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\begin{document}
\title{Perelman Proves Poincar\'e}
\author{Jonathan A.~Poritz}
\institute[CSU-P]{{\tt jonathan.poritz@gmail.com}\\
{\tt poritz.net/jonathan}\\
\ \\
Department of Mathematics \& Physics \\
Colorado State University, Pueblo \\
2200 Bonforte Blvd.\\
Pueblo, CO 81001-4901}
\date{\today}
\date{Math 495, 5 November 2014}
%\slideCaption{\textit{Jonathan A.~Poritz, CSUP, \today}}
%\Logo(8.3,7.78){\includegraphics{CSU-Pueblo-seal-flat-bw.eps}}
\begin{frame}
\titlepage
\end{frame}
\begin{frame}
\frametitle{The Poincar\'e Conjecture}
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\begin{minipage}[t]{\MiniPageLeft}
\begin{figure}[t]
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\vskip2cm
Henri Poincar\'e, 1904:\newline
\\
\XXX Every {\bf compact},\newline
\XXX\hphantom{Every }{\bf connected},\newline
\XXX\hphantom{Every }{\bf simply-connected},\newline
\XXX\hphantom{Every }{\bf $3$-manifold},\newline
\XXX is the\, {\bf $3$-sphere}.
\end{minipage}
\end{frame}
\begin{frame}
\frametitle{Manifolds [Roughly]}
\noindent $\RR^n$ is the set $\{(x_1,\dots,x_n)\}$ of $n$-tuples of real
numbers. \emph{E.g.:}
\begin{itemize}
\item $\RR^1$ is the usual ``number line''
\item $\RR^2$ is Descartes' version of the Euclidean plane
\item $\RR^3$ is ``three-dimensional space''
\end{itemize}
\noindent An {\bf $n$-manifold} $M$ \emph{[without boundary]} is a set such
around every point of $M$ there can be found a sufficiently small
neighborhood which looks like part of $\RR^n$. \emph{E.g., when $n=2$:}
\begin{itemize}
\item all of $\RR^2$
\item an open disk in $\RR^2$ (but {\bf not} a closed disk)
\item \emph{punctured} $\RR^2$ (or a punctured disk)
\item several of the above examples (perhaps repeated copies of one) next to
each other
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{Topology}
\noindent
A {\bf metric} $d$ on a space \emph{[set]} $X$ is a function $d:X\times X\to\RR$
satisfying:
\begin{itemize}
\item $\forall x,y\in X$, $d(x,y)=d(y,x)$ \emph{[symmetry]}
\item $\forall x,y\in X$, $d(x,y)\ge0$ and $d(x,y)=0$ if and only if $x=y$
\item $\forall x,y,z\in X$, $d(x,z)\le d(x,y)+d(y,z)$ \emph{[the triangle
inequality]}
\end{itemize}
\noindent
A {\bf topology} on a space \emph{[set]} $X$ is a collection $\Oo$ of subsets
of $X$ (called the {\bf open sets of $X$} satisfying:
\begin{itemize}
\item $\emptyset\in\Oo$ and $X\in\Oo$
\item $\forall A,B\in\Oo$, $A\cap B\in\Oo$ \emph{[closed under finite
intersections]}
\item $\forall\Cc\subseteq\Oo$ of open sets, $\bigcup_{C\in\Cc} C\in\Oo$
\emph{[closed under arbitrary unions]}
\end{itemize}
A {\bf topological space} is a space $X$ together with a topology on $X$.
\noindent
Metrics induce topologies, but not every topology is ``metrizable.''
\noindent
A {\bf continuous function} $f:X\to Y$ between topological spaces is one for
which $f^{-1}$ of every open set in $Y$ is open in $X$.
\end{frame}
\begin{frame}
\frametitle{Connected \& Compact}
A topological space $X$ is {\bf connected} if you can get from any point of $X$
to any other point by an continuous path. \emph{E.g.:}
\begin{itemize}
\item Any of the above \emph{single} examples.
\item Any manifold which doesn't fall into pieces when you pick it up.
