A Taste of the 2012 Joint Mathematics Awards and Prizes

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William Thurston Received the Leroy P. Steele Prize for Seminal Contribution to Research

William Thurston, past Fields Medal recipient: Image by Bergman, George M

Over recent decades, William Thurston has written a number of papers concerning low-dimensional topology. His 1986 paper, Hyperbolic structures on 3-manifolds. (I.) Deformation of acylindrical manifolds, is said to have “revolutionized 3-manifold theory”.

Thurston continued to develop connections between low dimensional topology and complex analysis, hyperbolic geometry and dynamical systems.

The citation notes that Thurston had made a conjecture that gave a complete picture of all compact 3-manifolds, but proved his conjecture only for a large number of examples. Other mathematicians, notably Grigori Perelman, completed the proofs using alternative approaches.

Decoded Science asked Allyn Jackson of the American Mathematical Society a few additional questions about this award:

Decoded Science: “William Thurston’s “seminal contribution to research” dates from 1986… He is also the only recipient named as winning the Fields medal. That was in 1982, also for 3-manifolds. Is there any significance to the long time between that award and this”?

Allyn Jackson: “The Steele Prize for a Seminal Contribution to Research, which is what Thurston got, honors work that with hindsight one can see had a major impact on the field.  So it is not unusual that winners of this Steele Prize are honored for work that is a few decades old.”

Decoded Science: “Which recipient’s work would spark an ‘aha!’ moment for a Decoded Science reader, who might be a knowledgeable amateur rather than a professional mathematician?”

Allyn Jackson: “Well, this is very difficult to say, as ‘aha!’ moments vary so much from individual to individual.  I can only state my personal opinion. To me, Thurston’s work is monumental for the way it established a sweeping new vision of low-dimensional topology.  His work gets at the heart of the basic notion of shape.

However, the proof of the conjecture in full generality came only with the work of Grigory Perelman.  Perelman as a mathematician is very different from Thurston, with a very different approach and style.  He carried out a program that had been initiated by Richard Hamilton and succeeded in proving the full Geometrization Conjecture, using means from mathematical analysis.  Thurston, by contrast, is a geometer/topologist, and his work does not use so much analysis.  This is an example of something I mentioned earlier, about very exciting breakthroughs coming from connections between seemingly disparate areas.”

Understanding Topology and Geometry

Topology is a relative of geometry, but studies “the properties that are preserved through deformations, twistings, and stretchings of objects” without tearing. Geometers tease topologists by saying that the topologists cannot distinguish a diamond ring from a doughnut that has a hole.

A manifold is a “topological space that is locally Euclidean”. The planet Earth is an example of a simple 3-dimensional manifold. The surface of the Earth seems “flat” over a football field, so the surface is “locally Euclidean”. As seen from the Moon, of course, the Earth is a sphere and the surface is convex, not flat.

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