Mathematical Sciences: Classification Problems in Geometric Topology

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The principal investigator intends to focus on classifying certain kinds of differentiable manifolds or group actions on manifolds using a combination of methods from surgery and stable homotopy theory. In particular, he would like to study the diffeomorphism classification of certain homogeneous spaces and complete intersections, give a stable classification of four- manifolds by spectral invariants, and determine the relationship between spherical involutions and Mahowald root invariants. These classification questions all involve extracting geometric information from algebraic information about the manifolds or group actions in question.