Update on Overleaf.

main.tex

@@ -63,8 +63,24 @@ code-for-last-col = \color{blue}
 
 \end{frame}
 
-\begin{frame}{introduction}
-    121fdaaeegag
+\begin{frame}{Introduction}
+
+
+\begin{figure}[H]
+	\centering
+	\includegraphics[width=0.5\linewidth]{mug-neighbourhoods.png}
+	\caption{The prototypical example of a manifold -- a mug. Image source: [13]}
+	\label{fig:mug-neighbourhoods}
+\end{figure}
+
+\begin{figure}[H]
+	\centering
+   %$ \captionsetup{width=.75\linewidth}
+	\includegraphics[width=0.27\linewidth]{Contractible loops.png}
+	\caption{Diagram showing two double tori with (non)-contractible paths.  Image source[7]}
+	\label{fig:CoLoop}
+\end{figure}
+
 \end{frame}
 
 \begin{frame}{Preliminaries- quotient groups (Put actual title later)}
@@ -127,14 +143,31 @@ $$(\Delta + E_\beta^M)\psi_\beta^{M, i} = 0$$
 
 \begin{frame}{CMB radiation in an inhomogeneous spherical space}
 
-Manifolds
+\begin{itemize}
+    \item Multi-connected space: it has non-contractable loops
+    \item Inhomogeneous space: it does not look the same from every point on the 
+\end{itemize}
 
 \end{frame}
 
 \begin{frame}{Test for anisotropy in the mean of the CMB temperature fluctuation in spherical harmonic space}
+\begin{itemize}
+    \item this section goes through an overview of a mathematical method for computing and interpreting CMB temperature fluctuations in spherical spaces, incorporating statistical isotropy, covariance structures, and Monte Carlo simulations, which are later compared with the Wilkinson Microwave Anisotropy Probe (WMAP) seven-year observation data. 
     \textit{statistical isotropy}, meaning that the statistical properties of the CMB should be the same in all directions. Testing this assumption is essential, as deviations from isotropy could indicate alternative topologies for the universe.
+\end{itemize}
+
 \end{frame}
 
+\begin{frame}{Test for anisotropy in the mean of the CMB temperature fluctuation in spherical harmonic space}
+    \begin{figure} [h!]
+	\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}
+
+
 \begin{frame}{Conclusion}
     akbbfbsKJBKJBLJs
 \end{frame}