Poincare Conjecture and Elliptization Conjecture

\section{Poincare Conjecture}
\begin{thm}[Poincare Conjecture]
If 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 $S^3$?
\end{thm}
\section{Thurston Elliptization Conjecture}
\begin{thm}[Thurston Elliptization Conjecture]
Every 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.
\end{thm}

\section{History}
\begin{itemize}
\item 1961, Stephen Smale, $n>4$;
\item 1982
\begin{itemize}
\item　Michael Freedman, $n=4$;
\item　William Thurston, Geometrization conjecture;
\item　Richard Hamilton, Ricci flow method;
\end{itemize}
\item　2006, Grisha Perelman, proved Geometrization conjecture.
\end{itemize}
\section{Papers of Perelman}
\begin{itemize}
\item The entropy formula for the Ricci flow and its geometric applications
\item Ricci flow with surgery on three-manifolds
\item Finite extinction time for the solutions to the Ricci flow on certain three-manifolds
\end{itemize}

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