Remove matroid thingy

main.tex

@@ -75,16 +75,7 @@ code-for-last-col = \color{blue}
 	\end{enumerate}
 \end{frame}
 
-% should we make is so that these bullet points appear one after another? Yes
-\begin{frame}{Abstracting Independence - Motivation}
-	\begin{itemize}
-		\item 1
-		\item 2
-	\end{itemize}
-\end{frame}
-
 \include{prerequisites}
-
 \include{isospectral}
 
 \begin{frame}[fragile]{CMB Anisotropy of Homogeneous Spherical Spaces}
@@ -92,9 +83,9 @@ code-for-last-col = \color{blue}
 		\item Manifolds $\textcolor{blue}{M} := \mathbb{S}^3 /_ \sim$ where $\sim $ identifies the orbits of finite $H \leq SO(4)$
 
 		\item Helmholtz equation on $\textcolor{blue}{M}$ given by
-		\begin{align*}
+		      \begin{align*}
-		    (\Delta + E_{\textcolor{red}{\beta}}^{\textcolor{blue}{M}})\psi_{\textcolor{red}{\beta}}^{{\textcolor{blue}{M}}, i} = 0 
+			      (\Delta + E_{\textcolor{red}{\beta}}^{\textcolor{blue}{M}})\psi_{\textcolor{red}{\beta}}^{{\textcolor{blue}{M}}, i} = 0
-		\end{align*}
+		      \end{align*}
 
 		\item In fact $E_{\textcolor{red}{\beta}}^m = {\textcolor{red}{\beta}}^2-1$ for $\textcolor{red}{\beta} \in \mathbb{N}$ we call $\textcolor{red}{\beta}$ a wave number
 
@@ -138,15 +129,15 @@ code-for-last-col = \color{blue}
 \begin{frame}{CMB radiation in an inhomogeneous spherical space}
 
 	\begin{itemize}
-        \item Manifolds of the form $\mathbb S/\Gamma$ with $\Gamma$ a group acting on $\mathbb S^3$.
+		\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 Fixing $|\Gamma|=8$, we have three multi-connected manifolds, up to equivalence:
-        \begin{enumerate}
+		      \begin{enumerate}
-            \item homogeneous: $N3$ and $L(8,1)$
+			      \item homogeneous: $N3$ and $L(8,1)$
-            \item inhomogeneous: $N2 \equiv L(8,3)$
+			      \item inhomogeneous: $N2 \equiv L(8,3)$
-        \end{enumerate}
+		      \end{enumerate}
-        \item Results: inhomogeneous spaces have more variety in CMB anisotropies than homogeneous spaces, because observer dependency changes the suppression of CMB anisotropies.
+		\item Results: inhomogeneous spaces have more variety in CMB anisotropies than homogeneous spaces, because observer dependency changes the suppression of CMB anisotropies.
 	\end{itemize}
 
 
@@ -155,34 +146,34 @@ code-for-last-col = \color{blue}
 \section{Test for anisotropy in the mean of the CMB temperature fluctuation in spherical harmonic space}
 
 \begin{frame}{The setup}
-        From the perspective of an Earth-based observer, we can view CMB as a function defined on the celestial sphere $\mathbb{S}^2$. \\
+	From the perspective of an Earth-based observer, we can view CMB as a function defined on the celestial sphere $\mathbb{S}^2$. \\
-        So, $\Delta T(\hat{n})$ expanded using \textit{spherical harmonics}, with $a_{\ell m}$ its coefficients.\\
+	So, $\Delta T(\hat{n})$ expanded using \textit{spherical harmonics}, with $a_{\ell m}$ its coefficients.\\
-        We assume the fluctuations: 
+	We assume the fluctuations:
-        \begin{itemize}
+	\begin{itemize}
-            \item Statistically isotropic and homogeneous in the mean. 
+		\item Statistically isotropic and homogeneous in the mean.
-            \item Gaussian distribution.
+		\item Gaussian distribution.
-            \item $\textcolor{blue}{a_{\ell m}}$ are independent Gaussian variables. 
+		\item $\textcolor{blue}{a_{\ell m}}$ are independent Gaussian variables.
-        \end{itemize}
+	\end{itemize}
-        Coefficients should satisfy:$\langle \textcolor{blue}{a_{\ell m}} \rangle = 0$ \\
+	Coefficients should satisfy:$\langle \textcolor{blue}{a_{\ell m}} \rangle = 0$ \\
 
-        \textbf{\textcolor{red}{Remark:}} $\langle \textcolor{blue}{a_{\ell m}} \rangle \neq 0$, \textbf{suggests possible anisotropies or preferred directions in the universe}
+	\textbf{\textcolor{red}{Remark:}} $\langle \textcolor{blue}{a_{\ell m}} \rangle \neq 0$, \textbf{suggests possible anisotropies or preferred directions in the universe}
 \end{frame}
 
 \begin{frame}{The setup}
-Real CMB observations are affected by instrumental noise and \textit{sky masking}. So, estimating $C_\ell$ accurately requires simulations. 
+	Real CMB observations are affected by instrumental noise and \textit{sky masking}. So, estimating $C_\ell$ accurately requires simulations.
 
-Given $C^{th}_\ell$, used to generate synthetic realizations of the CMB sky
+	Given $C^{th}_\ell$, used to generate synthetic realizations of the CMB sky
-\[
+	\[
-	\textcolor{blue}{a_{\ell m}} \sim \mathcal{N} (0, C_\ell^{\text{th}}).
+		\textcolor{blue}{a_{\ell m}} \sim \mathcal{N} (0, C_\ell^{\text{th}}).
-\]
+	\]
-However, this introduces bias, which must be corrected using a decorrelation matrix  $W$, and a \textit{sky mask} matrix  $M$ which accounts for issues caused by masking effects. 
+	However, this introduces bias, which must be corrected using a decorrelation matrix  $W$, and a \textit{sky mask} matrix  $M$ which accounts for issues caused by masking effects.
 
-Giving us the test statistic (tests the assumption of statistical isotropy):
+	Giving us the test statistic (tests the assumption of statistical isotropy):
-\[
+	\[
-	S_i = \sum_{j} W_{ij} M_j,
+		S_i = \sum_{j} W_{ij} M_j,
-\]
+	\]
-Which examines $\langle \textcolor{blue}{a_{\ell m}} \rangle$ across multipole ranges. \\ 
+	Which examines $\langle \textcolor{blue}{a_{\ell m}} \rangle$ across multipole ranges. \\
-\textbf{\textcolor{red}{Recall:}} $\langle a_{\ell m} \rangle \neq 0$, \textbf{suggests possible anisotropies or preferred directions in the universe}
+	\textbf{\textcolor{red}{Recall:}} $\langle a_{\ell m} \rangle \neq 0$, \textbf{suggests possible anisotropies or preferred directions in the universe}
 
 \end{frame}
 
@@ -192,7 +183,7 @@ Which examines $\langle \textcolor{blue}{a_{\ell m}} \rangle$ across multipole r
 		\includegraphics[width=0.7\linewidth]{DSE-Test Graph}
 		\caption{The decorrelated band mean spectrum obtained by the seven-year WMAP data [9].}
 	\end{figure}
-\textcolor{red}{Remark}In a positively curved space certain multipoles scales being preferred or suppressed in the CMB power spectrum.
+	\textcolor{red}{Remark}In a positively curved space certain multipoles scales being preferred or suppressed in the CMB power spectrum.
 
 \end{frame}