Poincaré conjecture

The Poincaré conjecture states that every closed, simply connected three-dimensional manifold is homeomorphic to the three-dimensional sphere. Grigori Perelman proved it using Richard Hamilton’s Ricci flow program.

Ricci flow: A geometric evolution equation that smooths a manifold’s metric by making curvature drive its change. Hamilton’s flow deforms three-dimensional geometry toward canonical shapes relevant to the conjecture.
Three-manifold: A topological space that locally resembles ordinary three-dimensional Euclidean space. The conjecture concerns the global topology of spaces built from locally three-dimensional neighborhoods.
Henri Poincaré: A French mathematician whose work helped establish topology as a mathematical field. He posed the three-dimensional problem in 1904 while investigating the topology of manifolds.
Grigori Perelman: A Russian mathematician who proved the geometrization conjecture using Ricci flow. His papers established the estimates and extinction results that complete the Poincaré proof.
Millennium Prize Problems: Seven mathematical problems announced by the Clay Mathematics Institute in 2000, each carrying a million-dollar prize. The conjecture was the first of these problems to be solved, though Perelman declined its prize.
Ricci flow with surgery: A method that cuts out developing singular regions during Ricci flow and continues the evolution on the remaining pieces. Perelman used surgery to control neck-pinches without losing the topological information needed for the proof.
Closed manifold: A compact manifold without boundary. Closedness rules out boundary behavior and noncompact ends in the conjecture.
Analysis Situs: Henri Poincaré’s foundational series of papers on topology, published from 1895 onward. The conjecture grew from questions Poincaré raised in his foundational study of spaces and their invariants.
Richard S. Hamilton: An American mathematician who introduced Ricci flow and developed it as a tool in geometric analysis. His program supplied the central method Perelman later advanced to completion.
Fields Medal: An international mathematics award given every four years to mathematicians under forty. Perelman was awarded the medal for his work, then declined it.

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