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Helicity, cohomology, and configuration spaces

Presented by: 
J Parsley [Wake Forest University, USA]
Thursday 25th October 2012 - 11:30 to 12:30
INI Seminar Room 1
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.)
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Presentation Material: 
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons