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Helicity, cohomology, and configuration spaces

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If you have a question about this talk, please contact Mustapha Amrani.

Topological Dynamics in the Physical and Biological Sciences

We realize helicity as an integral over the compactified configuration space of 2 points on a domain M in R^3. This space is the appropriate domain for integration, as the traditional helicity integral is improper along the diagonal MxM. Further, this configuration space contains a two-dimensional cohomology class, which we show represents helicity and which immediately shows the invariance of helicity under SDiff actions on M. This topological approach also produces a general formula for how much the helicity of a 2-form 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 is joint work with Jason Cantarella.)

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

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