Principal Investigator: R.L. Cohen, S.P. Kerckhoff, R.J. Milgram
This project consists of research programs in a broad range of topics in Algebraic and Geometric Topology. The principal investigators, Professors R.L. Cohen, S.P. Kerckhoff, and R.J. Milgram are senior topologists whose research interests cover a large number of areas of topology and related fields. Included in these projects are the study of core topological questions such as the study of 3 - dimensional geometry, the study of the topology of complex and related structures on manifolds, and topological invariants of finite groups. In addition it includes applications of topological methods to engineering questions in the field of Robotics.
The topology of 3-dimensional manifolds is intimately connected to differential geometry, particularly to homogeneous structures like Euclidean, spherical, and hyperbolic geometry. Of these, hyperbolic geometry is the most prevalent. On a closed 3-manifold a hyperbolic structure, if it exists, is unique by the Mostow Rigidity Theorem. Thus, any geometric invariant is a topological invariant and can be used to distinguish different 3-manifolds. When the 3-manifold has boundary, it can have a large parameter space of hyperbolic structures; in this case, the qualitativeq properties of the structures as one varies over the parameter space are of particular interest. Kerckhoff studies hyperbolic structures on 3-manifolds, trying to describe the behavior of these geometric structures under a topological procedure, called Dehn surgery, and in the presence of singularities modeled on a cone. R.L. Cohen is pursuing several projects that involve studying algebraic topological aspects of questons stemming from geometry. These projects lie under the following general headings. 1. The topology of moduli spaces of holomorphic curves in complex and symplectic manifolds. 2. The homotopy type of the stable mapping class group and the Mumford Conjecture. 3. Properties and applications of holomorphic K-theory. R.J. Milgram will pursue questions involving the cohomology of finite simple groups, as well as applications of the topology of configuration spaces to questions in the design and behavior of robotic arms.
