Series: Mathematics Colloquium

Date: Thursday, February 12, 2004

Time: 4:00 - 5:00 PM

Place: 102 McAllister Building

Host: Mohammad Ghomi

Refreshments: 3:15 - 4:00 PM, in 212 McAllister

Speaker: Herman Gluck, University of Pennsylvania

Title: 

  The Gauss Linking Integral on the 3-Sphere and in Hyperbolic 3-Space

Abstract:  

  Gauss's integral formula (1833) for the linking number of two
  disjoint smooth closed curves serves as a cornerstone of geometric
  knot theory in Euclidean 3-space, with rich connections to molecular
  biology (especially DNA structure) through the related formula for
  the writhing number of a single curve, and to fluid dynamics,
  electrodynamics and plasma physics through the related formula for
  the helicity of a vector field.  Gauss was interested in computing
  the linking number of the earth's orbit with the orbits of certain
  asteroids, and although he presented his formula without proof, it
  is believed that he simply counted up how many times the vector from
  the earth to the asteroid covered the celestial sphere ... a
  degree-of-map argument.  Gauss undoubtedly knew another proof: run a
  current through the first loop, and calculate the circulation of the
  resulting magnetic field around the second loop.  By Ampere's Law,
  this circulation is equal to the total current enclosed by the
  second loop, which means the current flowing along the first loop,
  multiplied by the linking number of the two loops.  Then the
  Biot-Savart formula for the magnetic field leads directly to Gauss's
  linking integral.  We report here on the discovery and proof of the
  formulas for linking, writhing and helicity on the 3-sphere and in
  hyperbolic 3-space.  Gauss's degree-of-map proof does not work on
  the 3-sphere; instead, we develop there a rudimentary form of
  classical electrodynamics, including a Biot-Savart formula for the
  magnetic field and a corresponding Ampere's law, sufficient to lead
  us to the linking integral.  By contrast, a degree-of-map proof does
  work in hyperbolic 3-space.