#### William P. Thurston, Theoretical Mathematician, Dies at 65

William P. Thurston, a mathematician who revolutionized understanding of the structure of three-dimensional spaces and won the Fields Medal, often described as the equivalent of the Nobel Prize for mathematics, died on Tuesday in Rochester. He was 65.

The cause was cancer, his son Dylan said.

Dr. Thurston s fields of expertise were geometry and topology, the study of different possible shapes for multidimensional space.

Perhaps his greatest accomplishment in a lifetime of breakthroughs was his Geometrization conjecture, which postulated that all possible three-dimensional spaces are made up of eight types of geometric pieces, a discovery he likened to finding eight outfits that could fit anybody in the world.

For most of his professional life, Dr. Thurston was among a very rarefied group in his field that thinks deep theoretical thoughts with no particular practical application, a luxury he reveled in.

I don t do it for the bottom line, Dr. Thurston told The Wall Street Journal in 1983. The inner force that drives mathematicians isn t to look for applications; it is to understand the structure and inner beauty of mathematics.

John Milnor, co-director of the Institute for Mathematical Sciences at Stony Brook University on Long Island, acknowledged that Dr. Thurston delighted in working in a very esoteric realm. But he added that Dr. Thurston s work had made a tremendous difference in the way we look at many problems.

Without that work, a Russian mathematician, Grisha Perelman, would not have been able in 2003 to solve the Poincaré conjecture, which asserts that the sphere is the only three-dimensional shape in which every loop in its structure can be shrunk to a single point, without ripping or tearing either the loop or the space. The problem had challenged mathematicians for 100 years.

In addition, cosmologists have drawn on Dr. Thurston s discoveries in their search for the shape of the universe.

On a more unlikely note, his musings about the possible shapes of the universe inspired the designer Issey Miyake s 2010 ready-to-wear collection, a colorful series of draped and asymmetrical forms. The fashion Web site Style.com reported that after the show, the house s designer and Dr. Thurston wrapped themselves for the press in a long stretch of red tubing to make the point that something that looks random is actually (according to Thurston) beautiful geometry.

To his colleagues, Dr. Thurston s most unusual gift was his ability to visualize complex shapes and problems. They said he loved nothing more than to sit in a common room and help colleagues or students brainstorm on solutions to vexing issues. He could look at a problem and see simplicity where nobody else could find it, said Jeff Weeks, a mathematician who studied with Dr. Thurston at Princeton.

For example, he could think in multiple dimensions. People don t understand how I can visualize four or five dimensions, Dr. Thurston told The Journal. 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.

William Paul Thurston was born on Oct. 30, 1946, in Washington, to Paul and Margaret Thurston. His father was a Naval engineer. William showed aptitude in math early on, amazing preschool teachers with his ability to add two- and three-digit numbers in his head, his older brother, Robert, recalled. Asked how he did it, he replied that he counted on his fingers in his mind, Robert Thurston said.

Dr. Thurston received an undergraduate degree at New College in Florida and a Ph.D. in mathematics at the University of California, Berkeley, in 1972. He quickly earned a reputation as an original thinker and caught the attention of Princeton s renowned mathematics department, which recruited him to be a professor at the age of 27.

He was in his mid-30s when he received the Fields Medal in 1982 for his work in deepening the connection between geometry and topology. (The medal is awarded every four years by the International Mathematical Union.) He worked longest at Princeton but also held posts at Berkeley and the University of California, Davis. He was most recently at Cornell.

Dr. Thurston s first marriage, to Rachel Findley, ended in divorce. In addition to his son Dylan and his brother Robert, he is survived by his wife, Julian Muriel Thurston; their children Hannah Jade and Liam; two children from his first marriage, Nathaniel and Emily; his mother, Margaret; a sister, Jean Baker; a brother, George; and two grandchildren.

Dylan Thurston, also a mathematician, said that despite working in a realm of rather cold abstractions, his father was personally very warm.

Growing up there were many beautiful mathematical pictures in the house, he said. He was a very visual thinker; he had powers to see spaces that no one before him could, and he was always drawing pictures of what he could see and doodles in notebooks, and we would talk about it.

Math was always very fun for him.

**L Kaufman**

August 22, 2012 © NY Times