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Cohomology reveals when helicity is a diffeomorphism invariant

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Topological Dynamics in the Physical and Biological Sciences

We consider the helicity of a vector field, which calculates the average linking number of the fields flowlines. Helicity is invariant under certain diffeomorphisms of its domain we seek to understand which ones.

Extending to differential (k+1)-forms on domains R, we express helicity as a cohomology class. This topological approach allows us to find a general formula for how much helicity changes when the form is pushed forward by a diffeomorphism of the domain. We classify the helicity-preserving diffeomorphisms on a given domain, finding new ones on the two-holed solid torus and proving that there are no new ones on the standard solid torus. This approach also leads us to define submanifold helicities: differential (k+1)-forms on n-dimensional subdomains of Rm.

This talk is part of the Isaac Newton Institute Seminar Series series.

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