Mathematical Sciences: Geometric Questions in Algebraic Topology

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8901879 Mann Professor Mann will investigate two projects in algebraic topology which, in addition to their topological interest, have applications in mathematical physics, non-linear partial differential equations, and linear control theory. The first project, representing joint work with Professor C. P. Boyer, continues the ongoing program of Professors Boyer and Mann to study the geometry and topology of moduli spaces of instantons and monopoles which arise in the differential geometric formulation of Yang-Mills-Higgs gauge theories. The techniques developed in this program have applications to other moduli problems which will also be investigated. The second project, representing joint work with Professor R.J. Milgram, attempts to extend the results of Professors F.R. Cohen, R.L. Cohen, B.M. Mann, and R.J. Milgram on the more general case of rational functions from the Riemann sphere to Grassmannians and flag manifolds. The topology of such spaces is important in several areas, including linear control theory, and the results of this project will be useful in applications to other areas of mathematics.