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Cohomology reveals when helicity is a diffeomorphism invariant

Presented by: 
J Parsley Wake Forest University
Thursday 6th December 2012 - 09:00 to 09:20
INI Seminar Room 1
Session Title: 
Knots in mathematics: Tight Knots, etc.
Session Chair: 
Rob Kusner
We consider the helicity of a vector field, which calculates the average linking number of the field’s 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^{2k+1}, 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 R^m.
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University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons