- Title: The Sphere-Filling Curve Defined by the Figure-Eight Knot
Abstract: The biggest result in topology in the last fifty years
is Perelman’s proof of Thurston’s Geometrization Conjecture,
which claims that all $3$-manifolds admit a natural
rigid geometric structure. This conjecture includes the million-dollar
Poincare Conjecture, a prize that Perelman won but refused to accept.
I will explain the meaning of Thurston’s conjecture, though not
its long and difficult proof. The thoughts that led Thurston to
his conjecture are fascinating. The beginning for Thurston was
a claim of Robert Riley that the complement of the figure-eight knot admits
a geometric structure. Thurston had intuitive reasons for thinking
that Riley’s result could not possibly be true. But eventually Thurston changed
his mind and proceeded to show that almost all $3$-manifolds do
in fact admit natural geometric structures, a result eventually
subsumed by Perelman’s proof of the complete conjecture.
That Riley’s result was true led Thurston to another conjecture,
less important but fascinating, that the figure-eight knot defines a
natural $2$-sphere-filling curve (like the famous but artificial
space-filling curves of Peano, Hilbert, and Polya).