diff --git a/main.pdf b/main.pdf
index 0b659f2..cd55816 100644
Binary files a/main.pdf and b/main.pdf differ
diff --git a/main.tex b/main.tex
index 2babbeb..11fca55 100644
--- a/main.tex
+++ b/main.tex
@@ -234,7 +234,6 @@ code-for-last-col = \color{blue}
 
 \section{Conclusion}
 \begin{frame}{Conclusion}
-
 	\pause
 	\begin{enumerate}
 		\item We can infer the shape of the universe from its spectrum.
@@ -263,7 +262,7 @@ code-for-last-col = \color{blue}
 \end{frame}
 
 
-\begin{frame}{References 2 --- electric boogaloo}
+\begin{frame}{References 2 --- Electric Boogaloo}
 	\begin{itemize}
 
 		\item [6] N. Jarosik et. al. \textit{Seven-year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Sky Maps, Systematic Errors, and Basic Results}. The American Astronomical Society, 2011.
@@ -274,7 +273,7 @@ code-for-last-col = \color{blue}
 	\end{itemize}
 \end{frame}
 
-\begin{frame}{References 3 --- the references strike again}
+\begin{frame}{References 3 --- the References Strike Again}
 	\begin{itemize}
 		\item [11] E. A. Lauret and B. Linowitz. \textit{The spectral geometry of hyperbolic and spherical manifolds: analogies and open problems}. New York Journal of Mathematics, 2025.
 		\item [12] R. Lehoucq, J. Weeks, J. P. Uzan, E. Gausmann, and J.P. Luminet. \textit{Eigenmodes of three-dimensional spherical spaces and their application to cosmology}. Classical and Quantum Gravity, 2002.
diff --git a/prerequisites.tex b/prerequisites.tex
index 9411a5b..b53307a 100644
--- a/prerequisites.tex
+++ b/prerequisites.tex
@@ -1,5 +1,5 @@
 \section{Prerequisites}
-\begin{frame}{Manifolds \& Homotopy groups}
+\begin{frame}{Manifolds \& Homotopy Groups}
 	\begin{figure}[H]
 		\centering
 		\includegraphics[width=0.5\linewidth]{mug-neighbourhoods.png}
@@ -27,13 +27,13 @@
 	\end{itemize}
 \end{frame}
 
-\begin{frame}{Lens spaces}
+\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{frame}{Lens Spaces — the Explicit Construction}
 	\begin{definition}[Lens space]
 		Given $q \in \mathbb Z$ and $s \in \mathbb Z ^n$ elementwise coprime with $q$
 		\begin{align*}