Famed mathematician William Thurston died Tuesday 21 Aug 2012, at his home in Rochester, New York, from cancer. He was arguably one of the handful of 20th century mathematicians — pure or applied — who will be discussed in some detail in 22nd century histories of mathematics and science. As Edward Tenner wrote in the Atlantic Even as he contributed to theoretical physics, Bill’s work was proof that the most abstract math can have gorgeous practical applications.
Although he did work in several areas, the majority of Thurston’s research work was in geometry and topology, namely the branch of mathematics that focuses on the structure of surfaces and shapes in multidimensional space, and the extent to which they can be continuously deformed. He was a winner of the 1982 Fields medal for some of this work. Even among Fields medalists (all extraordinary mathematicians) he was held in some awe.
His most widely known accomplishment was his “geometrization conjecture,” which proposed that all possible three-dimensional spaces are constructed from eight general types of geometric objects. He once drew the analogy to finding that a combination of eight outfits could fit anyone in the world.
John Milnor, of the Institute for Mathematical Sciences at Stony Brook University, and a 1962 Fields medalist, emphasised that Thurston has changed the way mathematicians and physicists view many problems. Thurston’s work, for instance, laid the foundation for the 2003 proof of the Poincare conjecture by reclusive Russian mathematician Grisha Perelman. Perelman showed that the sphere is the only three-dimensional shape in which every loop in its structure can be shrunk to a single point, by continuous deformations that do not rip or tear the space. This was a problem that had challenged mathematicians for at least 100 years.
Thurston acknowledged that he had a special talent for thinking in multiple dimensions. “Five-dimensional shapes are hard to visualize — but it doesn’t mean you can’t think about them. Thinking is really the same as seeing.”
Thurston was quite vocal about the often disappointing quality of communication and teaching in the field of mathematics. His views on experimentation were consistent with those of the present bloggers. He wrote in Proof and Progress in Mathematics that
Mathematicians have developed habits of communication that are often dysfunctional. Organizers of colloquium talks everywhere exhort speakers to explain things in elementary terms. Nonetheless, most of the audience at an average colloquium talk gets little of value from it. Perhaps they are lost within the first 5 minutes, yet sit silently through the remaining 55 minutes. Or perhaps they quickly lose interest because the speaker plunges into technical details without presenting any reason to investigate them. At the end of the talk, the few mathematicians who are close to the field of the speaker ask a question or two to avoid embarrassment.
This pattern is similar to what often holds in classrooms, where we go through the motions of saying for the record what we think the students “ought” to learn, while the students are trying to grapple with the more fundamental issues of learning our language and guessing at our mental models. Books compensate by giving samples of how to solve every type of homework problem. Professors compensate by giving homework and tests that are much easier than the material “covered” in the course, and then grading the homework and tests on a scale that requires little understanding. We assume that the problem is with the students rather than with communication: that the students either just don’t have what it takes, or else just don’t care. Outsiders are amazed at this phenomenon, but within the mathematical community, we dismiss it with shrugs.
For additional details, see Leslie Kaufman’s New York Times article, from which some of the above material was adapted. Also see Evelyn Lamb’s Scientific American article, which has some additional notes on his life and work. Finally, it would be remiss to suggest that as a great thinker Bill Thurston avoided more routine academic duties or arduous administration. He directed 33 PhD students and was the third Director from 1992-1997 of MSRI at Berkeley, a pre-eminent mathematics research centre.