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.

Using his Ricci flow tool, Dr. Hamilton proceeded to tackle the challenging 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.

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