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\def\R{\mathbb R}
\def\Z{\mathbb Z}
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\def\O{\mathcal O}
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\def\U{\mathcal U}
\def\V{\mathcal V}
\def\G{\mathcal G}
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\newenvironment{pf}{{\it Proof:}\quad}{\square \vskip 12pt}
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\title{Classical geometry --- Syllabus}

\author{Danny Calegari}
\address{Department of Mathematics \\ Harvard \\ Cambridge, MA 02138}
\email{dannyc@math.harvard.edu}

\maketitle

\section{A crash course in group theory}
\subsection{Basic examples and definitions}
\subsubsection{Cyclic groups, Dihedral groups, symmetric groups}
\subsection{Products of groups, subgroups, normal subgroups}
\subsection{Homomorphisms, exact sequences}
\subsection{Commutators, centers, Abelian groups, 
Nilpotent groups, Solvable groups}
\subsection{Automorphisms (inner and outer), semi-direct products}
\subsection{Free groups, generators and relations, finitely 
generated/presented groups}
\subsection{Topological groups, Lie groups}
\section{Model geometries in dimension two}
\subsection{The Euclidean plane}
\subsubsection{Euclid's axioms}
\subsubsection{A closer look at the fourth postulate}
\subsubsection{A closer look at the parallel postulate}
\subsubsection{Symmetries of $\E^2$}
\subsection{The $2$--sphere}
\subsubsection{Elliptic geometry}
\subsubsection{Spherical trigonometry}
\subsubsection{The area of a spherical triangle}
\subsubsection{Kissing numbers --- the Newton--Gregory problem}
\subsubsection{Reflections, rotations, involutions; $SO(3)$}
\subsubsection{Algebraic groups}
\subsubsection{Quaternions and the group $\S^3$}
\subsection{The hyperbolic plane}
\subsubsection{The problem of models}
\subsubsection{The Poincar\' e (conformal) model} 
\subsubsection{The upper half--space model}
\subsubsection{The Klein (projective) model}
\subsubsection{Hyperbolic trigonometry}
\subsubsection{The area of a hyperbolic triangle}
\subsubsection{Projective geometry}
\subsubsection{Elliptic, parabolic, hyperbolic isometries}
\subsubsection{Horocircular geometry}
\section{Tessellations in dimension two}
\subsection{The topology of surfaces}
\subsubsection{Gluing polygons}
\subsubsection{The fundamental group and covering spaces}
\subsubsection{Discrete groups}
\subsection{Discrete subgroups of $\Isom^+(\E^2)$}
\subsubsection{Integral quadratic forms}
\subsubsection{Continued fractions and $SL(2,\Z)$}
\subsubsection{Aperiodic tilings}
\subsubsection{Penrose tiles}
\subsubsection{Foliations and laminations}
\subsubsection{Euclidean wallpaper groups}
\subsubsection{Moduli of Euclidean orbifolds}
\subsection{Finite subgroups of $SO(3)$ and $\S^3$}
\subsubsection{The ``fair dice''}
\subsubsection{Spherical orbifolds}
\subsubsection{Reflection groups, Coxeter diagrams}
\subsubsection{``Bad'' orbifolds}
\subsection{Discrete subgroups of $PSL(2,\R)$}
\subsubsection{Mapping--class groups, Teichm\"uller space}
\subsubsection{Dehn twists and Lickorish's theorem}
\section{Appendix --- What is Geometry?}
\subsection{Klein's ``Erlanger Programm''}
\subsubsection{Category theory}
\subsection{Metric geometry}
\subsubsection{Metric spaces}
\subsubsection{Isometries, Lipschitz maps}
\subsection{Differential geometry}
\subsubsection{Smooth Manifolds}
\subsection{Topology}
\subsubsection{Open and closed sets, continuous maps}
\subsubsection{Subspace topology, quotient topology}
\subsubsection{Homeomorphisms}
\subsubsection{Connected, locally connected, compact, locally compact}
\subsubsection{Hausdorff}
\subsubsection{Manifolds}

\begin{thebibliography}{99}
\bibitem{jM87}
	J. Montesinos,
	\emph{Classical tessellations and three--manifolds},
	Springer--Verlag Universitext (1987)
\bibitem{jS00}
	J. Stallings,
	\emph{Category language},
	notes; available at {\tt http://www.math.berkeley.edu/\~\!stall}
\bibitem{wT97}
	W. Thurston,
	\emph{Three--dimensional geometry and topology, vol. 1},
	Princeton University Press, Princeton Math. Series 35 (1997)
\end{thebibliography}

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