\end{itemize}
A topological space $X$ is {\bf compact} if any infinite sequence of points from
$X$ ``piles up'' at \emph{[has a limit point which is]} some point of
$X$. \emph{E.g., when $n=2$:}
\begin{itemize}
\item the $2$-sphere and its evil twin $\RR\PP^2$
\item the torus $T^2$ and its evil twin the Klein bottle $\widetilde{T}^2$
\item the higher genus Riemann surfaces $\Sigma_g$, for $g>1$, and
their evil twins $\widetilde{\Sigma}_g$
\end{itemize}
But {\bf not} all of $\RR^2$ or a Riemann surface of infinite genus.
\end{frame}
\begin{frame}
\frametitle{Homotopy}
Two maps $f_0,f_1:X\to Y$ between topological spaces are
{\bf homotopic} if there exists a continuous map $F:X\times[0,1]\to Y$ such that
$\forall x\in X$, $F(x,0)=f_0(x)$ and $F(x,1)=f_1(x)$. We write $f_0\simeq f_1$.
\vskip2mm
Topological spaces $X$ and $Y$ are {\bf homotopic} if there exist maps
$f:X\to Y$ and $g:Y\to X$ such that $f\circ g\simeq\id_Y$ and
$g\circ f\simeq\id_X$.
\vskip2mm
Homotopy of topological spaces is a very weak notion: it merely means that
one can be pulled and stretched and twisted without tearing
\emph{[continuously deformed]} to become the other. {\it E.g.,} Topologists
famously confuse their coffee cups with their doughnuts because they are
homotopic.
\vskip1mm
[A stronger notion is {\bf homeomorphism}: a continuous map with a continuous
inverse.]
\end{frame}
\begin{frame}
\frametitle{The Fundamental Group}
Start with a connected topological space $X$ and one particular point $p$ of
$X$. Look at the set of all continuous loops on $X$ which start and
end at $p$. Consider two such loops to be ``the same'' if they are
homotopic. The set of distinct loops that remains is called the
{\bf fundamental group of $M$}, written $\pi_1(X)$.
\vskip2mm
If the only loop (up to homotopy, remember) in $\pi_1(X)$ is the trivial loop
(which starts, ends, and in fact spends it whole time at $p$), then we say
$X$ is {\bf simply-connected}.\ \ \ \ \emph{Examples:}
\begin{itemize}
\item $\RR^2$ and $S^2$ are simply-connected
\item $\pi_1(\RR\PP^2)\cong\ZZ_2$
\item $\pi_1(T^2)\cong\ZZ\oplus\ZZ$
\item $\pi_1(\Sigma_g)$ is infinite but has a simple description
\end{itemize}
Equivalently, $X$ is simply-connected if any closed loop based at $p$ can be
continuously deformed to the trivial loop at $p$.
\end{frame}
\begin{frame}
\frametitle{{\bf Smooth} Manifolds and other additional structures}
To complete the definition of a {\bf smooth} manifold $M$, we require that the
maps from neighborhoods of $M$ to $\RR^n$ satisfy the compatibility condition
that on overlaps the compose to make an infinitely differentiable function from
$\RR^n$ to $\RR^n$. This enables us to take derivatives of functions on
manifolds, to solve differential equations, and, with some additional
structure (a notion of the length of a tangent vector to a smooth curve on a
manifold (called a {\bf Riemann metric})) to compute the lengths of curves.
\vskip2mm
On a manifold endowed with a Riemannian metric (called a {\bf Riemann
manifold}), we say a curve is a {\bf geodesic} if it is the shortest curve
connecting every pair of its points.
\vskip2mm
Another kind of additional structure was emphasized by the German mathematician
Felix Klein in his {\it Erlangenprogramm}, to wit:
\end{frame}
\begin{frame}
\frametitle{Transformation Groups}
A collection $G$ of transformations of a manifold $M$ is called
a {\bf transformation group acting on} $M$ if:
\begin{itemize}\addtolength{\itemsep}{-0.5\baselineskip}
\item $G$ contains the trivial (=''identity'') transformation;\newline
\item whenever $f$ and $g$ are in $G$, so is their composition
$f\circ g$;\newline
\item whenever $f$ is in $G$, so is its inverse $f^{-1}$.
