- 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).