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main.pdf
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main.tex
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\section{Conclusion} \section{Conclusion}
\begin{frame}{Conclusion} \begin{frame}{Conclusion}
\pause \pause
\begin{enumerate} \begin{enumerate}
\item We can infer the shape of the universe from its spectrum. \item We can infer the shape of the universe from its spectrum.
@ -263,7 +262,7 @@ code-for-last-col = \color{blue}
\end{frame} \end{frame}
\begin{frame}{References 2 --- electric boogaloo} \begin{frame}{References 2 --- Electric Boogaloo}
\begin{itemize} \begin{itemize}
\item [6] N. Jarosik et. al. \textit{Seven-year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Sky Maps, Systematic Errors, and Basic Results}. The American Astronomical Society, 2011. \item [6] N. Jarosik et. al. \textit{Seven-year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Sky Maps, Systematic Errors, and Basic Results}. The American Astronomical Society, 2011.
@ -274,7 +273,7 @@ code-for-last-col = \color{blue}
\end{itemize} \end{itemize}
\end{frame} \end{frame}
\begin{frame}{References 3 --- the references strike again} \begin{frame}{References 3 --- the References Strike Again}
\begin{itemize} \begin{itemize}
\item [11] E. A. Lauret and B. Linowitz. \textit{The spectral geometry of hyperbolic and spherical manifolds: analogies and open problems}. New York Journal of Mathematics, 2025. \item [11] E. A. Lauret and B. Linowitz. \textit{The spectral geometry of hyperbolic and spherical manifolds: analogies and open problems}. New York Journal of Mathematics, 2025.
\item [12] R. Lehoucq, J. Weeks, J. P. Uzan, E. Gausmann, and J.P. Luminet. \textit{Eigenmodes of three-dimensional spherical spaces and their application to cosmology}. Classical and Quantum Gravity, 2002. \item [12] R. Lehoucq, J. Weeks, J. P. Uzan, E. Gausmann, and J.P. Luminet. \textit{Eigenmodes of three-dimensional spherical spaces and their application to cosmology}. Classical and Quantum Gravity, 2002.

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prerequisites.tex
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\section{Prerequisites} \section{Prerequisites}
\begin{frame}{Manifolds \& Homotopy groups} \begin{frame}{Manifolds \& Homotopy Groups}
\begin{figure}[H] \begin{figure}[H]
\centering \centering
\includegraphics[width=0.5\linewidth]{mug-neighbourhoods.png} \includegraphics[width=0.5\linewidth]{mug-neighbourhoods.png}
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\end{itemize} \end{itemize}
\end{frame} \end{frame}
\begin{frame}{Lens spaces} \begin{frame}{Lens Spaces}
\begin{itemize} \begin{itemize}
\item Lens spaces are obtained by taking the quotient of some $n$-sphere by a cyclic group. \item Lens spaces are obtained by taking the quotient of some $n$-sphere by a cyclic group.
\item They cannot be distinguished by their homotopy group alone. \item They cannot be distinguished by their homotopy group alone.
\end{itemize} \end{itemize}
\end{frame} \end{frame}
\begin{frame}{Lens spaces — the explicit construction} \begin{frame}{Lens Spaces — the Explicit Construction}
\begin{definition}[Lens space] \begin{definition}[Lens space]
Given $q \in \mathbb Z$ and $s \in \mathbb Z ^n$ elementwise coprime with $q$ Given $q \in \mathbb Z$ and $s \in \mathbb Z ^n$ elementwise coprime with $q$
\begin{align*} \begin{align*}