Add section about isospectral pairs

.gitignore

@@ -4,6 +4,10 @@
 *.fdb_latexmk
 *.fls
 *.log
+*.nav
+*.snm
+*.toc
+*.vrb
 *.out
 *.synctex
 *.synctex(busy)

isospectral.tex

@@ -0,0 +1,37 @@
+\section{ The spectral geometry of hyperbolic and spherical spaces}
+\begin{frame}{The Laplace-Beltrami operator}
+	\begin{enumerate}
+		\begin{definition}[Laplace-Beltrami operator]
+			a generalization of the Laplace operator to more general spaces
+			\begin{align*}
+				\Delta f \coloneq \divergence \nabla f
+			\end{align*}
+		\end{definition}
+		\item The spectrum of this operator is a subset of $[0, \infty)$. We call it the \emph{spectrum} of the manifold.
+		\item The spectrum determines many things about the space (like its volume).
+		\item When two distinct (up to isomorphism) manifolds share a spectrum, they for an \emph{isospectral pair}.
+	\end{enumerate}
+\end{frame}
+
+\begin{frame}{Can isospectral pairs approximate your mom}
+	\begin{itemize}
+		\item One of the problems the paper studies is finding volume-maximizing isospectral pairs of spherical forms (i.e. the spaces we are interested in).
+		\item Isospectral pairs share the order of their homotopy groups.
+		\item The paper classifies spherical forms with homotopy group of order $<24$.
+		\item It follows that any such isospectral pairs (with group order $<24$) are lens spaces.
+	\end{itemize}
+\end{frame}
+
+\begin{frame}{Combining lenses into larger spaces}
+	\begin{itemize}
+		\item We can combine lens spaces by concatenating their lists of indices.
+		\item Together with a computational search, the paper builds a table of the solutions for a bunch of dimensions. The general case is still an open question.
+	\end{itemize}
+\end{frame}
+
+\begin{frame}{The silver lining}
+	\begin{itemize}
+		\item Isospectral pairs are cool and all, but they cannot occur in dimension $3$.
+		\item We can thus attempt to infer the shape of our universe based on its spectrum.
+	\end{itemize}
+\end{frame}

main.tex

@@ -12,11 +12,17 @@
 \usepackage{tikz}
 \usepackage{pgfplots}
 \usepackage{verbatim}
+\usepackage{mathtools}
 \pgfplotsset{compat = newest}
 \usetikzlibrary{matrix}
 \usepackage[dvipsnames]{xcolor}
 \usetikzlibrary{perspective}
 
+\DeclareMathOperator{\divergence}{div}
+\DeclareMathOperator{\lensop}{L}
+\DeclareMathOperator{\rotmat}{R}
+\newcommand*{\lens}[2]{\lensop\left(#1,#2\right)} %
+
 % cool color
 
 \usepackage{xcolor}
@@ -43,6 +49,8 @@ code-for-last-col = \color{blue}
 
 \maketitle
 
+\section{Introduction & preliminaries}
+
 \begin{frame}{Outline}
 	%Should we make an outline?
 	\begin{enumerate}%[<+->]
@@ -56,10 +64,10 @@ code-for-last-col = \color{blue}
 
 % should we make is so that these bullet points appear one after another? Yes
 \begin{frame}{Abstracting Independence - Motivation}
-\begin{itemize}
+	\begin{itemize}
 		\item 1
 		\item 2
-\end{itemize}
+	\end{itemize}
 
 \end{frame}
 
@@ -71,49 +79,51 @@ code-for-last-col = \color{blue}
 	ddgagagagaga
 \end{frame}
 
+\include{isospectral}
+
 \begin{frame}[fragile]{CMB Anisotropy of Homogeneous Spherical Spaces}
-\begin{itemize}
-\item Manifolds $M := \mathbb{S}^3 /_ \sim$ where $\sim $ identifies the orbits of finite $H \leq SO(4)$ finite
+	\begin{itemize}
+		\item Manifolds $M := \mathbb{S}^3 /_ \sim$ where $\sim $ identifies the orbits of finite $H \leq SO(4)$ finite
 
-\item Helmholtz equation on $M$ given by
-$$(\Delta + E_\beta^M)\psi_\beta^{M, i} = 0$$
+		\item Helmholtz equation on $M$ given by
+		      $$(\Delta + E_\beta^M)\psi_\beta^{M, i} = 0$$
 
-\item In fact $E_\beta^m = \beta^2-1$ for $\beta \in \mathbb{N}$ we call $\beta$ a wave number
+		\item In fact $E_\beta^m = \beta^2-1$ for $\beta \in \mathbb{N}$ we call $\beta$ a wave number
 
