Lens spaces and whatnot

isospectral.tex

@@ -13,6 +13,7 @@
 	\end{enumerate}
 \end{frame}
 
+\begin{comment}
 \begin{frame}{Can isospectral pairs approximate your mom}
 	\begin{itemize}
 		\item One of the problems the paper studies is finding volume-maximizing isospectral pairs of spherical forms (i.e. the spaces we are interested in).
@@ -28,6 +29,7 @@
 		\item Together with a computational search, the paper builds a table of the solutions for a bunch of dimensions. The general case is still an open question.
 	\end{itemize}
 \end{frame}
+\end{comment}
 
 \begin{frame}{The silver lining}
 	\begin{itemize}

main.tex

@@ -2,8 +2,6 @@
 \usetheme{Warsaw}
 \usecolortheme{lily}
 
-
-
 % boadilla seems to be the only one with enough space for stuff.compile it now it is very toxic but works
 
 \usepackage{graphicx} % Required for inserting images
@@ -20,8 +18,12 @@
 
 \DeclareMathOperator{\divergence}{div}
 \DeclareMathOperator{\lensop}{L}
-\DeclareMathOperator{\rotmat}{R}
+\DeclareMathOperator{\rotmatop}{R}
+\DeclareMathOperator{\soop}{SO}
 \newcommand*{\lens}[2]{\lensop\left(#1,#2\right)} %
+\newcommand*{\so}[1]{\soop\left(#1\right)}
+\newcommand*{\rotmat}[1]{\rotmatop\left(#1\right)}
+\renewcommand{\S}{\mathbb{S}}
 
 % cool color
 
@@ -49,9 +51,12 @@ code-for-last-col = \color{blue}
 \setbeamertemplate{headline}{%
   \leavevmode%
   \begin{beamercolorbox}[wd=\paperwidth,ht=2.5ex,dp=1.5ex]{section in head/foot}%
-    \hspace{.5em}\strut\insertsectionhead\hfill\mbox{}%
+    \mbox{}\hspace{.5em}\strut\insertsectionhead\hfill%
   \end{beamercolorbox}%
 }
+
+\setbeamercolor{block title}{fg=white, bg=purple!50!black}
+\setbeamercolor{block body}{fg=black, bg=pink!20}
 \begin{document}
 
 
@@ -76,10 +81,10 @@ code-for-last-col = \color{blue}
 		\item 1
 		\item 2
 	\end{itemize}
-
 \end{frame}
 
 \include{prerequisites}
+
 \include{isospectral}
 
 \begin{frame}[fragile]{CMB Anisotropy of Homogeneous Spherical Spaces}
@@ -123,7 +128,6 @@ code-for-last-col = \color{blue}
 		\includegraphics[width=0.4\linewidth]{binary-octahedron.png}
 		\caption{Graphical representation of the binary tetrahedral group
 				[5]}
-		\label{fig:binary-octahedron}
 	\end{figure}
 
 \end{frame}
@@ -158,7 +162,6 @@ code-for-last-col = \color{blue}
 		\centering
 		\includegraphics[width=0.6\linewidth]{DSE-Test Graph}
 		\caption{The decorrelated band mean spectrum obtained by the seven-year WMAP data [9].}
-		\label{fig:Decorrelated Statistical Spectrum}
 	\end{figure}
 \end{frame}
 

prerequisites.tex

@@ -1,9 +1,9 @@
 \section{Prerequisites}
-\begin{frame}{Introduction}
+\begin{frame}{Manifolds \& Homotopy groups}
 	\begin{figure}[H]
 		\centering
 		\includegraphics[width=0.5\linewidth]{mug-neighbourhoods.png}
-		\caption{The prototypical example of a manifold a mug. Image source: [13]}
+		\caption{The prototypical example of a manifold a mug [13].}
 		\label{fig:mug-neighbourhoods}
 	\end{figure}
 
@@ -11,11 +11,43 @@
 		\centering
 		%$ \captionsetup{width=.75\linewidth}
 		\includegraphics[width=0.2\linewidth]{Contractible loops.png}
-		\caption{Diagram showing two double tori with (non)-contractible paths.  Image source[7]}
+		\caption{Two double tori with (non)-contractible paths [7].}
 		\label{fig:CoLoop}
 	\end{figure}
 \end{frame}
 
-\begin{frame}{Preliminaries- quotient groups (Put actual title later)}
-	ddgagagagaga
+\begin{frame}{Quotients of the $3$-sphere}
+	\begin{itemize}
+		\item $\so 4$ is isomorphic to the isometry group of $\S^3$
+		\item Subgroups $\Gamma \leqslant \so 4$ define equivalence classes of orbits on $\S^3$ by the standard action of $\so 4$ on $\R^3$, i.e. $x \sim y$ iff $x = My$ for some $M \in \Gamma$.
+		\item The obtain space is (\emph{sometimes}) a manifold. In particular, it is well defined and spherical for finite $\Gamma$.
+		\item This can be easily generalised to the $n$-sphere.
+	\end{itemize}
+\end{frame}
+
+\begin{frame}{Lens spaces}
+	\begin{itemize}
+		\item Lens spaces are obtained by taking the quotient of some $n$-sphere by a cyclic group.
+		\item They cannot be distinguished by their homotopy group alone.
+	\end{itemize}
+\end{frame}
+\begin{frame}{Lens spaces — the explicit construction}
+	\begin{definition}[Lens space]
+		Given $q \in \mathbb N$ and $s \in \mathbb Z ^n$ each coprime with $q$
+		\begin{align*}
+			\lens q s \coloneq \S^{2n-1}/\langle M_{q,s} \rangle,
+		\end{align*}
+		where $\langle M \rangle$ is the group generated by
+		\begin{align*}
+			M_{q,s} \coloneq
+			\begin{pmatrix}
+				\rotmat{2 \pi s_1 / q} &                        &        &                        \\
+				                       & \rotmat{2 \pi s_2 / q} &        &                        \\
+				                       &                        & \ddots &                        \\
+				                       &                        &        & \rotmat{2 \pi s_n / q}
+			\end{pmatrix}
+		\end{align*}
+	\end{definition}
+
+	In the $\S^3$ case, we denote $\lens p q \coloneq \lens p {(1, q)}$ for coprime $p, q \in \mathbb \mathbb Z$.
 \end{frame}