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isospectral.tex
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@ -44,7 +44,7 @@
\begin{frame}{The silver lining}
\begin{itemize}
\item Isospectral pairs are intriguing in their own right, but they cannot occur in dimension $3$.
\item Isospectral manifolds are intriguing in their own right, but in dimension 3 all isospectral manifolds are isometric.
\item We can thus attempt to infer the shape of our universe based on its spectrum.
\end{itemize}
\end{frame}

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main.tex
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@ -44,9 +44,9 @@ code-for-last-col = \color{blue}
}
\title{Computing CMB temperature fluctuations for spherical spaces}
\title{Computing CMB Temperature Fluctuations for Spherical Spaces}
\author{Adriel Matei, Béla Schneider, Javier Vela, Juš Kocutar}
%\institute{Presenting: Javier, Juš}
\institute{University of Groningen}
\date{March 24, 2025}
@ -97,7 +97,7 @@ code-for-last-col = \color{blue}
(\Delta + E_{\textcolor{red}{\beta}}^{\textcolor{blue}{M}})\psi_{\textcolor{red}{\beta}}^{{\textcolor{blue}{M}}, i} = 0.
\end{align*}
\pause
\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.
\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.
\pause
\item The set of all possible wave numbers [for which a nonzero solution exists] depends on $\Gamma$ - usually a proper subset of $\mathbb{{N}}$.
\pause
@ -141,6 +141,7 @@ code-for-last-col = \color{blue}
\begin{frame}{Inhomogeneous spherical space}
\begin{itemize}
\pause
\item Manifolds of the form $\S^3/\Gamma$ with $\Gamma$ a group acting on $\S^3$.
\item \textcolor{red}{Multi-connected} space: it has non-contractable loops.
\item \textcolor{red}{Inhomogeneous} space: it does not look identical from every point in space.
@ -170,7 +171,7 @@ code-for-last-col = \color{blue}
\Delta T(\hat{n}) = \sum_{\ell=0}^{\infty} \sum_{m=-\ell}^{\ell} \textcolor{blue}{a_{\ell m}} Y_{\ell m}(\hat{n})
\]
\pause
\item If the universe is isotropic, the \textbf{mean of the harmonic coefficients} should satisfy:
\item If the universe is \textcolor{red}{isotropic, the \textbf{mean of the harmonic coefficients} should satisfy:
\[
\langle \textcolor{blue}{a_{\ell m}} \rangle = 0
\]
@ -227,7 +228,7 @@ code-for-last-col = \color{blue}
\textbf{Possible Explanations:}
\begin{itemize}
\item A real cosmological signal? → A finite universe or new physics.
\item A systematic effect? → Foreground contamination or instrumental noise.
\item A systematic effect? → Technological or theory related limitations.
\end{itemize}
\end{columns}
\end{frame}

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prerequisites.tex
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@ -3,7 +3,7 @@
\begin{figure}[H]
\centering
\includegraphics[width=0.5\linewidth]{mug-neighbourhoods.png}
\caption{The prototypical example of a manifold a mug [13].}
\caption{The prototypical example of a manifold -- a mug [13].}
\label{fig:mug-neighbourhoods}
\end{figure}
\pause