Mathematical Sciences: Topology

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This project covers a wide range of topics in algebraic and geometric topology. More specifically, the researchers plan to study the following: branched coverings and rational functions; finiteness and other arithmetic-like properties of certain discrete groups; geometric invariants of groups; Steenrod's problem in various equivariant contexts, including that of RO(G)- graded homology; homotopy G-actions; equivariant smoothing and triangulation theory; smooth, equivariant Rn-bundles; free products and surgery on knots; laminations in 3-manifolds; controlled simple homotopy theory; parameter spaces for iterations of complex maps. Such studies are basic to understanding the qualitative behavior of many models of physical or biological or even social systems. One often wishes to know about stability, periodicity, and other qualitative properties of models as much or more than one wants explicit and detailed numerical solutions. Whenever qualitative behavior is at issue, topology is sure to be involved at some level.