or more variables). The extra flexibility inherent in a larger number of variables means that many more tools can be brought to bear on problems. The geometry of the expanding networks that are important in communications theory and the theory of dynamical systems (i.e.,change under symmetries of a system) are particularly important tools for the project. Throughout the period of the project the principal investigator intends to mentor young scientists through their involvement in the research, and to continue to work on structural improvements to mathematical education and research in Hawaii.
One of the most important invariants of a manifold is its fundamental group, which governs the way loops can be formed inside the manifold. Associated to fundamental groups are algebraic group rings and analytic group operator algebras. The theory of linear algebra over group rings and group operator algebras is organized by their K-theory, and understanding the associated K-groups is a difficult problem that is fundamental to the understanding of higher dimensional manifolds via such problems as the Novikov and Borel conjectures in topological classification and the Gromov-Lawson conjecture in differential geometry. The Baum-Connes and Farrell-Jones conjectures postulate techniques for computing K-theory and anchor the current approach to the problems in manifold theory mentioned earlier. The principal investigator intends to use techniques from index theory of elliptic differential operators, operator algebras, topological dynamical systems, metric geometry of expanding networks, and representation theory to study these conjectures and related issues.