\end{itemize}
\vskip2mm
In the above situation, we can define the {\bf quotient} $M/G$ of $M$
by the action of $G$ as the space $M$ again, but now with a new notion
that two points $p$ and $q$ will be considered ``the same'' if $q=f(p)$
for some $f$ in $G$.
\vskip2mm
\emph{E.g.,} $T^2=\RR^2/\ZZ^2$\newline
\hphantom{\emph{E.g.,} }$\Sigma_g = \Hh^2/\Gamma_g$\quad \emph{[What's
that?]}
\end{frame}
\begin{frame}
\frametitle{The Hyperbolic Plane}
\setlength{\MiniPageLeft}{0.7\textwidth}
As a set, the {\bf hyperbolic plane} $\Hh^2$ is the upper half
of $\RR^2$. It is given a metric so that the following are all
straight lines
\vskip1cm
\center{\includegraphics[width=\MiniPageLeft]{modular_grp.eps}}
\end{frame}
\begin{frame}
\frametitle{The Hyperbolic Disk}
\setlength{\MiniPageLeft}{0.4\textwidth}
The hyperbolic plane is isometric to the {\bf hyperbolic disk}
which consists of the interior of the unit disk in $\RR^2$,
endowed with a metric so that the straight lines are now
\vskip5mm
\center{\includegraphics[width=\MiniPageLeft]{hypdisk.eps}}
\end{frame}
\begin{frame}
\frametitle{Angels and Demons}
\setlength{\MiniPageLeft}{0.4\textwidth}
[Was that last picture reminiscent of
\vskip5mm
\centerline{\includegraphics[width=\MiniPageLeft]{escher.eps}}
\vskip-2.7cm\hskip9cm ?\ \ \ ]
\end{frame}
\begin{frame}
\frametitle{Curvature}
$S^2$ is said to have {\bf positive curvature}, because when it
sits in $\RR^3$, at each point there are two perpendicular directions
which both curve to the same side of the tangent plane. Triangles
on $S^2$ have interior angle sums which always {\bf exceed} $180^\circ$.
$T^2$ is said to be {\bf flat}, because its intrinsic geometry is
classically Euclidean (although it cannot be embedded in $\RR^3$
with this metric). Triangles on $T^2$ have interior angle sums which
always {\bf equal} $180^\circ$.
$\Sigma_g$, for any $g>1$, is said to have {\bf negative curvature}
when it is built by identifications of sides of polygons in $\Hh^2$;
it cannot be embedded in $\RR^3$ with this metric. Triangles on $\Sigma_g$
have interior angle sums which are always {\bf less than} $180^\circ$.
Actually, all of these are {\bf constant curvature} manifolds.
\end{frame}
\begin{frame}
\frametitle{Classification in Two Dimensions}
All compact, connected $2$-dimensions are homotopic (so, ``the same
as'', to a topologist; actually, they are also \emph{homeomorphic} and
even \emph{diffeomorphic}) to one of the manifolds
$$
S^2, \RR\PP^2, T^2, \widetilde{T}^2, \Sigma_g, \widetilde{\Sigma}_g
$$
where $g$ is any positive integer.
This is essentially something known in complex analysis as {\bf The
Uniformization Theorem}. It can be proven in several ways, for example
by putting some metric on the $2$-manifold and then solving a differential
equation (similar to the heat equation) to reform the metric until it has
constant curvature, then using the very simple geometry of constant curvature
spaces.
\end{frame}
\begin{frame}
\frametitle{Wait, what? -- Proof strategies}
A topology is very little structure on a space. There just isn't enough to
get your fingernails into in order to prove theorems. [This is also perhaps
why it is hard to prove theorems about prime numbers.]
\vskip2mm
One strategy to prove theorems on a topological space (or smooth manifold)
then is to endow it with additional structure (like a Riemann metric), then
to reform that structure (maybe by solving a differential equation), so that
in the end it is such a nice and rich structure many things can be
calculated about it.
\vskip2mm
An equation used in this way is one similar to the {\it heat equation}, giving
\begin{itemize}
\item for $n=1$, the ``curve shortening flow'' ... curves get round (and
short)
\item for $n=2$, this gives in the limit a constant curvature metric and\newline
\XXX\XXX\ \,therefore geometric structure on $\Sigma_g$
\item for $n=3$, Hamilton and then Perelman studied the {\bf Ricci flow}.
