Add pauses

isospectral.tex

@@ -8,6 +8,7 @@
 			\end{align*}
 		\end{definition}
 		\item The spectrum of this operator is a subset of $[0, \infty)$. We call it the \emph{spectrum} of the manifold.
+		      \pause
 		\item The spectrum determines many things about the space (like its volume).
 		\item When two distinct (up to isomorphism) manifolds share a spectrum, they for an \emph{isospectral pair}.
 	\end{enumerate}
@@ -32,6 +33,12 @@
 \end{comment}
 
 \begin{frame}{The silver lining}
+	\begin{tikzpicture}
+		\clip (0,0) rectangle (\paperwidth,\paperheight);
+		\fill[color=orange] (\paperwidth-10pt,0) rectangle (\paperwidth,\paperheight);
+		% Added
+		\fill[color=orange](0,0) rectangle (10pt,\paperheight);
+	\end{tikzpicture}
 	\begin{itemize}
 		\item Isospectral pairs are cool and all, but they cannot occur in dimension $3$.
 		\item We can thus attempt to infer the shape of our universe based on its spectrum.

prerequisites.tex

@@ -6,6 +6,7 @@
 		\caption{The prototypical example of a manifold a mug [13].}
 		\label{fig:mug-neighbourhoods}
 	\end{figure}
+	\pause
 
 	\begin{figure}[H]
 		\centering
@@ -20,6 +21,7 @@
 	\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 $\R^4$, i.e. $x \sim y$ iff $x = My$ for some $M \in \Gamma$.
+		      \pause
 		\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}
@@ -49,5 +51,6 @@
 		\end{align*}
 	\end{definition}
 
+	\pause
 	In the $\S^3$ case, we denote $\lens p q \coloneq \lens p {(1, q)}$ for coprime $p, q \in \mathbb Z$.
 \end{frame}