Mathematical Sciences: Geometric Questions in Algebraic Topology

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Professor Mann will investigate two independent projects in algebraic topology. These projects study moduli spaces of instantons and piecewise linear group actions on piecewise linear manifolds and bordism modules. The first project, representing joint work with Professor C.P. Boyer, continues the ongoing program of Professors Boyer and Mann to study the geometry of moduli spaces of instantons, which arise in the differential geometric formulation of Yang-Mills gauge theory, by using the theory of iterated loop spaces and their associated homology operations. The second project, representing joint work with Professor E.Y. Miller, is based in large part on the work of Professors Mann and Miller on equivariant function spaces, classifying spaces for equivariant piecewise linear structures and associated action maps. This work will allow Professor Mann to answer geometric questions which, while well-known for smooth manifolds, are now unknown for piecewise linear manifolds. It is gratifying to find theoretical physics and topology stimulating each other as in the first project.