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Jason Parsley : Helicity, Configuration Spaces, & Characteristic Classes

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The helicity of a vector field in R^3, an analog to linking number, measures the extent to which its flowlines coil and wrap around one another. Helicity turns out to be invariant under volume-preserving diffeomorphisms that are isotopic to the identity. Motivated by Bott-Taubes integration, we provide a new proof of this invariance using configuration spaces. We then present a new topological explanation for helicity, as a characteristic class. Among other results, this point of view allows us to completely characterize the diffeomorphisms under which helicity is invariant and give an explicit formula for the change in helicity under a diffeomorphism under which helicity is not invariant. (joint work with Jason Cantarella, U. of Georgia)

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