-\item The set of all possible wave numbers [for which there exists a non-zero solution] depends on $H$
+		\item The set of all possible wave numbers [for which there exists a non-zero solution] depends on $H$
 
-\item Solutions to the Helmholtz equation are calculated for each $H$ and CMB anisotropies $\frac{\delta T}{T}$ computed
-\end{itemize}
+		\item Solutions to the Helmholtz equation are calculated for each $H$ and CMB anisotropies $\frac{\delta T}{T}$ computed
+	\end{itemize}
 \end{frame}
 
 
 \begin{comment}
 
 \begin{frame}[fragile]{CMB Anisotropy of Homogeneous Spherical Spaces}
-\begin{itemize}
-\item If $H = \{1\}$ the set of wave numbers is $\mathbb{N}$
+	\begin{itemize}
+		\item If $H = \{1\}$ the set of wave numbers is $\mathbb{N}$
 		\item If $H = \mathbb Z/m\mathbb Z$ for odd $m$ the set of wave numbers is $$\{1,3,\cdots, m\}\cup \{m+1, m+2, m+3, \cdots \}.$$
 		\item  If $H = O^*$, the binary tetrahedral group --- a subgroup of $SO(4)$ of size 24, the set of wave numbers is $$\{ 1,7,9 \} \cup \{13,15,17,\cdots \}.$$
-\end{itemize}
+	\end{itemize}
 \end{frame}
 
 \end{comment}
 
 
 \begin{frame}[fragile]{CMB Anisotropy of Homogeneous Spherical Spaces - Conclusion}
-\begin{itemize}
-\item For the majority of groups $H$ the anisotropies $\frac{\delta T}{T}$ do not coincide with observations
-\item The only for which it does are $H = O^*$ and $H = I^*.$ the \textit{binary octahedral} and \textit{binary icosahedral} groups of order 48 and 120
+	\begin{itemize}
+		\item For the majority of groups $H$ the anisotropies $\frac{\delta T}{T}$ do not coincide with observations
+		\item The only for which it does are $H = O^*$ and $H = I^*.$ the \textit{binary octahedral} and \textit{binary icosahedral} groups of order 48 and 120
 
 
-\end{itemize}
-\begin{figure}[H]
+	\end{itemize}
+	\begin{figure}[H]
 		\centering
-\includegraphics[width=0.4\linewidth]{binary-octahedron.png}
+		\includegraphics[width=0.4\linewidth]{binary-octahedron.png}
 		\caption{Graphical representation of the binary tetrahedral group
 				[5]}
 		\label{fig:binary-octahedron}
-\end{figure}
+	\end{figure}
 
 \end{frame}
 
@@ -127,7 +137,7 @@ $$(\Delta + E_\beta^M)\psi_\beta^{M, i} = 0$$
 
 \begin{frame}{CMB radiation in an inhomogeneous spherical space}
 
-Manifolds
+	Manifolds
 
 \end{frame}
 
@@ -141,23 +151,23 @@ Manifolds
 
 \begin{frame}{To summerize}
 	egqgaaf
-\begin{itemize}
+	\begin{itemize}
 		\item 1
 		\item 2
 		\item 3
-\end{itemize}
+	\end{itemize}
 
 \end{frame}
 
 
 \begin{frame}{References}
-\begin{itemize}
+	\begin{itemize}
 		\item R. Aurich, P. Kramer, and S. Lustig. \textit{Cosmic microwave background radiation in an inhomogeneous spherical space}. Physica Scripta, 2011.
 		\item R. Aurich, P. Kramer, and S. Lustig. \textit{A survey of lens spaces and large-scale CMB anisotropy}. Monthly Notices of the Royal Astronomical Society, 2012.
 		\item  R. Aurich, S. Lustig, and F. Steiner. \textit{CMB anisotropy of spherical spaces}. Classical and Quantum Gravity, 2005.
 		\item P. Bielewicz, K. M. G´orski, and A. J. Banday. \textit{Low-order multipole maps of cosmic microwave background anisotropy derived from WMAP}. Oxford University Press, 2004.
 		\item G. Egan. \textit{Symmetries and the 24-cell}. 2021. Accessed at https://www.gregegan.net/SCIENCE/24-cell/24-cell.html.
-\end{itemize}
+	\end{itemize}
 \end{frame}
 
 
@@ -183,9 +193,9 @@ Manifolds
 
 
 \begin{frame}{}
-\begin{center}
-\huge Thank You!
-\end{center}
+	\begin{center}
+		\huge Thank You!
+	\end{center}
 \end{frame}
 
 \end{document}