Mathematical Sciences: Studies in Geometric Topology

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Professor Floyd plans to work in geometric group theory and topology. In an attempt to make some progress on the geometric group theory of nonpositively curved manifolds, he plans to study graph amalgamation products of finite groups over finite graphs. This will be joint work with Walter Parry. He intends to continue his research on the growth functions of hyperbolic groups and make the computer a more effective tool for studying growth functions. His other research topic is a joint project with Edwin E. Floyd on constructing (and computing the homology of) classifying spaces of finite groups via their Euclidean actions. Professor Quinn plans to work in geometric topology, and related topics in algebra and geometry. Specifically he plans to complete the development of controlled versions of K- and L-theory. This is expected to yield information about assembly maps from homology groups to K- and L- groups. In turn this should allow further computations of K- and L- groups of infinite groups with torsion, especially discrete subgroups of Lie groups. It should also shed light on the topological structure of stratified spaces, like algebraic varieties and quotients of group actions. In short, Floyd and Quinn are investigating the fundamental structure of geometric objects known as manifolds (spheres, toruses, and other more highly connected objects) as well as their symmetry properties, which can be described algebraically by groups.