August 28, 2012

The Scientist: Prof. William Thurston’s Mathematical Legacy (1946-2012)

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Prof. William “Bill” Thurston was a world renowned mathematician who made  several strides in the field of low-dimensional topology, the study of how two-, three- and four-dimensional objects are oriented in space. Often referred to as a mathematical magician, Thurston was known for his ability to visualize the potential for complex shapes to fit into multidimensional spaces.  His work in 3-manifolds, which deals with phenomena occurring in three-dimensions, earned him the 1982 Fields Medal — an award often considered the Nobel Prize of Mathematics.

The Geometrization Conjecture

Topology takes a qualitative look at shapes and how they fit in multi-dimensional spaces — and how they stay the same when stretched.  One shape studied in low-dimensional topology is 2-manifolds, or two-dimensional surfaces.  Two-manifolds often act as the boundaries for solid objects when they are mathematically glued together, much like the surface of a sphere. The two-manifolds have coordinates that can be navigated in two-dimensions, X and Y, much like using latitude and longitude to navigate a globe.

Mathematicians create 3-manifolds by mathematically sticking the boundaries around three-dimensional spaces together. Thurston’s best-known achievement in a lifetime of breakthroughs, is the “Geometrization Conjecture,” which deals with 3-manifolds, shapes and spaces that can be navigated using three coordinates, such as X, Y and Z, or length, width and depth. According to the “Geometrization Conjecture,” Thurston postulated that any closed three-dimensional space could be decomposed into eight different geometric pieces.   Thurston compared the geometrization conjecture to finding eight outfits that could fit any person in the world, regardless of their body size.

“A big surprise in Thurston’s work on his Geometization Conjecture is that the most prevalent type of geometry to show up in 3-manifolds is not the usual ‘Euclidean’ geometry we are most familiar with,” said Prof. Tim Riley, mathematics, “But rather it is “hyperbolic” geometry — a geometry in which parallel lines do not obey the usual rules.”

Thurston was awarded the Fields Medal when he showed that the “Geometrization Conjecture” could be applied to large complex manifolds known as the Haken manifolds.  Thurston’s work inspired another renowned mathematician Grigori Perelman, to provide proofs for the “Geometrization Conjecture” and the famous hundred-year old “Poincaré Conjecture.”

Reflections From “Bill’s” Colleagues

Prof. John Hubbard, mathematics, said in an email that he was present at what might have been the first mention of the geometrization conjecture.

“I remember Bill saying: ‘there has been a lot of work on the Poincaré conjecture, and it hasn’t been very successful. Maybe that isn’t the right way to think of 3-manifolds. Instead, we should think of all 3-manifolds, and try to see what geometry they do have.’  He went on to propose eight possible geometries.” Thurston’s impact in topology was substantial.   Hubbard went on to say that, “In low-dimensional topology, there are two periods: BT (before Thurston) and AT.”

Prof. Daina Taimina, mathematics, reflected on one of Thurston’s lectures in which he was explaining structures in multi-dimensional spaces to his audience, she said he ventured into very large dimensions that most people couldn’t begin to comprehend. She wrote that, upon realizing that Thurston had lost most of his audience, he said, ‘Oh, I think I got too far; this was in 27-dimensions. Let us look at an easier example – let us consider only 10-dimensions.’ Taimina noted that this was her first encounter with Thurston’s way of visualizing mathematics, and that for a mathematical genius like him, dropping seventeen dimensions was his way of making things easier to see.

Thurston’s Legacy

In addition to Thurston’s many strides within topology, one of his most popular contributions was his eversion of the sphere. He turned a sphere inside out without producing any tears, creases or sharp bends.  His mathematical method for turning the sphere inside-out included folding it into parallel and alternating ridges and grooves and then twisting the sphere and unfolding the ridges, in a feat of mathematical genius that some say borders magic.

But Thurston is remembered for more than just his mathematical genius—he’s credited with changing the way a generation of mathematicians thought about geometry and topology.

On his MathOverflow profile, Thurston describes the study of mathematics as being “a process of staring hard enough with enough perseverance at the fog of muddle and confusion to eventually break through to improved clarity.” Taimina adds that Thurston “always stressed that mathematicians are searching for new results in order to uncover the internal beauty of mathematics.”

Check out Thurston’s sphere eversion video here

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Original Author: Nicholas St. Fleur