Richard Hamilton, an inventive mathematician who devised the Ricci flow, a groundbreaking equation that helped advance understanding of the fundamental nature of three-dimensional space, died on Sept. 29 in Manhattan. He was 81.
The death, in a hospital, was confirmed by his son, Andrew. Dr. Hamilton had taught at Columbia University since 1998.
In 1982, Dr. Hamilton published “Three-manifolds with positive Ricci curvature” in The Journal of Differential Geometry. The article laid out his revolutionary theory: a kind of geometric analog to the heat equation in physics.
While the heat equation described how heat diffuses throughout space, as hot spots gradually merge with cooler regions, resulting in temperature equilibrium, the Ricci flow (named after the 19th-century Italian mathematician Gregorio Ricci-Curbastro) offered a model for understanding how irregular shapes can smooth themselves out, evolving into spheres.
Dr. Hamilton then went on to tackle an even more challenging problem: the Poincaré conjecture, which sought to understand the basic shape of three-dimensional space.
Initially posed by the French polymath Henri Poincaré in 1904, the conjecture hypothesized that any three-dimensional shape that was finite and closed, without any holes, could be deformed or stretched into a perfect sphere. In 2000, the nonprofit Clay Mathematics Institute made it a Millennium Prize problem, offering $1 million for a successful solution.
For years, Dr. Hamilton, struggled to find that solution. By the late 1990s, he was stuck.
As he told his students, it was important to choose a worthwhile problem — and “regardless of which problem you work on, you’re always going to get stuck,” said Lani Wu, a former Ph.D. student at the University of California, San Francisco, who worked with Dr. Hamilton.
In November 2002, Dr. Hamilton discovered that someone else had found the solution — but not without his help.
A paper by the Russian mathematician Grigori Perelman posted online offered a “sketch of an eclectic proof” of the Poincaré conjecture.
Over the years, Dr. Hamilton had spoken openly with Dr. Perelman about his efforts to solve the problem, and Dr. Perelman was generous about sharing credit with him. (He had also sent Dr. Hamilton an advance copy of his paper by email, but Dr. Hamilton, who paid little attention to email, never saw it.)
When Dr. Hamilton read the paper, he was impressed. “I’m about as surprised as anyone to see this all working,” he said later. “I’m enormously grateful to Grisha Perelman for finishing it off.”
By 2006, Dr. Perelman’s work had been verified by mathematicians and was celebrated in The New York Times as “a landmark not just of mathematics, but of human thought.” It was regarded as a crowning achievement of geometric analysis, and a milestone for the mathematical community.
But in 2010, Dr. Perelman turned down the Millennium Prize. “I had quite a lot of reasons both for and against,” he explained to a Russian news agency.
One reason not to accept: He believed Dr. Hamilton should share the credit.
Dr. Hamilton, ever the contrarian, joked that Dr. Perelman should have simply accepted the prize and given him half the money.
“The credit for the Poincaré conjecture should really be shared between Hamilton and Perelman,” said Gerhard Huisken, the director of the Oberwolfach Research Institute for Mathematics, in Germany, and a specialist in geometric analysis. “Whenever I talk about the proof of the Poincaré conjecture, I call it the Hamilton-Perelman proof.”
Richard Streit Hamilton was born on Jan. 10, 1943, in Cincinnati to William Selden Hamilton, a surgeon, and Hester Streit.
He taught himself advanced algebra in the fourth grade and skipped his last year at Walnut Hills High School in Cincinnati to attend Yale University when he was 16. He graduated with a bachelor’s degree in 1963. He received a Ph.D. in mathematics from Princeton in 1966 and began teaching at Cornell.
By the mid-1970s, Dr. Hamilton had begun thinking about the Ricci flow, inspired by the pioneering work of the Cornell mathematician James Eells Jr. and his collaborator, J.H. Sampson. Dr. Hamilton’s line of research evolved from the same mathematics — partial differential equations — underlying Einstein’s theory of relativity.
“While Einstein’s equations have shown us how curvature of space-time can be used to describe gravity phenomena,” Dr. Huisken said, “Ricci flow has opened deep and fascinating new links between concepts of curvature and diffusion phenomena in our physical world.”
