Update on Overleaf.

isospectral.tex

@@ -4,7 +4,7 @@
 		\begin{definition}[Laplace-Beltrami operator]
 			a generalization of the Laplace operator to more general spaces
 			\begin{align*}
-				\Delta f \coloneq \divergence \nabla f
+				\Delta f \coloneq \divergence \nabla f.
 			\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.

main.tex

@@ -4,6 +4,7 @@
 \usecolortheme{lily}
 \setbeamertemplate{navigation symbols}{}
 
+
 % 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
@@ -120,7 +121,7 @@ code-for-last-col = \color{blue}
 		\pause
 		\item For the majority of groups $\Gamma$, the anisotropies $\frac{\delta T}{T}$ do not coincide with observations.
 		      \pause
-		\item The only groups for which it does are $\Gamma = O^*$ and $\Gamma = I^*$ — the \textcolor{red}{binary octahedral} and \textcolor{red}{binary icosahedral} groups of order 48 and 120 respectively.
+		\item The only groups for which they do are $\Gamma = O^*$ and $\Gamma = I^*$ — the \textcolor{red}{binary octahedral} and \textcolor{red}{binary icosahedral} groups of order 48 and 120 respectively.
 		      \pause
 	\end{itemize}
 	\begin{figure}[H]
@@ -158,7 +159,7 @@ code-for-last-col = \color{blue}
 \begin{frame}{Statistical Isotropy and Hypothesis}
 	\small
 	\textbf{Theoretical Expectation:}\\
-	From the perspective of an Earth-based observer, we can view the CMB as a function defined on the celestial sphere $\mathbb{S}^2$.\\
+	From the perspective of an Earth-based observer, we can view the CMB as a function defined on the celestial sphere $\mathbb{S}^2.$\\
 	\begin{itemize}
 		\pause
 		\item The CMB temperature fluctuations can be expanded as: