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

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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