It soon became apparent to some physicists that the Ricci flow was important to string theory — the “theory of everything,” as it’s called, that portrays point-like particles as one-dimensional vibrating strings. “We don’t fully understand the implications of this relationship, but we do know that it is pretty deep,” Edward Witten, a physicist at the Institute for Advanced Study in Princeton, N.J., said in an email.
Mathematicians in the know had an “immediate and positive reaction,” said Karen Uhlenbeck, a professor at the University of Texas, Austin and a distinguished visiting professor at the Institute for Advanced Study.
But many geometers were slow to appreciate the significance, said Richard Schoen, a professor emeritus at Stanford. “Richard spent quite a long time working on his own, developing some of the basic techniques that were used later to do the many things that the Ricci flow has accomplished,” he said.
Using the Ricci flow, Dr. Hamilton showed that sometimes the curvature of smooth objects, or manifolds, could evolve into spheres, almost as if you were inflating a collapsed beach ball. But occasionally, the curvature would blow up to infinity, swelling into a so-called singularity that possessed an extremely high curvature.
In pure mathematics, this realm of topology — or rubber-sheet geometry, as it is sometimes called — can be theoretically manipulated with a cut-and-paste procedure called surgery, and a new manifold can be created from an existing one. For the Ricci flow, Dr. Hamilton designed a type of surgery that “was completely novel,” said Simon Brendle, a professor of mathematics at Columbia.
At lectures, Dr. Hamilton liked to entertain audiences with stories of how “a manifold goes into the hospital,” and how he would then treat the manifold as if it were an appendix that had become inflamed. “See, my dad was actually a real surgeon,” he said in one lecture. “And when I was a kid, he told me about taking out appendices.”
“You want to cut it out and throw away the appendix,” he continued. “And then you sew on a nice cap. And the idea is that you have reduced the inflammation, because you got rid of where the curvature was real big.”
But even to Dr. Hamilton, some problems seemed intractable. For instance, he worried that the Ricci flow might produce a cigar-shaped singularity that could not be altered and would pose a major obstacle in his quest to solve the Poincaré problem.
“His goal was to understand how singularities can form,” Dr. Brendle said. “That was at the heart of Richard’s vision for attacking the Poincaré conjecture.”
In 1965, Dr. Hamilton married Sally Harper Swigert; they divorced in around 1968. In addition to their son, he is survived by Susan Harris, his longtime partner, whom he met in the mid-1980s while teaching at the University of California, San Diego, where she was studying mathematics.
Dr. Hamilton became a visiting professor at the University of Hawaii at Manoa in 1988, and an honorary adjunct professor there in 2022. He bought a house in Haleiwa, on the north shore of Oahu, where he surfed, water-skied, wind surfed, went scuba diving, swam and snorkeled with dolphins when he wasn’t doing math.
Moving to Columbia University in 1998, he continued to work on his own exposition of the Poincaré proof — simplifying parts, combining techniques, adding new ideas.
Among the awards Dr. Hamilton won were the 2011 Shaw Prize in Mathematics, which he shared with Demetrios Christodoulou of ETH Zurich, and a Basic Science Lifetime Award, worth $700,000, which he received in July.
The day after Dr. Hamilton’s death, Mu-Tao Wang, a Columbia colleague, announced the news at the Simons Laufer Mathematical Sciences Institute, in Berkeley, where more than 100 researchers from around the world had convened to attend programs on curvature, flows, general relativity, special geometric structures and the like — subjects to which Dr. Hamilton had made foundational contributions.
“Richard has equipped mathematicians with new tools to take on fundamental questions that were out of reach before,” Dr. Brendle said in an interview.
The Poincaré proof is only one among many major applications of the Ricci flow.
In 2007, Dr. Brendle and Dr. Schoen used the Ricci flow to prove “the differentiable sphere theorem” — a way to identify sphere-like shapes in higher dimensions. And in April, Dr. Brendle won the Mathematics Breakthrough Prize, which comes with a cash award of $3 million, in part for work inspired by Dr. Hamilton’s research.
Shing-Tung Yau, a professor emeritus at Harvard and the director of the Yau Mathematical Sciences Center at Tsinghua University in Beijing, was Dr. Hamilton’s friend and collaborator for half a century. He once joked that his friend was a “madman” for his Ricci flow idea. (Dr. Hamilton, naturally, took this as a compliment.)
Eventually, he said, he realized that Dr. Hamilton was a hero for the advancements he made to modern geometry. “I was totally amazed by his creativity and power,” he said.
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