Merge branch 'master' of https://git.overleaf.com/67dbe68df0414d63b309f98a

main.tex

@@ -24,6 +24,7 @@
 \newcommand*{\so}[1]{\soop\left(#1\right)}
 \newcommand*{\rotmat}[1]{\rotmatop\left(#1\right)}
 \renewcommand{\S}{\mathbb{S}}
+\newcommand{\R}{\mathbb{R}}
 
 % cool color
 
@@ -121,19 +122,20 @@ code-for-last-col = \color{blue}
 
 \end{frame}
 
+\section{CMB radiation in an inhomogeneous spherical space}
 
-\begin{frame}{CMB radiation in an inhomogeneous spherical space}
+\begin{frame}{Inhomogeneous spherical space}
 
 	\begin{itemize}
-		\item Manifolds of the form $\mathbb S/\Gamma$ with $\Gamma$ a group acting on $\mathbb S^3$.
-		\item Multi-connected space: it has non-contractable loops.
-		\item Inhomogeneous space: it does not look identical from every point in space.
+		\item Manifolds of the form $\S^3/\Gamma$ with $\Gamma$ a group acting on $\S^3$.
+		\item Multi-connected space: it has non-contractable loops
+		\item Inhomogeneous space: it does not look identical from every point in space
 		\item Fixing $|\Gamma|=8$, we have three multi-connected manifolds, up to equivalence:
-		      \begin{enumerate}
-			      \item homogeneous: $N3$ and $L(8,1)$
-			      \item inhomogeneous: $N2 \equiv L(8,3)$
-		      \end{enumerate}
-		\item Results: inhomogeneous spaces have more variety in CMB anisotropies than homogeneous spaces, because different observation points have.
+        \begin{enumerate}
+            \item homogeneous: $N3$ and $L(8,1)$
+            \item inhomogeneous: $N2 \equiv L(8,3)$
+        \end{enumerate}
+        \item Results: inhomogeneous spaces have more variety in CMB anisotropies than homogeneous spaces, because the strength of anisotropy suppression is observer dependent.
 	\end{itemize}