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{Three dimensions}
There is a $3$-dimensional analogue of hyperbolic space defined as
follows: $\Hh^3$ is the set of points $(x,y,z)$ in $\RR^3$ satisfying
$z>0$ (the ``upper half-space''). To find the distance between
two points in $\Hh^3$, or to connect them with a straight line,
consider the vertical half-plane in $\RR^3$ containing the points.
The distance between the points, and the straight line connecting them,
will be the distance and line in that half-plane thought of as $\Hh^2$.
\vskip1mm
There is also a spherical geometry $S^3$ (the usual round sphere in $\RR^4$)
and a flat geometry $T^3=\RR^3/\ZZ^3$ (or a cube with opposite faces
identified) in three dimensions.
\vskip1mm
In fact, there are {\bf five more exotic} geometries in $3$-dimensions.
All together these geometries can be called ``Thurston's $8$-fold Path''.
\end{frame}
\begin{frame}
\frametitle{Thurston Geometrization}
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\begin{minipage}[t]{\MiniPageLeft}
\begin{figure}[t]
\begin{center}
\includegraphics[width=\MiniPageLeft]{thurston.eps}
\end{center}
\end{figure}
\end{minipage}%
\begin{minipage}[t]{\MiniPageRight}
\vskip1.5cm
\ \ \ \ Wild Bill Thurston, late 1970's:\newline
\\
\ \ \ \ \XXX Every $3$-manifold can be canoni-\newline
\ \ \ \ \XXX cally decomposed into pieces,\newline
\ \ \ \ \XXX each of which admits a geometric\newline
\ \ \ \ \XXX structure from the $8$-fold path.
\end{minipage}
\end{frame}
\begin{frame}
\frametitle{Perelman's Proof}
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\begin{minipage}[t]{\MiniPageLeft}
\begin{figure}[t]
\begin{center}
\vskip1.4cm
\includegraphics[width=\MiniPageLeft]{perelman.eps}
\end{center}
\end{figure}
\end{minipage}%
\begin{minipage}[t]{\MiniPageRight}
\vskip1.5cm
\ \ \ \ Grigori (Grisha) Perelman, late 2003/4:\newline
\vskip2mm
\ \ \ \ Thurston's Geometrization Conjecture\newline
\ \ \ \ \XXX\XXX\XXX {\bf is true}\newline
\vskip2mm
\ \ \ \ so also the Poincar\'e Conjecture\newline
\ \ \ \ \XXX\XXX\XXX {\bf is true}\newline
\end{minipage}
\end{frame}
\begin{frame}
\frametitle{Higher Dimensions}
FYI: the situation is quite different in higher dimensions than three.
\vskip2mm
First, in four dimensions (think: spacetimes?) the situation is very
complex and while great work was done in the 1980s, yielding very
surprising results, this area has somewhat stalled.
\vskip2mm
Oddly enough, dimensions even higher, so $5$ and up, the situation is
much, much simpler. There is so much room in higher-dimensional spaces
that the analogue of the Poincar\'e conjecture was proven around $40$
years ago.
\end{frame}
\begin{frame}
\frametitle{Morals}
\begin{enumerate}
\item {\bf Mathematical Strategy:} Often it is hard to prove theorems about
mathematical objects with very simple definitions. If so, it can help to add
extra structure and then to solve an equation for a particularly nice case of
that structure.
\item Physics can loan us nice differential equations -- {\it e.g.,} the heat
equation and the Ricci flow (which is related to General Relatively).
\item {\bf Thurston's Geometrization Conjecture is true!} It should be the
start of a very exciting period in three-dimensional geometry....
\item Grish Perelman is a very unusual and admirable character ({\it e.g.,} he
turned down the Fields Medal, a \$1million prize from the Clay Institute, and
jobs at famous universities because ``the work is what is important; I did it
and now it should just stand on its own... and I don't need the money or the
job''). See various recent books about him if you are interested, such as
the one by Masha Gessen.
\end{enumerate}
\end{frame}
\end{document}