History of Science: Russian mathematician quietly publishes paper – and solves one of mathematics’ most famous unsolved conjectures – November 11, 2002

On a cold day in November, a man living a quiet life in Russia posted a paper on a public server.

This book, published by “Grisha Perelman” and titled “The Ricci-style entropy formula and its geometric applications,” became the basis for one of the most important mathematical proofs.

The paper was the first of three published the following year to resolve the long-standing Poincaré conjecture, posed nearly a century earlier by Henri Poincaré.

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Simply put, Poincaré hypothesized that if you took any kind of 3D space, from a cat to the Empire State Building, and drew a 2D loop over it, if you could reduce that loop to a point without breaking either the loop or its shape, then the space would be mathematically equivalent to a sphere.

Proving this conjecture was crucial to topology, the mathematical study of shapes. Mathematician Stephen Smale solved this conjecture in five dimensions in 1961, winning mathematics’ prestigious Fields Medal in the process. However, the 3D case proved to be the most difficult to handle.

In the 1980s, Columbia University mathematician Richard Hamilton proposed solving this conjecture using a mathematical technique called Ricci flow, which was instrumental in Einstein’s theory of general relativity and string theory.

In 2006, New York Times reporter Dennis Overby compared Rich Flow technology to using the heat of a hair dryer to smooth out shrink wrap. Similarly, Ricci Flow can smooth out wrinkles and curves and reduce complex shapes to more basic shapes.

Rich flow worked to simplify round shapes into spheres, but singularities (points of infinite density) continued to appear in more complex shapes. Topologists could perform a kind of “surgery” to remove these singularities, but there was still a chance that they would continue to appear forever. Researchers were stuck.

Perelman’s work solved the singularity problem. Perelman (whose first name is Grigori, also spelled Grigori; Grisha is his nickname) has spent the past decade doing postdoctoral research at several institutions in the United States. In the mid-1990s, he turned down highly prestigious mathematics fellowships in the United States and Europe and returned to St. Petersburg to take up a post at the Steklov Institute of Mathematics.

Poincaré Conjecture – Numberphile – YouTube

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The friendly but shy and “other-worldly” mathematician “looked like Rasputin, with long hair and nails,” and told colleagues that he liked hiking and picking mushrooms in the forests around St. Petersburg, UCLA mathematician Robert Greene told Overby in 2006. Colleagues reported that he seemed to have no interest in wealth or material success.

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Perelman fell into obscurity after returning to Russia in the mid-to-late 1990s, and many of his colleagues thought he had left mathematics altogether.

Perelman then published a 2002 paper. The following year, he published two more papers and gave a series of lectures at several universities on the East Coast, explaining his process. Then he faded into the background again.

Perelman’s work showed that all singularities actually reduce to simple shapes such as spheres and tubes, and that if we could follow the Ricci process all the way to the end, we would find that 3D shapes can be reduced to spheres. Although he had proved the Poincaré conjecture, it took several more years for mathematicians to comb through his brilliant, ingenious, and highly technical proofs and confirm that the big problems about geography had actually been solved.

In 2006, mathematicians John Morgan and Gan Tian published a 473-page paper showing that Perelman’s work, which built on Hamilton’s work, did indeed prove the elusive conjecture.

Perelman was awarded the prestigious Fields Medal and the Clay Millennium Mathematics Prize, which comes with a $1 million prize. He reportedly turned them down because he objected to how credit should be given for problem-solving.

Mr. Perelman stepped down from his position at the Stekulov Institute in 2005 and has fiercely avoided the spotlight ever since. It’s unclear whether he still works on math in his St. Petersburg apartment, but in the early 2010s, neighbors said he was caring for his elderly mother.

When a reporter tried to contact him in 2010, he refused an interview, saying, “I’m in the way. I’m mushroom foraging.”