\\section{Poincare Conjecture}
\n\\begin{thm}[Poincare Conjecture]
\nIf a compact three-dimensional manifold $M^3$ has the property that every simple closed curve within the manifold can be deformed continuously to a point, does it follow that $M^3$ is homeomorphic to the sphere $S^3$?
\n\\end{thm}
\n\\section{Thurston Elliptization Conjecture}
\n\\begin{thm}[Thurston Elliptization Conjecture]
\nEvery closed 3-manifold with finite fundamental group has a metric of constant positive curvature and hence is homeomorphic to a quotient $S^3\/\\Gamma$, where $\\mathrm{\\Gamma} \\subset \\mathrm{SO(4)}$ is a finite group of rotations that acts freely on $S^3$. The Poincare Conjecture corresponds to the special case where the group $\\Gamma \\cong \\pi_1(M^3)$ is trivial.
\n\\end{thm}
\n
\n\\section{History}
\n\\begin{itemize}
\n\\item 1961, Stephen Smale, $n>4$;
\n\\item 1982
\n\\begin{itemize}
\n\\item\u3000Michael Freedman, $n=4$;
\n\\item\u3000William Thurston, Geometrization conjecture;
\n\\item\u3000Richard Hamilton, Ricci flow method;
\n\\end{itemize}
\n\\item\u30002006, Grisha Perelman, proved Geometrization conjecture.
\n\\end{itemize}
\n\\section{Papers of Perelman}
\n\\begin{itemize}
\n\\item The entropy formula for the Ricci flow and its geometric applications
\n\\item Ricci flow with surgery on three-manifolds
\n\\item Finite extinction time for the solutions to the Ricci flow on certain three-manifolds
\n\\end{itemize}<\/p>\n","protected":false},"excerpt":{"rendered":"
\\section{Poincare Conjecture} \\begin{thm …<\/p>\n