The Necessity of Abstraction: How the blind can help the sighted with 3-d visualization issues associated with Stokes Theorem

"The Necessity of Abstraction: How the blind can help the sighted with 3-D visualization issues associated with Stokes Theorem”

By

P. R. Kotiuga
ECE Department, Boston University, USA

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   Date

2. Location

Abstract

Given a closed contour and a solenoidal vector field, finding the flux linkage “through the contour”, is an inverse problem involving Stokes’ theorem. Typically one has a surface, and its boundary is easy to find, but here one has the boundary and it is the orientable surface which is sought. The same type of surface also serves as a “cut” for a magnetic scalar potential exterior to a knotted current carrying wire. In knot theory, such an orientable embedded surface is called a Seifert surface.

Finding cuts for magnetic scalar potentials amounts to realizing certain generators for relative homology groups as orientable manifolds. For knotted current paths, these cuts can be highly unintuitive. Although finite element based algorithms for finding such surfaces have been around for about two decades, the results are not necessarily intuitive. In order to aid visualization, there is a temptation to find the “simplest cuts” by imposing additional constraints, such as seeking:

1) embedded surfaces

2) surfaces which are level sets of solutions to elliptic equations

3) minimal area surfaces

4) minimal genus surfaces

Orientable embedded manifolds are easily realized as level sets of maps into the circle. Level sets of harmonic maps into the circle are readily computed, are a great aid in visualization, but articulating why is difficult. In contrast, seeking minimal area surfaces can lead to erroneous results, whereas finding minimal genus surfaces can take an exponential amount of work. This talk will show that the necessary abstractions for dealing with these dilemmas in three-dimensional visualization can be traced back to the seminal work of blind topologists and geometers! Notably, Plateau, and Pontryagin. To bring the discussion up to date, the role of cuts in defining isotopy invariant boundary conditions which render the curl operator self-adjoint on a multiply-connected domain will be considered. This problem arises in formulating boundary value problems for force-free magnetic fields in plasma physics, and relates to the seminal work in the area of contact structures by another blind topologist: Emmanuel Giroux.