skip to content
 

Cohomology reveals when helicity is a diffeomorphism invariant

Presented by: 
J Parsley Wake Forest University
Date: 
Thursday 6th December 2012 - 09:00 to 09:20
Venue: 
INI Seminar Room 1
Session Title: 
Knots in mathematics: Tight Knots, etc.
Session Chair: 
Rob Kusner
Abstract: 
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.
The video for this talk should appear here if JavaScript is enabled.
If it doesn't, something may have gone wrong with our embedded player.
We'll get it fixed as soon as possible.
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons