Mathematical Sciences: Chern-Weil Theory for Topological Groups

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David Stone will work on geometric problems whose motivation comes from computational physics. The techniques to be used come from geometry and topology. He will be applying combinatorial methods where numerical integration has been used in the past. Stone's research concerns the general problem of extracting global topological information about a space from local geometric data. In classical differential geometry the data are presented in infinitesimal form, and the Chern-Weil theory constructs from them further infinitesimals which, when integrated, yield the desired characteristic classes. Stone will develop a systematic analogue of the entire Chern-Weil theory that does not use infinitesimals but employs techniques of algebraic topology. This work promises to have physical applications in the theory of lattice gauge fields.