Henri Poincaré | Raymond Poincaré | Poincaré sphere | Goldbach's conjecture | Poincaré | ''n''! conjecture | n! conjecture | Kato's conjecture | Calabi conjecture | Weil conjecture | ''Uncle Petros and Goldbach's Conjecture'' by Apostolos Doxiadis | Uncle Petros and Goldbach's Conjecture | Schanuel's conjecture | Pollock's conjecture | Poincaré disk model | Poincare | Mumford conjecture | Kepler conjecture | Heawood conjecture | Franc Poincaré | Chang's conjecture | Catalan's conjecture | Blattner's conjecture | Beal's conjecture |
Henri Poincaré's 1895 paper Analysis Situs studied three-and-higher-dimensional manifolds(which he called "varieties"), giving rigorous definitions of homology, homotopy (which had originally been defined in the context of late nineteenth-century knot theory, developed by Maxwell and others), and Betti numbers and raised a question, today known as the Poincaré conjecture, based his new concept of the fundamental group.
The solution by Smale, in 1961, of the Poincaré conjecture in higher dimensions made dimensions three and four seem the hardest; and indeed they required new methods, while the freedom of higher dimensions meant that questions could be reduced to computational methods available in surgery theory.
He has advised Grigori Perelman, who solved the Poincaré conjecture, one of the seven Millennium Prize Problems.
In 1999, the Clay Mathematics Institute announced the Millennium Prize Problems: $1,000,000 prizes for the proof of any of seven conjectures, including the Poincaré conjecture.